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pretraining/mathematica/geometry/solids/12333.txt

@@ -0,0 +1,32 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -1.618 & 0. \\
+ 0. & 1.618 & 0. \\
+ -1.618 & 0. & -0.862 \\
+ 0.851 & 0. & 0.526 \\
+ 0.263 & -0.809 & 0.526 \\
+ 0.263 & 0.809 & 0.526 \\
+ 1.618 & 0. & -0.862 \\
+ -0.951 & -1.309 & 0. \\
+ -0.951 & 1.309 & 0. \\
+ 0.951 & -1.309 & 0. \\
+ 0.951 & 1.309 & 0. \\
+ -0.688 & -0.5 & 0.526 \\
+ -0.688 & 0.5 & 0.526 \\
+ -0.5 & -1.539 & -0.862 \\
+ -0.5 & 1.539 & -0.862 \\
+ 0.5 & -1.539 & -0.862 \\
+ 0.5 & 1.539 & -0.862 \\
+ -1.309 & -0.951 & -0.862 \\
+ -1.309 & 0.951 & -0.862 \\
+ 1.309 & -0.951 & -0.862 \\
+ 1.309 & 0.951 & -0.862 \\
+ -1.539 & -0.5 & 0. \\
+ -1.539 & 0.5 & 0. \\
+ 1.539 & -0.5 & 0. \\
+ 1.539 & 0.5 & 0. \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$55.$

pretraining/mathematica/geometry/solids/12389.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.064 & 0.507 & 0.238 \\
+ 0.402 & 0.211 & 0.734 \\
+ 0.439 & 0.945 & 0.298 \\
+ 0.708 & 0.111 & 0.274 \\
+ 0.186 & 0.874 & 0.851 \\
+ 0.979 & 0.675 & 0.189 \\
+ 0.774 & 0.115 & 0.839 \\
+ 0.743 & 0.183 & 0.028 \\
+\end{array}
+\right)$. 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: $2.1$ 
+Solid Angle: $1.59$ 
+Volume: $0.21$

pretraining/mathematica/geometry/solids/13234.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{9.975,-8.791,4.543\}$ has radius $7.241$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $1590.62$ 
+Surface Area: $658.96$

pretraining/mathematica/geometry/solids/13657.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.753 & 0.541 & 0.893 \\
+ 0.841 & 0.806 & 0.22 \\
+ 0.366 & 0.332 & 0.816 \\
+ 0.522 & 0.834 & 0.297 \\
+ 0.983 & 0.02 & 0.617 \\
+ 0.844 & 0.962 & 0.339 \\
+ 0.587 & 0.276 & 0.324 \\
+ 0.828 & 0.017 & 0.347 \\
+\end{array}
+\right)$. 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: $1.83$ 
+Volume: $0.11$ 
+Surface Area: $1.41$

pretraining/mathematica/geometry/solids/1421.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.449 & 0.002 & 0.477 \\
+ 0.702 & 0.448 & 0.742 \\
+ 0.154 & 0.255 & 0.514 \\
+ 0.728 & 0.066 & 0.541 \\
+ 0.544 & 0.994 & 0.28 \\
+ 0.108 & 0.489 & 0.536 \\
+ 0.844 & 0.118 & 0.586 \\
+ 0.159 & 0.572 & 0.7 \\
+\end{array}
+\right)$. 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: $0.07$ 
+Surface Area: $1.15$ 
+Solid Angle: $0.84$

pretraining/mathematica/geometry/solids/14776.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.063 & 0.116 & 0.097 \\
+ 0.499 & 0.363 & 0.583 \\
+ 0.446 & 0.382 & 0.928 \\
+ 0.725 & 0.475 & 0.831 \\
+ 0.968 & 0.291 & 0.481 \\
+ 0.781 & 0.471 & 0.558 \\
+ 0.666 & 0.239 & 0.015 \\
+ 0.665 & 0.244 & 0.819 \\
+ 0.553 & 0.094 & 0.043 \\
+\end{array}
+\right)$. 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: $0.31$ 
+Volume: $0.06$ 
+Surface Area: $1.21$

