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pretraining/mathematica/geometry/solids/10464.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.553 & 0.084 & 0.703 \\
+ 0.818 & 0.495 & 0.754 \\
+ 0.469 & 0.771 & 0.914 \\
+ 0.506 & 0.778 & 0.588 \\
+ 0.789 & 0.204 & 0.103 \\
+ 0.794 & 0.174 & 0.135 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.04$ 
+Surface Area: $0.85$ 
+Solid Angle: $0.71$

pretraining/mathematica/geometry/solids/10518.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.413 & 0.652 & 0.964 \\
+ 0.516 & 0.991 & 0.482 \\
+ 0.603 & 0.618 & 0.975 \\
+ 0.481 & 0.938 & 0.77 \\
+ 0.145 & 0.732 & 0.074 \\
+ 0.644 & 0.487 & 0.783 \\
+ 0.892 & 0.926 & 0.622 \\
+ 0.567 & 0.441 & 0.068 \\
+ 0.748 & 0.747 & 0.048 \\
+ 0.671 & 0.797 & 0.041 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Solid Angle: $1.57$ 
+Surface Area: $1.47$

pretraining/mathematica/geometry/solids/10918.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.106 & 0.91 & 0.128 \\
+ 0.959 & 0.337 & 0.415 \\
+ 0.721 & 0.958 & 0.147 \\
+ 0.877 & 0.904 & 0.958 \\
+ 0.861 & 0.013 & 0.309 \\
+ 0.164 & 0.948 & 0.893 \\
+ 0.945 & 0.561 & 0.279 \\
+ 0.219 & 0.714 & 0.233 \\
+ 0.607 & 0.031 & 0.456 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.23$ 
+Surface Area: $2.35$ 
+Solid Angle: $0.97$

pretraining/mathematica/geometry/solids/1219.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.686 & 0.838 & 0.121 \\
+ 0.211 & 0.257 & 0.169 \\
+ 0.576 & 0.458 & 0.818 \\
+ 0.401 & 0.772 & 0.672 \\
+ 0.071 & 0.335 & 0.934 \\
+ 0.543 & 0.014 & 0.639 \\
+ 0.257 & 0.761 & 0.569 \\
+ 0.559 & 0.818 & 0.716 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.53$ 
+Volume: $0.13$ 
+Solid Angle: $0.65$

pretraining/mathematica/geometry/solids/13260.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.199 & 0.9 & 0.39 \\
+ 0.494 & 0.877 & 0.127 \\
+ 0.627 & 0.014 & 0.678 \\
+ 0.825 & 0.467 & 0.517 \\
+ 0.896 & 0.466 & 0.892 \\
+ 0.612 & 0.838 & 0.308 \\
+ 0.524 & 0.045 & 0.154 \\
+ 0.34 & 0.893 & 0.916 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.14$ 
+Surface Area: $1.79$ 
+Solid Angle: $1.49$

pretraining/mathematica/geometry/solids/15477.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.633 & 0.267 & 0.358 \\
+ 0.91 & 0.794 & 0.1 \\
+ 0.74 & 0.039 & 0.081 \\
+ 0.949 & 0.136 & 0.322 \\
+ 0.932 & 0.922 & 0.437 \\
+ 0.245 & 0.861 & 0.96 \\
+ 0.962 & 0.69 & 0.979 \\
+ 0.434 & 0.359 & 0.132 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.97$ 
+Volume: $0.16$ 
+Solid Angle: $5.72$

pretraining/mathematica/geometry/solids/16771.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.962 & 0.798 & 0.939 \\
+ 0.99 & 0.911 & 0.595 \\
+ 0.147 & 0.646 & 0.835 \\
+ 0.729 & 0.323 & 0.084 \\
+ 0.842 & 0.896 & 0.925 \\
+ 0.108 & 0.11 & 0.641 \\
+ 0.719 & 0.981 & 0.637 \\
+ 0.231 & 0.929 & 0.55 \\
+ 0.701 & 0.132 & 0.003 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.18$ 
+Surface Area: $2.09$ 
+Solid Angle: $1.61$

pretraining/mathematica/geometry/solids/20864.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.371 & 0.974 & 0.53 \\
+ 0.75 & 0.37 & 0.515 \\
+ 0.727 & 0.881 & 0.96 \\
+ 0.153 & 0.086 & 0.268 \\
+ 0.145 & 0.298 & 0.189 \\
+ 0.868 & 0.41 & 0.699 \\
+ 0.306 & 0.964 & 0.603 \\
+ 0.591 & 0.781 & 0.029 \\
+ 0.706 & 0.699 & 0.397 \\
+ 0.012 & 0.79 & 0.536 \\
+ 0.81 & 0.641 & 0.81 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Surface Area: $1.8$ 
+Solid Angle: $3.91$

