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pretraining/mathematica/geometry/solids/10465.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.85 & 0.197 & 0.264 \\
+ 0.351 & 0.854 & 0.89 \\
+ 0.383 & 0.001 & 0.188 \\
+ 0.845 & 0.393 & 0.026 \\
+ 0.624 & 0.512 & 0.988 \\
+ 0.514 & 0.207 & 0.89 \\
+ 0.009 & 0.053 & 0.612 \\
+ 0.064 & 0.19 & 0.405 \\
+ 0.273 & 0.318 & 0.166 \\
+ 0.958 & 0.682 & 0.53 \\
+ 0.346 & 0.467 & 0.122 \\
+\end{array}
+\right)$. 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: $3.07$ 
+Volume: $0.23$ 
+Surface Area: $2.18$

pretraining/mathematica/geometry/solids/11604.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.604 & 0.936 & 0.754 \\
+ 0.648 & 0.726 & 0.371 \\
+ 0.631 & 0.386 & 0.853 \\
+ 0.104 & 0.937 & 0.15 \\
+\end{array}
+\right)$. 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: $0.72$ 
+Surface Area: $0.63$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/11927.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.633 & 1. & 0.068 \\
+ 0.574 & 0.901 & 0.444 \\
+ 0.625 & 0.096 & 0.634 \\
+ 0.317 & 0.28 & 0.852 \\
+\end{array}
+\right)$. 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: $0.64$ 
+Volume: $0.01$ 
+Solid Angle: $0.09$

pretraining/mathematica/geometry/solids/13012.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.831 & 0.282 & 0.758 \\
+ 0.116 & 0.438 & 0.527 \\
+ 0.081 & 0.125 & 0.328 \\
+ 0.002 & 0.345 & 0.146 \\
+ 0.098 & 0.985 & 0.623 \\
+ 0.368 & 0.305 & 0.141 \\
+ 0.653 & 0.279 & 0.156 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $0.48$ 
+Volume: $0.09$ 
+Surface Area: $1.37$

pretraining/mathematica/geometry/solids/13020.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.437 & 0.115 & 0.812 \\
+ 0.575 & 0.704 & 0.891 \\
+ 0.91 & 0.897 & 0.332 \\
+ 0.28 & 0.112 & 0.434 \\
+ 0.98 & 0.127 & 0.447 \\
+ 0.32 & 0.873 & 0.646 \\
+ 0.682 & 0.563 & 0.896 \\
+ 0.811 & 0.811 & 0.708 \\
+\end{array}
+\right)$. 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: $1.77$ 
+Volume: $0.14$ 
+Surface Area: $1.6$

pretraining/mathematica/geometry/solids/13347.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-8.988,-6.447,7.867\}$ has radius $0.514$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $3.32$ 
+Volume: $0.57$

pretraining/mathematica/geometry/solids/14503.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.773 & 0.575 & 0.265 \\
+ 0.11 & 0.776 & 0.449 \\
+ 0.899 & 0.025 & 0.07 \\
+ 0.478 & 0.775 & 0.532 \\
+ 0.175 & 0.804 & 0.056 \\
+ 0.431 & 0.688 & 0.614 \\
+\end{array}
+\right)$. 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: $2.06$ 
+Surface Area: $0.94$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/16981.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.105 & 0.87 & 0.532 \\
+ 0.58 & 0.091 & 0.093 \\
+ 0.045 & 0.265 & 0.579 \\
+ 0.938 & 0.071 & 0.752 \\
+\end{array}
+\right)$. 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: $1.15$ 
+Solid Angle: $0.29$