pretraining/mathematica/geometry/solids/17799.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.414 & 0.921 & 0.69 \\
+ 0.715 & 0.344 & 0.361 \\
+ 0.997 & 0.395 & 0.005 \\
+ 0.591 & 0.871 & 0.005 \\
+\end{array}
+\right)$. 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: $0.7$ 
+Volume: $0.01$ 
+Solid Angle: $0.09$

pretraining/mathematica/geometry/solids/17842.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.308 & 0.639 & 0.498 \\
+ 0.332 & 0.779 & 0.691 \\
+ 0.205 & 0.559 & 0.336 \\
+ 0.924 & 0.963 & 0.297 \\
+ 0.576 & 0.774 & 0.441 \\
+ 0.354 & 0.641 & 0.11 \\
+\end{array}
+\right)$. 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: $0.01$ 
+Surface Area: $0.51$ 
+Solid Angle: $4.1$

pretraining/mathematica/geometry/solids/20440.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.248 & 0.924 & 0.122 \\
+ 0.67 & 0.971 & 0.766 \\
+ 0.351 & 0.106 & 0.106 \\
+ 0.254 & 0.215 & 0.501 \\
+ 0.008 & 0.121 & 0.212 \\
+ 0.303 & 0.286 & 0.04 \\
+\end{array}
+\right)$. 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: $0.06$ 
+Solid Angle: $0.5$ 
+Surface Area: $1.19$

pretraining/mathematica/geometry/solids/27311.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.049 & 0.894 & 0.657 \\
+ 0.832 & 0.312 & 0.219 \\
+ 0.596 & 0.201 & 0.811 \\
+ 0.51 & 0.472 & 0.294 \\
+ 0.132 & 0.643 & 0.712 \\
+ 0.563 & 0.181 & 0.701 \\
+\end{array}
+\right)$. 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: $0.17$ 
+Surface Area: $0.75$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/29441.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.364 & 0.135 & 0.11 \\
+ 0.723 & 0.883 & 0.074 \\
+ 0.685 & 0.323 & 0.028 \\
+ 0.398 & 0.677 & 0.983 \\
+ 0.815 & 0.707 & 0.639 \\
+ 0.259 & 0.161 & 0.455 \\
+ 0.766 & 0.131 & 0.537 \\
+ 0.366 & 0.485 & 0.041 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.67$ 
+Volume: $0.15$ 
+Solid Angle: $1.93$

pretraining/mathematica/geometry/solids/29779.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.118 & 0.134 & 0.038 \\
+ 0.927 & 0.067 & 0.935 \\
+ 0.181 & 0.225 & 0.706 \\
+ 0.089 & 0.898 & 0.739 \\
+ 0.859 & 0.968 & 0.498 \\
+ 0.494 & 0.805 & 0.908 \\
+ 0.219 & 0.727 & 0.145 \\
+ 0.223 & 0.508 & 0.012 \\
+ 0.916 & 0.806 & 0.956 \\
+ 0.022 & 0.36 & 0.137 \\
+\end{array}
+\right)$. 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: $2.7$ 
+Volume: $0.29$ 
+Solid Angle: $1.15$

pretraining/mathematica/geometry/solids/30154.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.646 & 0.41 & 0.747 \\
+ 0.706 & 0.648 & 0.626 \\
+ 0.965 & 0.448 & 0.427 \\
+ 0.949 & 0.499 & 0.519 \\
+ 0.435 & 0.666 & 0.962 \\
+ 0.486 & 0.402 & 0.586 \\
+ 0.98 & 0.438 & 0.595 \\
+ 0.143 & 0.526 & 0.347 \\
+ 0.719 & 0.634 & 0.242 \\
+ 0.024 & 0.978 & 0.936 \\
+\end{array}
+\right)$. 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: $3.27$ 
+Surface Area: $1.25$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/32308.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.533 & 0.142 & 0.701 \\
+ 0.673 & 0.174 & 0.758 \\
+ 0.613 & 0.855 & 0.995 \\
+ 0.66 & 0.425 & 0.877 \\
+ 0.812 & 0.249 & 0.174 \\
+ 0.14 & 0.701 & 0.224 \\
+\end{array}
+\right)$. 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: $0.07$ 
+Solid Angle: $2.07$ 
+Surface Area: $1.22$