pretraining/mathematica/geometry/solids/21922.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.823 & 0.849 & 0.552 \\
+ 0.744 & 0.403 & 0.318 \\
+ 0.275 & 0.929 & 0.085 \\
+ 0.058 & 0.884 & 0.566 \\
+ 0.043 & 0.135 & 0.546 \\
+ 0.762 & 0.005 & 0.691 \\
+ 0.697 & 0.83 & 0.098 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.51$ 
+Volume: $0.15$ 
+Surface Area: $1.9$

pretraining/mathematica/geometry/solids/21937.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.873 & 0.257 & 0.422 \\
+ 0.007 & 0.94 & 0.904 \\
+ 0.527 & 0.129 & 0.243 \\
+ 0.019 & 0.463 & 0.373 \\
+ 0.695 & 0.389 & 0.897 \\
+ 0.442 & 0.136 & 0.81 \\
+ 0.135 & 0.336 & 0.172 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.1$ 
+Solid Angle: $0.95$ 
+Surface Area: $1.53$

pretraining/mathematica/geometry/solids/22103.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.524 & 0.987 & 0.657 \\
+ 0.219 & 0.67 & 0.832 \\
+ 0.284 & 0.831 & 0.642 \\
+ 0.939 & 0.208 & 0.312 \\
+ 0.458 & 0.899 & 0.348 \\
+ 0.299 & 0.634 & 0.365 \\
+ 0.735 & 0.716 & 0.867 \\
+ 0.852 & 0.057 & 0.606 \\
+ 0.177 & 0.299 & 0.356 \\
+ 0.012 & 0.023 & 0.971 \\
+ 0.124 & 0.279 & 0.978 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $2.08$ 
+Volume: $0.21$ 
+Solid Angle: $1.95$

pretraining/mathematica/geometry/solids/24193.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.478 & 0.219 & 0.029 \\
+ 0.085 & 0.378 & 0.46 \\
+ 0.732 & 0.807 & 0.528 \\
+ 0.453 & 0.613 & 0.887 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $0.75$ 
+Volume: $0.02$ 
+Solid Angle: $0.15$

pretraining/mathematica/geometry/solids/25588.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.915 & 0.162 & 0.876 \\
+ 0.605 & 0.964 & 0.501 \\
+ 0.205 & 0.448 & 0.091 \\
+ 0.267 & 0.295 & 0.894 \\
+ 0.022 & 0.764 & 0.773 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.39$ 
+Surface Area: $1.52$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/28866.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.352 & 0.244 & 0.962 \\
+ 0.861 & 0.37 & 0.783 \\
+ 0.041 & 0.144 & 0.047 \\
+ 0.535 & 0.709 & 0.89 \\
+ 0.142 & 0.958 & 0.991 \\
+ 0.856 & 0.348 & 0.466 \\
+ 0.889 & 0.783 & 0.074 \\
+ 0.683 & 0.964 & 0.134 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $2.5$ 
+Solid Angle: $1.73$ 
+Volume: $0.26$

pretraining/mathematica/geometry/solids/3218.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.21 & 0.58 & 0.689 \\
+ 0.473 & 0.829 & 0.852 \\
+ 0.919 & 0.723 & 0.762 \\
+ 0.741 & 0.867 & 0.954 \\
+ 0.952 & 0.901 & 0.386 \\
+ 0.274 & 0.004 & 0.651 \\
+ 0.052 & 0.903 & 0.369 \\
+ 0.41 & 0.444 & 0.022 \\
+ 0.455 & 0.134 & 0.92 \\
+ 0.914 & 0.889 & 0.194 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.22$ 
+Surface Area: $2.25$ 
+Solid Angle: $4.74$

pretraining/mathematica/geometry/solids/32898.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.343 & 0.048 & 0.288 \\
+ 0.687 & 0.458 & 0.012 \\
+ 0.555 & 0.162 & 0.2 \\
+ 0.38 & 0.452 & 0.472 \\
+ 0.739 & 0.309 & 0.974 \\
+ 0.082 & 0.431 & 0.076 \\
+ 0.858 & 0.375 & 0.838 \\
+ 0.364 & 0.983 & 0.124 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Solid Angle: $1.51$ 
+Surface Area: $1.54$