pretraining/mathematica/geometry/solids/1781.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.741 & 0.535 & 0.339 \\
+ 0.693 & 0.968 & 0.439 \\
+ 0.228 & 0.438 & 0.056 \\
+ 0.558 & 0.249 & 0.69 \\
+ 0.851 & 0.11 & 0.534 \\
+\end{array}
+\right)$. 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: $0.86$ 
+Solid Angle: $3.99$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/18337.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.139 & 0.036 & 0.358 \\
+ 0.197 & 0.133 & 0.086 \\
+ 0.011 & 0.696 & 0.897 \\
+ 0.515 & 0.817 & 0.648 \\
+ 0.631 & 0.742 & 0.033 \\
+ 0.503 & 0.238 & 0.591 \\
+ 0.079 & 0.455 & 0.333 \\
+ 0.1 & 0.55 & 0.864 \\
+ 0.555 & 0.784 & 0.211 \\
+\end{array}
+\right)$. 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: $1.51$ 
+Solid Angle: $1.54$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/20974.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.546 & 0.488 & 0.204 \\
+ 0.594 & 0.269 & 0.699 \\
+ 0.7 & 0.665 & 0.041 \\
+ 0.499 & 0.691 & 0.823 \\
+ 0.944 & 0.846 & 0.57 \\
+\end{array}
+\right)$. 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: $0.03$ 
+Solid Angle: $1.25$

pretraining/mathematica/geometry/solids/2108.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.859 & 0.26 & 0.956 \\
+ 0.989 & 0.446 & 0.814 \\
+ 0.995 & 0.968 & 0.265 \\
+ 0.396 & 0.976 & 0.951 \\
+ 0.262 & 0.328 & 0.915 \\
+ 0.125 & 0.41 & 0.521 \\
+ 0.851 & 0.301 & 0.484 \\
+ 0.174 & 0.252 & 0.462 \\
+ 0.048 & 0.91 & 0.203 \\
+ 0.411 & 0.18 & 0.7 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $1.76$ 
+Surface Area: $2.46$ 
+Volume: $0.27$

pretraining/mathematica/geometry/solids/22071.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.949 & 0.899 & 0.097 \\
+ 0.9 & 0.193 & 0.705 \\
+ 0.866 & 0.98 & 0.744 \\
+ 0.613 & 0.742 & 0.027 \\
+ 0.07 & 0.223 & 0.506 \\
+ 0.547 & 0.987 & 0.101 \\
+\end{array}
+\right)$. 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: $1.47$ 
+Surface Area: $1.75$

pretraining/mathematica/geometry/solids/23156.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.561 & 0.991 & 0.076 \\
+ 0.981 & 0.006 & 0.099 \\
+ 0.128 & 0.675 & 0.36 \\
+ 0.732 & 0.071 & 0.296 \\
+ 0.746 & 0.435 & 0.94 \\
+ 0.986 & 0.831 & 0.696 \\
+ 0.789 & 0.83 & 0.119 \\
+ 0.5 & 0.269 & 0.077 \\
+ 0.278 & 0.106 & 0.484 \\
+\end{array}
+\right)$. 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: $1.36$ 
+Volume: $0.24$ 
+Surface Area: $2.27$

pretraining/mathematica/geometry/solids/23369.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.148 & 0.525 & 0.826 \\
+ 0.895 & 0.82 & 0.823 \\
+ 0.84 & 0.062 & 0.831 \\
+ 0.729 & 0.84 & 0.477 \\
+ 0.916 & 0.034 & 0.28 \\
+ 0.64 & 0.666 & 0.031 \\
+ 0.439 & 0.04 & 0.747 \\
+ 0.906 & 0.32 & 0.004 \\
+\end{array}
+\right)$. 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: $2.02$ 
+Volume: $0.18$ 
+Solid Angle: $0.95$