pretraining/mathematica/geometry/solids/33268.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.351 & 0.472 & 0.847 \\
+ 0.252 & 0.607 & 0.714 \\
+ 0.59 & 0.838 & 0.693 \\
+ 0.697 & 0.018 & 0.505 \\
+ 0.412 & 0.095 & 0.437 \\
+ 0.275 & 0.397 & 0.156 \\
+\end{array}
+\right)$. 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: $0.88$ 
+Solid Angle: $1.76$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/33393.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.351 & 0.789 & 0.416 \\
+ 0.013 & 0.321 & 0.453 \\
+ 0.263 & 0.861 & 0.469 \\
+ 0.201 & 0.812 & 0.243 \\
+ 0.197 & 0.778 & 0.671 \\
+ 0.015 & 0.877 & 0.349 \\
+ 0.464 & 0.73 & 0.637 \\
+\end{array}
+\right)$. 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: $0.02$ 
+Solid Angle: $2.89$ 
+Surface Area: $0.49$

pretraining/mathematica/geometry/solids/34748.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.833 & 0.888 & 0.2 \\
+ 0.497 & 0.838 & 0.891 \\
+ 0.223 & 0.499 & 0.783 \\
+ 0.165 & 0.246 & 0.786 \\
+ 0.292 & 0.099 & 0.816 \\
+ 0.085 & 0.081 & 0.169 \\
+\end{array}
+\right)$. 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: $0.19$ 
+Volume: $0.06$ 
+Surface Area: $1.41$

pretraining/mathematica/geometry/solids/36367.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & -\frac{5}{\sqrt{50-10 \sqrt{5}}} \\
+ 0 & 0 & \frac{5}{\sqrt{50-10 \sqrt{5}}} \\
+ -\sqrt{\frac{2}{5-\sqrt{5}}} & 0 & -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \sqrt{\frac{2}{5-\sqrt{5}}} & 0 & \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2} & -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{1+\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2} & -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{-1-\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2} & \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{-1-\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2} & \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{1-\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2} \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{1-\sqrt{5}}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2} \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & -\frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{\sqrt{5}-1}{2 \sqrt{10-2 \sqrt{5}}} & -\frac{1}{2} \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+ \frac{\sqrt{5}-1}{2 \sqrt{10-2 \sqrt{5}}} & \frac{1}{2} \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}} & \frac{1}{\sqrt{10-2 \sqrt{5}}} \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$5 \sqrt{3}$

pretraining/mathematica/geometry/solids/3684.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.294 & 0.118 & 0.615 \\
+ 0.053 & 0.589 & 0.917 \\
+ 0.78 & 0.335 & 0.125 \\
+ 0.837 & 0.161 & 0.998 \\
+ 0.659 & 0.528 & 0.891 \\
+ 0.006 & 0.561 & 0.221 \\
+ 0.424 & 0.092 & 0.815 \\
+ 0.311 & 0.418 & 0.128 \\
+\end{array}
+\right)$. 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: $3.19$ 
+Volume: $0.13$ 
+Surface Area: $1.73$

pretraining/mathematica/geometry/solids/38322.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.714 & 0.047 & 0.918 \\
+ 0.139 & 0.352 & 0.581 \\
+ 0.24 & 0.38 & 0.817 \\
+ 0.167 & 0.085 & 0.049 \\
+ 0.534 & 0.45 & 0.921 \\
+ 0.07 & 0.993 & 0.707 \\
+ 0.728 & 0.273 & 0.756 \\
+ 0.263 & 0.016 & 0.831 \\
+\end{array}
+\right)$. 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: $1.07$ 
+Surface Area: $1.53$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/38469.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.065 & 0.67 & 0.015 \\
+ 0.329 & 0.94 & 0.014 \\
+ 0.045 & 0.257 & 0.074 \\
+ 0.606 & 0.641 & 0.686 \\
+ 0.421 & 0.353 & 0.046 \\
+ 0.287 & 0.8 & 0.571 \\
+\end{array}
+\right)$. 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: $1.81$ 
+Volume: $0.06$ 
+Surface Area: $0.99$