pretraining/mathematica/geometry/solids/33244.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.951 & 0.068 & 0.453 \\
+ 0.815 & 0.642 & 0.698 \\
+ 0.546 & 0.72 & 0.075 \\
+ 0.963 & 0.835 & 0.397 \\
+ 0.628 & 0.551 & 0.033 \\
+ 0.539 & 0.522 & 0.261 \\
+ 0.112 & 0.83 & 0.911 \\
+ 0.613 & 0.917 & 0.918 \\
+ 0.446 & 0.827 & 0.927 \\
+ 0.759 & 0.707 & 0.15 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.59$ 
+Surface Area: $1.62$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/33568.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.265 & 0.408 & 0.042 \\
+ 0.883 & 0.637 & 0.576 \\
+ 0.773 & 0.682 & 0.946 \\
+ 0.477 & 0.755 & 0.534 \\
+ 0.055 & 0.688 & 0.879 \\
+ 0.369 & 0.288 & 0.986 \\
+ 0.859 & 0.832 & 0.931 \\
+ 0.579 & 0.146 & 0.727 \\
+ 0.855 & 0.75 & 0.204 \\
+ 0.716 & 0.078 & 0.289 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $2.08$ 
+Solid Angle: $1.12$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/33649.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.02 & 0.034 & 0.222 \\
+ 0.768 & 0.82 & 0.903 \\
+ 0.168 & 0.193 & 0.036 \\
+ 0.912 & 0.05 & 0.489 \\
+ 0.492 & 0.392 & 0.806 \\
+ 0.339 & 0.708 & 0.151 \\
+ 0.527 & 0.011 & 0.924 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Surface Area: $1.95$ 
+Solid Angle: $1.01$

pretraining/mathematica/geometry/solids/34508.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.817 & 0.73 & 0.902 \\
+ 0.18 & 0.317 & 0.188 \\
+ 0.064 & 0.813 & 0.2 \\
+ 0.694 & 0.11 & 0.532 \\
+ 0.232 & 0.233 & 0.635 \\
+ 0.193 & 0.885 & 0.058 \\
+ 0.538 & 0.517 & 0.431 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.39$ 
+Solid Angle: $0.4$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/34828.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.438 & 0.249 & 0.778 \\
+ 0.013 & 0.955 & 0.856 \\
+ 0.16 & 0.575 & 0.637 \\
+ 0.988 & 0.32 & 0.624 \\
+ 0.539 & 0.796 & 0.669 \\
+ 0.318 & 0.886 & 0.365 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.05$ 
+Solid Angle: $0.76$ 
+Surface Area: $1.02$

pretraining/mathematica/geometry/solids/36395.txt

@@ -0,0 +1,22 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 \\
+ 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 \\
+ \sqrt{\frac{1}{2}+\frac{1}{2 \sqrt{5}}} & 0 & \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & 0 \\
+ -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & 0 \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & 0 \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & 0 \\
+ -\sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} & \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & -\frac{1}{2} & 0 \\
+ -\sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & \frac{1}{2} & 0 \\
+ \sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & -\frac{1}{2} & 0 \\
+ \sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & \frac{1}{2} & 0 \\
+\end{array}
+\right)$. Determine the Circumcenter.
+Answer:
+$\left\{0,0,\frac{1}{8} \left(-3 \sqrt{2 \left(5-\sqrt{5}\right)}-\sqrt{10 \left(5-\sqrt{5}\right)}\right)\right\}$

pretraining/mathematica/geometry/solids/39293.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.628 & 0.166 & 0.868 \\
+ 0.82 & 0.868 & 0.966 \\
+ 0.499 & 0.777 & 0.06 \\
+ 0.822 & 0.908 & 0.805 \\
+ 0.29 & 0.169 & 0.711 \\
+ 0.735 & 0.122 & 0.371 \\
+ 0.354 & 0.765 & 0.784 \\
+ 0.875 & 0.088 & 0.376 \\
+ 0.313 & 0.575 & 0.972 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $2.54$ 
+Surface Area: $1.8$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/40979.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.227 & 0.576 & 0.861 \\
+ 0.279 & 0.249 & 0.711 \\
+ 0.078 & 0.19 & 0.361 \\
+ 0.308 & 0.245 & 0.175 \\
+ 0.896 & 0.204 & 0.923 \\
+ 0.265 & 0.648 & 0.591 \\
+ 0.484 & 0.792 & 0.548 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.08$ 
+Surface Area: $1.13$ 
+Solid Angle: $1.68$