pretraining/mathematica/geometry/solids/4077.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.868 & 0.88 & 0.948 \\
+ 0.234 & 0.807 & 0.924 \\
+ 0.393 & 0.732 & 0.628 \\
+ 0.114 & 0.529 & 0.597 \\
+ 0.706 & 0.148 & 0.407 \\
+ 0.988 & 0.723 & 0.183 \\
+ 0.01 & 0.425 & 0.371 \\
+ 0.194 & 0.041 & 0.144 \\
+\end{array}
+\right)$. 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: $1.99$ 
+Volume: $0.14$ 
+Solid Angle: $0.63$

pretraining/mathematica/geometry/solids/40910.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.287 & 0.602 & 0.371 \\
+ 0.177 & 0.003 & 0.608 \\
+ 0.832 & 0.588 & 0.385 \\
+ 0.774 & 0.477 & 0.842 \\
+ 0.969 & 0.343 & 0.812 \\
+ 0.048 & 0.576 & 0.539 \\
+ 0.825 & 0.087 & 0.303 \\
+ 0.416 & 0.144 & 0.333 \\
+ 0.257 & 0.706 & 0.794 \\
+\end{array}
+\right)$. 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: $0.14$ 
+Solid Angle: $2.81$ 
+Surface Area: $1.56$

pretraining/mathematica/geometry/solids/40929.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.561 & 0.764 & 0.752 \\
+ 0.809 & 0.549 & 0.443 \\
+ 0.901 & 0.612 & 0.706 \\
+ 0.167 & 0.918 & 0.63 \\
+ 0.694 & 0.633 & 0.781 \\
+ 0.689 & 0.88 & 0.192 \\
+\end{array}
+\right)$. 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: $0.61$ 
+Solid Angle: $2.94$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/42092.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.972 & 0.558 & 0.497 \\
+ 0.499 & 0.812 & 0.289 \\
+ 0.733 & 0.784 & 0.711 \\
+ 0.289 & 0.394 & 0.322 \\
+ 0.365 & 0.691 & 0.29 \\
+ 0.399 & 0.292 & 0.43 \\
+ 0.934 & 0.534 & 0.245 \\
+\end{array}
+\right)$. 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: $0.74$ 
+Solid Angle: $1.24$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/42764.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.944 & 0.792 & 0.824 \\
+ 0.412 & 0.124 & 0.803 \\
+ 0.251 & 0.276 & 0.98 \\
+ 0.579 & 0.901 & 0.932 \\
+ 0.349 & 0.386 & 0.346 \\
+ 0.019 & 0.741 & 0.984 \\
+\end{array}
+\right)$. 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: $1.25$ 
+Solid Angle: $0.5$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/42951.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.662 & 0.468 & 0.914 \\
+ 0.851 & 0.453 & 0.62 \\
+ 0.051 & 0.703 & 0.674 \\
+ 0.428 & 0.735 & 0.001 \\
+ 0.354 & 0.863 & 0.068 \\
+ 0.641 & 0.331 & 0.424 \\
+\end{array}
+\right)$. 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: $0.57$ 
+Volume: $0.05$ 
+Surface Area: $1.03$

pretraining/mathematica/geometry/solids/43094.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.737 & 0.309 & 0.896 \\
+ 0.926 & 0.499 & 0.841 \\
+ 0.312 & 0.1 & 0.754 \\
+ 0.431 & 0.04 & 0.593 \\
+ 0.357 & 0.469 & 0.18 \\
+ 0.421 & 0.627 & 0.201 \\
+ 0.807 & 0.02 & 0.905 \\
+ 0.876 & 0.213 & 0.099 \\
+\end{array}
+\right)$. 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: $0.11$ 
+Surface Area: $1.38$ 
+Solid Angle: $3.82$

pretraining/mathematica/geometry/solids/438.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.821 & 0.842 & 0.86 \\
+ 0.384 & 0.964 & 0.583 \\
+ 0.856 & 0.605 & 0.496 \\
+ 0.216 & 0.266 & 0.228 \\
+ 0.336 & 0.728 & 0.216 \\
+ 0.393 & 0.257 & 0.473 \\
+ 0.61 & 0.965 & 0.955 \\
+ 0.142 & 0.497 & 0.024 \\
+ 0.968 & 0.196 & 0.554 \\
+\end{array}
+\right)$. 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: $1.56$ 
+Volume: $0.11$ 
+Solid Angle: $1.58$

pretraining/mathematica/geometry/solids/46984.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.072 & 0.521 & 0.075 \\
+ 0.107 & 0.105 & 0.753 \\
+ 0.164 & 0.765 & 0.844 \\
+ 0.751 & 0.518 & 0.117 \\
+ 0.497 & 0.03 & 0.568 \\
+ 0.712 & 0.122 & 0.712 \\
+ 0.418 & 0.987 & 0.961 \\
+ 0.78 & 0.923 & 0.04 \\
+\end{array}
+\right)$. 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: $1.37$ 
+Surface Area: $2.32$ 
+Volume: $0.24$

pretraining/mathematica/geometry/solids/47863.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.178 & 0.06 & 0.684 \\
+ 0.9 & 0.737 & 0.669 \\
+ 0.802 & 0.97 & 0.803 \\
+ 0.57 & 0.824 & 0.252 \\
+ 0.596 & 0.976 & 0.505 \\
+ 0.064 & 0.243 & 0.431 \\
+ 0.914 & 0.547 & 0.192 \\
+ 0.4 & 0.455 & 0.874 \\
+ 0.852 & 0.651 & 0.909 \\
+ 0.03 & 0.889 & 0.313 \\
+\end{array}
+\right)$. 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: $1.98$ 
+Solid Angle: $0.97$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/48972.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.228 & 0.151 & 0.278 \\
+ 0.601 & 0.07 & 0.534 \\
+ 0.685 & 0.648 & 0.275 \\
+ 0.769 & 0.072 & 0.479 \\
+ 0.4 & 0.245 & 0.999 \\
+ 0.707 & 0.597 & 0.792 \\
+ 0.253 & 0.883 & 0.398 \\
+ 0.999 & 0.27 & 0.651 \\
+ 0.116 & 0.289 & 0.967 \\
+\end{array}
+\right)$. 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: $1.52$ 
+Surface Area: $1.75$

pretraining/mathematica/geometry/solids/49004.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.33 & 0.078 & 0.294 \\
+ 0.812 & 0.674 & 0.801 \\
+ 0.434 & 0.629 & 0.245 \\
+ 0.818 & 0.981 & 0.598 \\
+ 0.841 & 0.159 & 0.068 \\
+ 0.403 & 0.667 & 0.306 \\
+\end{array}
+\right)$. 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: $0.6$ 
+Surface Area: $1.1$ 
+Volume: $0.06$

pretraining/mathematica/geometry/solids/4902.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.066 & 0.966 & 0.998 \\
+ 0.104 & 0.892 & 0.206 \\
+ 0.354 & 0.957 & 0.182 \\
+ 0.494 & 0.528 & 0.382 \\
+ 0.447 & 0.192 & 0.465 \\
+ 0.714 & 0.546 & 0.595 \\
+\end{array}
+\right)$. 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: $1.16$ 
+Volume: $0.07$ 
+Solid Angle: $0.35$

pretraining/mathematica/geometry/solids/50401.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.974 & 0.068 & 0.805 \\
+ 0.015 & 0.477 & 0.998 \\
+ 0.863 & 0.769 & 0.669 \\
+ 0.465 & 0.815 & 0.868 \\
+ 0.662 & 0.068 & 0.598 \\
+ 0.657 & 0.668 & 0.278 \\
+\end{array}
+\right)$. 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: $1.42$ 
+Volume: $0.1$ 
+Solid Angle: $0.78$

pretraining/mathematica/geometry/solids/52287.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.173 & 0.377 & 0.683 \\
+ 0.477 & 0.451 & 0.759 \\
+ 0.968 & 0.051 & 0.275 \\
+ 0.031 & 0.849 & 0.255 \\
+ 0.327 & 0.017 & 0.655 \\
+ 0.487 & 0.805 & 0.646 \\
+ 0.465 & 0.247 & 0.046 \\
+ 0.896 & 0.67 & 0.591 \\
+ 0.861 & 0.374 & 0.604 \\
+\end{array}
+\right)$. 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: $1.85$ 
+Solid Angle: $2.68$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/52798.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.496 & 0.782 & 0.604 \\
+ 0.665 & 0.125 & 0.998 \\
+ 0.158 & 0.88 & 0.382 \\
+ 0.158 & 0.475 & 0.515 \\
+\end{array}
+\right)$. 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: $0.13$ 
+Volume: $0.$ 
+Surface Area: $0.51$