pretraining/mathematica/geometry/solids/43235.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -\frac{1}{2} & 0 \\
+ 0 & \frac{1}{2} & 0 \\
+ \frac{1}{5} \left(-1-\frac{1}{\sqrt{6}}-\sqrt{\frac{1}{3} \left(71-19 \sqrt{6}\right)}\right) & \frac{1}{2} & \frac{1}{5} \sqrt{\frac{1}{6}+6 \sqrt{6}-\frac{1}{3} \sqrt{538+18 \sqrt{6}}} \\
+ -\frac{1}{2} & 0 & \sqrt{\frac{1}{2} \left(1+\sqrt{6}\right)} \\
+ \frac{1}{5} \left(-1-\frac{1}{\sqrt{6}}-\sqrt{\frac{1}{3} \left(71-19 \sqrt{6}\right)}\right) & -\frac{1}{2} & \frac{1}{5} \sqrt{\frac{1}{6}+6 \sqrt{6}-\frac{1}{3} \sqrt{538+18 \sqrt{6}}} \\
+ \frac{1}{30} \left(6+\sqrt{6}+2 \sqrt{213-57 \sqrt{6}}\right) & \frac{1}{2} & \frac{1}{5} \sqrt{\frac{1}{6}+6 \sqrt{6}-\frac{1}{3} \sqrt{538+18 \sqrt{6}}} \\
+ \frac{1}{2} & 0 & \sqrt{\frac{1}{2} \left(1+\sqrt{6}\right)} \\
+ \frac{1}{30} \left(6+\sqrt{6}+2 \sqrt{213-57 \sqrt{6}}\right) & -\frac{1}{2} & \frac{1}{5} \sqrt{\frac{1}{6}+6 \sqrt{6}-\frac{1}{3} \sqrt{538+18 \sqrt{6}}} \\
+ 0 & \frac{1}{30} \left(9-\sqrt{6}+2 \sqrt{213-57 \sqrt{6}}\right) & \frac{1}{5} \sqrt{\frac{1}{6}+6 \sqrt{6}+\frac{1}{3} \sqrt{538+18 \sqrt{6}}} \\
+ 0 & \frac{1}{30} \left(-9+\sqrt{6}-2 \sqrt{213-57 \sqrt{6}}\right) & \frac{1}{5} \sqrt{\frac{1}{6}+6 \sqrt{6}+\frac{1}{3} \sqrt{538+18 \sqrt{6}}} \\
+ \frac{1}{60} \left(6+\sqrt{6}+2 \sqrt{213-57 \sqrt{6}}+2 \sqrt{3+108 \sqrt{6}-6 \sqrt{538+18 \sqrt{6}}}\right) & 0 & -\sqrt{\text{Root}\left[94371840000 \text{$\#$1}^8-376229068800 \text{$\#$1}^7+727828135936 \text{$\#$1}^6-828332834816 \text{$\#$1}^5+578722553856 \text{$\#$1}^4-243092221952 \text{$\#$1}^3+55632772864 \text{$\#$1}^2-5651497536 \text{$\#$1}+197037369\&,2\right]} \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$\sqrt{\text{Root}\left[45137758519296 \text{$\#$1}^8-110336743047168 \text{$\#$1}^7-191069246324736 \text{$\#$1}^6+209269081571328 \text{$\#$1}^5+364547659290624 \text{$\#$1}^4-58793017190400 \text{$\#$1}^3+3306865979520 \text{$\#$1}^2-1275399855936 \text{$\#$1}+1439671249\&,4\right]}$

pretraining/mathematica/geometry/solids/44108.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.39 & 0.433 & 0.997 \\
+ 0.611 & 0.269 & 0.235 \\
+ 0.397 & 0.326 & 0.005 \\
+ 0.796 & 0.906 & 0.605 \\
+ 0.552 & 0.16 & 0.418 \\
+ 0.125 & 0.963 & 0.379 \\
+ 0.549 & 0.883 & 0.399 \\
+ 0.125 & 0.031 & 0.513 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.78$ 
+Solid Angle: $1.14$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/4468.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.364 & 0.081 & 0.282 \\
+ 0.329 & 0.884 & 0.275 \\
+ 0.949 & 0.097 & 0.326 \\
+ 0.428 & 0.296 & 0.45 \\
+ 0.875 & 0.484 & 0.854 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.05$ 
+Surface Area: $1.04$ 
+Solid Angle: $0.71$

pretraining/mathematica/geometry/solids/45020.txt

@@ -0,0 +1,99 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ \frac{1}{3} \left(-5-\sqrt{5}\right) & 0 & \frac{1}{3} \left(2 \sqrt{5}-7\right) \\
+ \frac{1}{3} \left(-5-\sqrt{5}\right) & 0 & \frac{2}{3} \left(\sqrt{5}-2\right) \\
+ \frac{1}{6} \left(-7-3 \sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(-7-3 \sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{2 \sqrt{5}}{3}-1 \\
+ \frac{1}{6} \left(-7-3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(-7-3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{2 \sqrt{5}}{3}-1 \\
+ -\frac{2}{3} \left(1+\sqrt{5}\right) & 0 & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{1}{3} \left(-4-\sqrt{5}\right) & -\frac{1}{\sqrt{3}} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{3} \left(-4-\sqrt{5}\right) & \frac{1}{\sqrt{3}} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{6} \left(-5-3 \sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \frac{1}{3} \left(\sqrt{5}-6\right) \\
+ \frac{1}{6} \left(-5-3 \sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \frac{1}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(-5-3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-6\right) \\
+ \frac{1}{6} \left(-5-3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-3\right) \\
+ -1-\frac{\sqrt{5}}{3} & 0 & -\frac{7}{3} \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & -2 \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \sqrt{5}-1 \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & -2 \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \sqrt{5}-1 \\
+ \frac{1}{6} \left(-7-\sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{1}{6} \left(-7-\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ -\frac{2 \sqrt{5}}{3} & 0 & \sqrt{5}-\frac{2}{3} \\
+ \frac{1}{6} \left(-5-\sqrt{5}\right) & -\sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-5\right) \\
+ \frac{1}{6} \left(-5-\sqrt{5}\right) & -\sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-2\right) \\
+ \frac{1}{6} \left(-5-\sqrt{5}\right) & \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-5\right) \\
+ \frac{1}{6} \left(-5-\sqrt{5}\right) & \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-2\right) \\
+ \frac{1}{3} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{3} \left(-1-\sqrt{5}\right) & \sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{6} \left(-3-\sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \sqrt{5}-\frac{2}{3} \\
+ \frac{1}{6} \left(-3-\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & -\frac{8}{3} \\
+ \frac{1}{6} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & -\frac{8}{3} \\
+ \frac{1}{6} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \sqrt{5}-\frac{2}{3} \\
+ -\frac{\sqrt{5}}{3} & -\sqrt{\frac{5}{3}} & -\frac{7}{3} \\
+ -\frac{\sqrt{5}}{3} & -\frac{1}{\sqrt{3}} & \sqrt{5}-\frac{1}{3} \\
+ -\frac{\sqrt{5}}{3} & \frac{1}{\sqrt{3}} & \sqrt{5}-\frac{1}{3} \\
+ -\frac{\sqrt{5}}{3} & \sqrt{\frac{5}{3}} & -\frac{7}{3} \\
+ \frac{1}{6} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{1}{6} \left(-1-\sqrt{5}\right) & \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ -\frac{1}{3} & -\sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{3} \left(\sqrt{5}-6\right) \\
+ -\frac{1}{3} & -\sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{3} \left(\sqrt{5}-3\right) \\
+ -\frac{1}{3} & \sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{3} \left(\sqrt{5}-6\right) \\
+ -\frac{1}{3} & \sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(\sqrt{5}-3\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \sqrt{5}-\frac{1}{3} \\
+ \frac{1}{6} \left(\sqrt{5}-3\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \sqrt{5}-\frac{1}{3} \\
+ 0 & -\sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & -2 \\
+ 0 & -\sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & \sqrt{5}-1 \\
+ 0 & 0 & -3 \\
+ 0 & 0 & \sqrt{5} \\
+ 0 & \sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & -2 \\
+ 0 & \sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & \sqrt{5}-1 \\
+ \frac{1}{6} \left(3-\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & -\frac{8}{3} \\
+ \frac{1}{6} \left(3-\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & -\frac{8}{3} \\
+ \frac{1}{3} & -\sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{2}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{3} & -\sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{2 \sqrt{5}}{3}-1 \\
+ \frac{1}{3} & \sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{2}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{3} & \sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{2 \sqrt{5}}{3}-1 \\
+ \frac{1}{6} \left(1+\sqrt{5}\right) & -\sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{6} \left(1+\sqrt{5}\right) & \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{\sqrt{5}}{3} & -\sqrt{\frac{5}{3}} & \sqrt{5}-\frac{2}{3} \\
+ \frac{\sqrt{5}}{3} & -\frac{1}{\sqrt{3}} & -\frac{8}{3} \\
+ \frac{\sqrt{5}}{3} & \frac{1}{\sqrt{3}} & -\frac{8}{3} \\
+ \frac{\sqrt{5}}{3} & \sqrt{\frac{5}{3}} & \sqrt{5}-\frac{2}{3} \\
+ \frac{1}{6} \left(3+\sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & -\frac{7}{3} \\
+ \frac{1}{6} \left(3+\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & \sqrt{5}-\frac{1}{3} \\
+ \frac{1}{6} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & \sqrt{5}-\frac{1}{3} \\
+ \frac{1}{6} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & -\frac{7}{3} \\
+ \frac{1}{3} \left(1+\sqrt{5}\right) & -\sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{1}{3} \left(1+\sqrt{5}\right) & \sqrt{\frac{2}{3} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{1}{6} \left(5+\sqrt{5}\right) & -\sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(2 \sqrt{5}-7\right) \\
+ \frac{1}{6} \left(5+\sqrt{5}\right) & -\sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-2\right) \\
+ \frac{1}{6} \left(5+\sqrt{5}\right) & \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(2 \sqrt{5}-7\right) \\
+ \frac{1}{6} \left(5+\sqrt{5}\right) & \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-2\right) \\
+ \frac{2 \sqrt{5}}{3} & 0 & -\frac{7}{3} \\
+ \frac{1}{6} \left(7+\sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{6} \left(7+\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & -2 \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \sqrt{5}-1 \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & -2 \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \sqrt{5}-1 \\
+ \frac{1}{3} \left(3+\sqrt{5}\right) & 0 & \sqrt{5}-\frac{2}{3} \\
+ \frac{1}{6} \left(5+3 \sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \frac{2}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(5+3 \sqrt{5}\right) & -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & \frac{2 \sqrt{5}}{3}-1 \\
+ \frac{1}{6} \left(5+3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{2}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(5+3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{2 \sqrt{5}}{3}-1 \\
+ \frac{1}{3} \left(4+\sqrt{5}\right) & -\frac{1}{\sqrt{3}} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{1}{3} \left(4+\sqrt{5}\right) & \frac{1}{\sqrt{3}} & \frac{2}{3} \left(\sqrt{5}-1\right) \\
+ \frac{2}{3} \left(1+\sqrt{5}\right) & 0 & \frac{1}{3} \left(\sqrt{5}-7\right) \\
+ \frac{1}{6} \left(7+3 \sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-6\right) \\
+ \frac{1}{6} \left(7+3 \sqrt{5}\right) & -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{6} \left(7+3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-6\right) \\
+ \frac{1}{6} \left(7+3 \sqrt{5}\right) & \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{3} \left(\sqrt{5}-3\right) \\
+ \frac{1}{3} \left(5+\sqrt{5}\right) & 0 & \frac{1}{3} \left(\sqrt{5}-5\right) \\
+ \frac{1}{3} \left(5+\sqrt{5}\right) & 0 & \frac{1}{3} \left(\sqrt{5}-2\right) \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$90$