pretraining/mathematica/geometry/solids/53519.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.602 & 0.745 & 0.588 \\
+ 0.955 & 0.47 & 0.302 \\
+ 0.388 & 0.447 & 0.091 \\
+ 0.781 & 0.311 & 0.48 \\
+ 0.4 & 0.253 & 0.163 \\
+ 0.186 & 0.633 & 0.263 \\
+ 0.371 & 0.391 & 0.827 \\
+ 0.015 & 0.927 & 0.553 \\
+ 0.535 & 0.049 & 0.174 \\
+\end{array}
+\right)$. 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: $1.4$ 
+Solid Angle: $2.41$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/53933.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $4.392$ has its base centered at$\{4.814,9.487,8.098\}$ and its tip is at $\{3.328,6.965,2.785\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Centroid: $\{4.44,8.86,6.77\}$ 
+Volume: $122.52$ 
+Surface Area: $163.92$

pretraining/mathematica/geometry/solids/54032.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.135 & 0.736 & 0.038 \\
+ 0.407 & 0.694 & 0.504 \\
+ 0.326 & 0.555 & 0.827 \\
+ 0.229 & 0.297 & 0.938 \\
+ 0.303 & 0.599 & 0.888 \\
+\end{array}
+\right)$. 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: $0.01$ 
+Solid Angle: $0.05$ 
+Surface Area: $0.41$

pretraining/mathematica/geometry/solids/55111.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $7.781$ has its base centered at$\{8.11,1.577,3.967\}$ and its tip is at $\{2.658,3.115,4.398\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Volume: $360.18$ 
+Centroid: $\{6.75,1.96,4.07\}$ 
+Surface Area: $425.69$

pretraining/mathematica/geometry/solids/55646.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.126 & 0.151 & 0.463 \\
+ 0.371 & 0.111 & 0.54 \\
+ 0.047 & 0.538 & 0.252 \\
+ 0.598 & 0.7 & 0.035 \\
+ 0.375 & 0.739 & 0.981 \\
+ 0.571 & 0.375 & 0.924 \\
+ 0.255 & 0.848 & 0.759 \\
+ 0.783 & 0.866 & 0.217 \\
+\end{array}
+\right)$. 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: $1.61$ 
+Solid Angle: $1.51$

pretraining/mathematica/geometry/solids/55755.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.673 & 0.773 & 0.981 \\
+ 0.019 & 0.016 & 0.893 \\
+ 0.953 & 0.784 & 0.222 \\
+ 0.229 & 0.101 & 0.028 \\
+ 0.299 & 0.719 & 0.634 \\
+ 0.807 & 0.863 & 0.815 \\
+\end{array}
+\right)$. 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: $0.92$ 
+Surface Area: $1.97$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/59168.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} & -\frac{1}{2 \sqrt{3}} \\
+ -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} & \frac{1}{2 \sqrt{3}} \\
+ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & -\frac{1}{2 \sqrt{3}} \\
+ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{2 \sqrt{3}} \\
+ 0 & -\sqrt{\frac{2}{3}} & -\frac{1}{2 \sqrt{3}} \\
+ 0 & -\sqrt{\frac{2}{3}} & \frac{1}{2 \sqrt{3}} \\
+ 0 & 0 & -\frac{\sqrt{3}}{2} \\
+ 0 & 0 & \frac{\sqrt{3}}{2} \\
+ 0 & \sqrt{\frac{2}{3}} & -\frac{1}{2 \sqrt{3}} \\
+ 0 & \sqrt{\frac{2}{3}} & \frac{1}{2 \sqrt{3}} \\
+ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} & -\frac{1}{2 \sqrt{3}} \\
+ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} & \frac{1}{2 \sqrt{3}} \\
+ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & -\frac{1}{2 \sqrt{3}} \\
+ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{2 \sqrt{3}} \\
+\end{array}
+\right)$. Determine the Circumradius.
+Answer:
+$\frac{\sqrt{3}}{2}$

pretraining/mathematica/geometry/solids/59271.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.426 & 0.118 & 0.344 \\
+ 0.228 & 0.509 & 0.981 \\
+ 0.618 & 0.447 & 0.541 \\
+ 0.347 & 0.524 & 0.223 \\
+ 0.719 & 0.536 & 0.092 \\
+ 0.062 & 0.592 & 0.589 \\
+ 0.067 & 0.72 & 0.592 \\
+ 0.446 & 0.526 & 0.923 \\
+\end{array}
+\right)$. 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: $0.06$ 
+Solid Angle: $1.16$ 
+Surface Area: $1.03$

pretraining/mathematica/geometry/solids/59577.txt

@@ -0,0 +1,37 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -1.618 & -0.5 \\
+ 0. & -1.618 & 0.5 \\
+ 0. & 1.618 & -0.5 \\
+ 0. & 1.618 & 0.5 \\
+ -0.688 & -0.5 & 1.026 \\
+ -0.688 & 0.5 & 1.026 \\
+ 0.688 & -0.5 & -1.026 \\
+ 0.688 & 0.5 & -1.026 \\
+ -0.851 & 0. & -1.026 \\
+ 0.851 & 0. & 1.026 \\
+ -0.263 & -0.809 & -1.026 \\
+ -0.263 & 0.809 & -1.026 \\
+ 0.263 & -0.809 & 1.026 \\
+ 0.263 & 0.809 & 1.026 \\
+ -0.951 & -1.309 & -0.5 \\
+ -0.951 & -1.309 & 0.5 \\
+ -0.951 & 1.309 & -0.5 \\
+ -0.951 & 1.309 & 0.5 \\
+ 0.951 & -1.309 & -0.5 \\
+ 0.951 & -1.309 & 0.5 \\
+ 0.951 & 1.309 & -0.5 \\
+ 0.951 & 1.309 & 0.5 \\
+ -1.539 & -0.5 & -0.5 \\
+ -1.539 & -0.5 & 0.5 \\
+ -1.539 & 0.5 & -0.5 \\
+ -1.539 & 0.5 & 0.5 \\
+ 1.539 & -0.5 & -0.5 \\
+ 1.539 & -0.5 & 0.5 \\
+ 1.539 & 0.5 & -0.5 \\
+ 1.539 & 0.5 & 0.5 \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$27.77$

pretraining/mathematica/geometry/solids/59595.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.663 & 0.751 & 0.199 \\
+ 0.027 & 0.573 & 0.826 \\
+ 0.882 & 0.887 & 0.135 \\
+ 0.931 & 0.426 & 0.539 \\
+\end{array}
+\right)$. 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: $0.01$ 
+Surface Area: $0.67$ 
+Solid Angle: $2.04$

pretraining/mathematica/geometry/solids/59761.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.059 & 0.083 & 0.988 \\
+ 0.34 & 0.933 & 0.239 \\
+ 0.394 & 0.49 & 0.358 \\
+ 0.791 & 0.892 & 0.891 \\
+ 0.825 & 0.343 & 0.574 \\
+ 0.647 & 0.016 & 0.582 \\
+ 0.764 & 0.885 & 0.297 \\
+ 0.018 & 0.181 & 0.76 \\
+ 0.784 & 0.888 & 0.578 \\
+\end{array}
+\right)$. 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: $0.75$ 
+Volume: $0.16$ 
+Surface Area: $1.9$