pretraining/mathematica/geometry/solids/48704.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.209 & 0.99 & 0.656 \\
+ 0.68 & 0.209 & 0.842 \\
+ 0.393 & 0.196 & 0.057 \\
+ 0.815 & 0.912 & 0.737 \\
+ 0.866 & 0.901 & 0.754 \\
+ 0.607 & 0.112 & 0.265 \\
+ 0.031 & 0.309 & 0.429 \\
+ 0.2 & 0.607 & 0.978 \\
+ 0.742 & 0.345 & 0.622 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.18$ 
+Solid Angle: $1.38$ 
+Surface Area: $1.89$

pretraining/mathematica/geometry/solids/50256.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-1.2,5.997,-1.533\}$ has radius $5.557$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $388.01$ 
+Volume: $718.68$

pretraining/mathematica/geometry/solids/51097.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.617 & 0.106 & 0.089 \\
+ 0.488 & 0.504 & 0.653 \\
+ 0.643 & 0.137 & 0.025 \\
+ 0.689 & 0.81 & 0.609 \\
+ 0.622 & 0.148 & 0.558 \\
+ 0.769 & 0.744 & 0.511 \\
+ 0.982 & 0.01 & 0.983 \\
+ 0.527 & 0.604 & 0.146 \\
+ 0.208 & 0.787 & 0.398 \\
+ 0.494 & 0.945 & 0.707 \\
+ 0.744 & 0.102 & 0.862 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.66$ 
+Solid Angle: $2.68$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/51830.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.907 & 0.58 & 0.292 \\
+ 0.258 & 0.981 & 0.936 \\
+ 0.463 & 0.264 & 0.587 \\
+ 0.791 & 0.466 & 0.851 \\
+ 0.243 & 0.198 & 0.887 \\
+ 0.309 & 0.656 & 0.935 \\
+ 0.541 & 0.59 & 0.349 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.62$ 
+Volume: $0.07$ 
+Surface Area: $1.15$

pretraining/mathematica/geometry/solids/53802.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.974 & 0.149 & 0.831 \\
+ 0.316 & 0.321 & 0.573 \\
+ 0.985 & 0.648 & 0.502 \\
+ 0.516 & 0.059 & 0.841 \\
+ 0.478 & 0.617 & 0.171 \\
+ 0.741 & 0.202 & 0.509 \\
+ 0.509 & 0.86 & 0.895 \\
+ 0.462 & 0.855 & 0.919 \\
+ 0.685 & 0.363 & 0.335 \\
+ 0.713 & 0.677 & 0.001 \\
+ 0.423 & 0.809 & 0.337 \\
+ 0.662 & 0.648 & 0.999 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.76$ 
+Volume: $0.17$ 
+Solid Angle: $1.4$

pretraining/mathematica/geometry/solids/54002.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.564 & 0.431 & 0.889 \\
+ 0.148 & 0.941 & 0.487 \\
+ 0.22 & 0.141 & 0.278 \\
+ 0.31 & 0.513 & 0.933 \\
+ 0.896 & 0.476 & 0.428 \\
+ 0.109 & 0.062 & 0.703 \\
+ 0.605 & 0.146 & 0.256 \\
+ 0.172 & 0.908 & 0.289 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.64$ 
+Solid Angle: $2.1$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/54728.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.101 & 0.317 & 0.853 \\
+ 0.693 & 0.687 & 0.012 \\
+ 0.825 & 0.739 & 0.314 \\
+ 0.416 & 0.925 & 0.813 \\
+ 0.066 & 0.539 & 0.801 \\
+ 0.187 & 0.377 & 0.007 \\
+ 0.852 & 0.434 & 0.742 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.7$ 
+Volume: $0.13$ 
+Solid Angle: $1.25$

pretraining/mathematica/geometry/solids/56354.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.594 & 0.848 & 0.586 \\
+ 0.26 & 0.419 & 0.895 \\
+ 0.249 & 0.956 & 0.02 \\
+ 0.078 & 0.182 & 0.973 \\
+ 0.953 & 0.313 & 0.415 \\
+ 0.03 & 0.428 & 0.738 \\
+ 0.142 & 0.683 & 0.957 \\
+ 0.593 & 0.64 & 0.102 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.13$ 
+Surface Area: $1.77$ 
+Solid Angle: $2.56$

pretraining/mathematica/geometry/solids/5760.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.928 & 0.918 & 0.292 \\
+ 0.084 & 0.373 & 0.133 \\
+ 0.509 & 0.266 & 0.002 \\
+ 0.509 & 0.774 & 0.373 \\
+ 0.592 & 0.27 & 0.516 \\
+ 0.567 & 0.712 & 0.476 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.37$ 
+Volume: $0.05$ 
+Surface Area: $0.89$

pretraining/mathematica/geometry/solids/58257.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.853 & 0.199 & 0.95 \\
+ 0.525 & 0.372 & 0.993 \\
+ 0.381 & 0.914 & 0.837 \\
+ 0.484 & 0. & 0.47 \\
+ 0.872 & 0.988 & 0.378 \\
+ 0.298 & 0.46 & 0.196 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.05$ 
+Surface Area: $1.66$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/62111.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.706 & 0.767 & 0.259 \\
+ 0.872 & 0.945 & 0.084 \\
+ 0.967 & 0.137 & 0.563 \\
+ 0.551 & 0.429 & 0.374 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.61$ 
+Volume: $0.$ 
+Surface Area: $0.4$

pretraining/mathematica/geometry/solids/62175.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.78 & 0.455 & 0.015 \\
+ 0.195 & 0.547 & 0.013 \\
+ 0.706 & 0.971 & 0.296 \\
+ 0.648 & 0.104 & 0.961 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.14$ 
+Volume: $0.06$ 
+Surface Area: $1.15$

pretraining/mathematica/geometry/solids/6271.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.686 & 0.306 & 0.489 \\
+ 0.401 & 0.933 & 0.923 \\
+ 0.896 & 0.407 & 0.312 \\
+ 0.605 & 0.149 & 0.056 \\
+ 0.677 & 0.074 & 0.327 \\
+ 0.265 & 0.627 & 0.154 \\
+ 0.725 & 0.873 & 0.73 \\
+ 0.545 & 0.905 & 0.387 \\
+ 0.902 & 0.58 & 0.176 \\
+ 0.478 & 0.02 & 0.192 \\
+ 0.759 & 0.686 & 0.13 \\
+ 0.48 & 0.334 & 0.597 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $5.17$ 
+Volume: $0.14$ 
+Surface Area: $1.6$

pretraining/mathematica/geometry/solids/68138.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.268 & 0.275 & 0.062 \\
+ 0.783 & 0.362 & 0.981 \\
+ 0.501 & 0.148 & 0.091 \\
+ 0.135 & 0.015 & 0.138 \\
+ 0.947 & 0.234 & 0.891 \\
+ 0.539 & 0.873 & 0.55 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.3$ 
+Volume: $0.07$ 
+Solid Angle: $2.13$

pretraining/mathematica/geometry/solids/68978.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.006 & 0.78 & 0.471 \\
+ 0.973 & 0.606 & 0.266 \\
+ 0.32 & 0.329 & 0.203 \\
+ 0.047 & 0.634 & 0.768 \\
+ 0.866 & 0.145 & 0.017 \\
+ 0.553 & 0.408 & 0.366 \\
+ 0.851 & 0.792 & 0.335 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.99$ 
+Volume: $0.06$ 
+Surface Area: $1.22$

pretraining/mathematica/geometry/solids/69068.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.025 & 0.312 & 0.409 \\
+ 0.839 & 0.548 & 0.815 \\
+ 0.598 & 0.785 & 0.047 \\
+ 0.131 & 0.266 & 0.43 \\
+ 0.902 & 0.266 & 0.99 \\
+ 0.793 & 0.334 & 0.73 \\
+ 0.637 & 0.267 & 0.521 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.48$ 
+Surface Area: $1.08$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/69498.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.238 & 0.738 & 0.926 \\
+ 0.494 & 0.774 & 0.103 \\
+ 0.871 & 0.301 & 0.491 \\
+ 0.007 & 0.257 & 0.044 \\
+ 0.469 & 0.055 & 0.107 \\
+ 0.494 & 0.991 & 0.674 \\
+ 0.523 & 0.305 & 0.07 \\
+ 0.678 & 0.872 & 0.769 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.84$ 
+Solid Angle: $1.01$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/69583.txt