pretraining/mathematica/geometry/solids/62127.txt

@@ -0,0 +1,87 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{4} \sqrt{1-\frac{2}{\sqrt{5}}} & -\frac{1}{4} & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{1-\frac{2}{\sqrt{5}}} & \frac{1}{4} & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1-\frac{2}{\sqrt{5}}} & -\frac{1}{4} & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1-\frac{2}{\sqrt{5}}} & \frac{1}{4} & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & 0 & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{32}-\frac{1}{32 \sqrt{5}}} & -\frac{3}{8} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \sqrt{\frac{1}{32}-\frac{1}{32 \sqrt{5}}} & \frac{3}{8} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(-2-\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \frac{1}{4} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(2+\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ \frac{1}{4} \sqrt{5+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(-2-\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \sqrt{5+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(2+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \sqrt{\frac{5}{32}+\frac{11}{32 \sqrt{5}}} & -\frac{3}{8} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{5}{32}+\frac{11}{32 \sqrt{5}}} & \frac{3}{8} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{13}{32}+\frac{19}{32 \sqrt{5}}} & \frac{1}{8} \left(-5-\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{13}{32}+\frac{19}{32 \sqrt{5}}} & \frac{1}{8} \left(5+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{13}{32}+\frac{29}{32 \sqrt{5}}} & \frac{1}{8} \left(-3-\sqrt{5}\right) & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{13}{32}+\frac{29}{32 \sqrt{5}}} & \frac{1}{8} \left(3+\sqrt{5}\right) & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{17}{32}+\frac{31}{32 \sqrt{5}}} & \frac{1}{8} \left(-5-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{17}{32}+\frac{31}{32 \sqrt{5}}} & \frac{1}{8} \left(5+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{25}{32}+\frac{41}{32 \sqrt{5}}} & \frac{1}{8} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{25}{32}+\frac{41}{32 \sqrt{5}}} & \frac{1}{8} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & 0 & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{2} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ \frac{1}{4} \sqrt{13+\frac{22}{\sqrt{5}}} & -\frac{1}{4} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \sqrt{13+\frac{22}{\sqrt{5}}} & \frac{1}{4} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \sqrt{\frac{29}{32}+\frac{61}{32 \sqrt{5}}} & \frac{1}{8} \left(1-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{29}{32}+\frac{61}{32 \sqrt{5}}} & \frac{1}{8} \left(\sqrt{5}-1\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ -\sqrt{1+\frac{2}{\sqrt{5}}} & 0 & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(-2-\sqrt{5}\right) & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{4} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(2+\sqrt{5}\right) & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{1+\frac{2}{\sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\frac{1}{4} \sqrt{5+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(-2-\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{5+\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(2+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{25}{2}+\frac{41}{2 \sqrt{5}}} & \frac{1}{8} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{25}{2}+\frac{41}{2 \sqrt{5}}} & \frac{1}{8} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{13+\frac{22}{\sqrt{5}}} & -\frac{1}{4} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{13+\frac{22}{\sqrt{5}}} & \frac{1}{4} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{29}{2}+\frac{61}{2 \sqrt{5}}} & \frac{1}{8} \left(1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{4} \sqrt{\frac{29}{2}+\frac{61}{2 \sqrt{5}}} & \frac{1}{8} \left(\sqrt{5}-1\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{8} \left(1-\sqrt{5}\right) & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{8} \left(\sqrt{5}-1\right) & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{8} \left(1-\sqrt{5}\right) & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{8} \left(\sqrt{5}-1\right) & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(65+19 \sqrt{5}\right)} & \frac{1}{8} \left(-5-\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(65+19 \sqrt{5}\right)} & \frac{1}{8} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(85+31 \sqrt{5}\right)} & \frac{1}{8} \left(-5-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(85+31 \sqrt{5}\right)} & \frac{1}{8} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} \\
+ -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & -\frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & 0 & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{2} \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(-3-\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & 0 & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \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{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{13}{32}+\frac{29}{32 \sqrt{5}}} & \frac{1}{8} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{13}{32}+\frac{29}{32 \sqrt{5}}} & \frac{1}{8} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{32}+\frac{11}{32 \sqrt{5}}} & -\frac{3}{8} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{32}+\frac{11}{32 \sqrt{5}}} & \frac{3}{8} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & -\frac{3}{8} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{3}{8} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+\end{array}
+\right)$. Determine the Circumradius.
+Answer:
+$\sqrt{\frac{9}{8}+\frac{3 \sqrt{5}}{8}}$

pretraining/mathematica/geometry/solids/62552.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.91 & 0.858 & 0.252 \\
+ 0.97 & 0.463 & 0.568 \\
+ 0.235 & 0.7 & 0.776 \\
+ 0.059 & 0.677 & 0.281 \\
+ 0.539 & 0.727 & 0.216 \\
+ 0.755 & 0.005 & 0.833 \\
+\end{array}
+\right)$. 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: $0.54$ 
+Volume: $0.08$ 
+Surface Area: $1.4$