@@ -0,0 +1,25 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} & -\frac{1}{2} \\
+ -\frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} & \frac{1}{2} \\
+ -\frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} & -\frac{1}{2} \\
+ -\frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} & \frac{1}{2} \\
+ -\frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \\
+ -\frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \\
+ -\frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \\
+ -\frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \\
+ \frac{(-1)^{11/18}}{(-1)^{2/9}-1} & 0 & -\frac{1}{2} \\
+ \frac{(-1)^{11/18}}{(-1)^{2/9}-1} & 0 & \frac{1}{2} \\
+ \frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \\
+ \frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \\
+ \frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \\
+ \frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \\
+ \frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \\
+ \frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \\
+ \frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \\
+ \frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \\
+\end{array}
+\right)$. Determine the Circumdiameter.
+Answer:
+$\sqrt{1-\frac{4 (-1)^{2/9}}{\left((-1)^{2/9}-1\right)^2}}$

pretraining/mathematica/geometry/solids/69824.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.093 & 0.276 & 0.48 \\
+ 0.651 & 0.955 & 0.836 \\
+ 0.852 & 0.833 & 0.235 \\
+ 0.682 & 0.847 & 0.989 \\
+ 0.024 & 0.851 & 0.934 \\
+ 0.998 & 0.081 & 0.726 \\
+ 0.559 & 0.004 & 0.863 \\
+ 0.103 & 0.438 & 0.178 \\
+ 0.293 & 0.374 & 0.251 \\
+ 0.305 & 0.028 & 0.591 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.29$ 
+Surface Area: $2.57$ 
+Solid Angle: $3.82$

pretraining/mathematica/geometry/solids/7076.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.89 & 0.282 & 0.42 \\
+ 0.508 & 0.799 & 0.228 \\
+ 0.4 & 0.327 & 0.689 \\
+ 0.895 & 0.919 & 0.13 \\
+ 0.804 & 0.782 & 0.775 \\
+ 0.802 & 0.097 & 0.251 \\
+ 0.785 & 0.994 & 0.046 \\
+ 0.329 & 0.097 & 0.269 \\
+ 0.875 & 0.594 & 0.964 \\
+ 0.328 & 0.506 & 0.979 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.17$ 
+Solid Angle: $4.03$ 
+Surface Area: $1.9$

pretraining/mathematica/geometry/solids/72706.txt

@@ -0,0 +1,31 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.934 & 0. & -0.178 \\
+ -0.756 & 0. & -0.577 \\
+ -0.756 & -0.5 & 0.289 \\
+ -0.756 & 0.5 & 0.289 \\
+ -0.577 & 0. & 0.756 \\
+ -0.467 & -0.809 & 0.178 \\
+ -0.467 & 0.809 & 0.178 \\
+ -0.289 & -0.5 & -0.756 \\
+ -0.289 & 0.5 & -0.756 \\
+ -0.178 & 0. & 0.934 \\
+ -0.178 & -0.809 & -0.467 \\
+ -0.178 & 0.809 & -0.467 \\
+ 0.178 & -0.809 & 0.467 \\
+ 0.178 & 0.809 & 0.467 \\
+ 0.178 & 0. & -0.934 \\
+ 0.289 & -0.5 & 0.756 \\
+ 0.289 & 0.5 & 0.756 \\
+ 0.467 & -0.809 & -0.178 \\
+ 0.467 & 0.809 & -0.178 \\
+ 0.577 & 0. & -0.756 \\
+ 0.756 & -0.5 & -0.289 \\
+ 0.756 & 0.5 & -0.289 \\
+ 0.756 & 0. & 0.577 \\
+ 0.934 & 0. & 0.178 \\
+\end{array}
+\right)$. Determine the Circumradius.
+Answer:
+$0.95$

pretraining/mathematica/geometry/solids/73101.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.34 & 0.639 & 0.101 \\
+ 0.919 & 0.337 & 0.272 \\
+ 0.974 & 0.737 & 0.267 \\
+ 0.673 & 0.935 & 0.695 \\
+ 0.248 & 0.498 & 0.209 \\
+ 0.881 & 0.48 & 0.589 \\
+ 0.601 & 0.529 & 0.832 \\
+ 0.866 & 0.779 & 0.837 \\
+ 0.795 & 0.133 & 0.583 \\
+ 0.333 & 0.656 & 0.789 \\
+ 0.736 & 0.34 & 0.84 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.57$ 
+Solid Angle: $1.54$ 
+Volume: $0.14$