pretraining/mathematica/geometry/solids/24561.txt

@@ -0,0 +1,57 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -2 \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ -2 \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ -2 \sqrt{\frac{2}{3}} & 0 & 0 \\
+ -2 \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ -2 \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & -2 \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & -2 \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & -\frac{2}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & 0 \\
+ -\sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & \frac{2}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & 0 & -\sqrt{3} \\
+ -\sqrt{\frac{2}{3}} & 0 & -\frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & 0 & \sqrt{3} \\
+ -\sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & -\frac{2}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & 0 \\
+ -\sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & \frac{2}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & 2 \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & 2 \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ 0 & -2 \sqrt{\frac{2}{3}} & 0 \\
+ 0 & -\sqrt{\frac{2}{3}} & -\sqrt{3} \\
+ 0 & -\sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ 0 & -\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ 0 & -\sqrt{\frac{2}{3}} & \sqrt{3} \\
+ 0 & 0 & -\frac{2}{\sqrt{3}} \\
+ 0 & 0 & \frac{2}{\sqrt{3}} \\
+ 0 & \sqrt{\frac{2}{3}} & -\sqrt{3} \\
+ 0 & \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ 0 & \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ 0 & \sqrt{\frac{2}{3}} & \sqrt{3} \\
+ 0 & 2 \sqrt{\frac{2}{3}} & 0 \\
+ \sqrt{\frac{2}{3}} & -2 \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & -2 \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & -\frac{2}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & 0 \\
+ \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & \frac{2}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & 0 & -\sqrt{3} \\
+ \sqrt{\frac{2}{3}} & 0 & -\frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & 0 & \sqrt{3} \\
+ \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & -\frac{2}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & 0 \\
+ \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & \frac{2}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & 2 \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & 2 \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ 2 \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ 2 \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ 2 \sqrt{\frac{2}{3}} & 0 & 0 \\
+ 2 \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ 2 \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+\end{array}
+\right)$. Determine the Inradius.
+Answer:
+$\sqrt{\frac{2}{3}}$

pretraining/mathematica/geometry/solids/25090.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.493 & 0.205 & 0.792 \\
+ 0.452 & 0.941 & 0.988 \\
+ 0.574 & 0.623 & 0.304 \\
+ 0.838 & 0.302 & 0.015 \\
+ 0.577 & 0.258 & 0.848 \\
+ 0.929 & 0.187 & 0.23 \\
+ 0.199 & 0.55 & 0.557 \\
+ 0.42 & 0.218 & 0.851 \\
+ 0.979 & 0.421 & 0.18 \\
+ 0.006 & 0.922 & 0.897 \\
+\end{array}
+\right)$. 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: $1.69$ 
+Solid Angle: $4.23$

pretraining/mathematica/geometry/solids/25311.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.337 & 0.826 & 0.432 \\
+ 0.246 & 0.45 & 0.693 \\
+ 0.418 & 0.132 & 0.944 \\
+ 0.094 & 0.254 & 0.229 \\
+ 0.297 & 0.626 & 0.404 \\
+ 0.252 & 0.239 & 0.839 \\
+ 0.096 & 0.233 & 0.451 \\
+\end{array}
+\right)$. 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: $0.02$ 
+Surface Area: $0.6$ 
+Solid Angle: $0.21$

pretraining/mathematica/geometry/solids/25458.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.566 & 0.773 & 0.776 \\
+ 0.701 & 0.109 & 0.075 \\
+ 0.095 & 0.587 & 0.052 \\
+ 0.285 & 0.214 & 0.027 \\
+ 0.336 & 0.13 & 0.38 \\
+ 0.825 & 0.37 & 0.545 \\
+ 0.627 & 0.35 & 0.076 \\
+\end{array}
+\right)$. 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: $0.41$ 
+Volume: $0.08$ 
+Surface Area: $1.21$

pretraining/mathematica/geometry/solids/2833.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.095 & 0.814 & 0.733 \\
+ 0.43 & 0.743 & 0.44 \\
+ 0.342 & 0.659 & 0.394 \\
+ 0.902 & 0.417 & 0.129 \\
+ 0.947 & 0.015 & 0.881 \\
+\end{array}
+\right)$. 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: $1.07$ 
+Solid Angle: $0.15$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/28847.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.082 & 0.459 & 0.762 \\
+ 0.366 & 0.875 & 0.206 \\
+ 0.925 & 0.831 & 0.533 \\
+ 0.068 & 0.149 & 0.287 \\
+ 0.886 & 0.836 & 0.135 \\
+ 0.267 & 0.071 & 0.802 \\
+ 0.525 & 0.991 & 0.993 \\
+\end{array}
+\right)$. 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: $2.02$ 
+Volume: $0.18$ 
+Solid Angle: $2.8$

pretraining/mathematica/geometry/solids/29034.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.286 & 0.933 & 0.205 \\
+ 0.096 & 0.612 & 0.221 \\
+ 0.127 & 0.774 & 0.577 \\
+ 0.595 & 0.649 & 0.551 \\
+ 0.759 & 0.47 & 0.125 \\
+ 0.153 & 0.149 & 0.858 \\
+\end{array}
+\right)$. 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: $0.08$ 
+Solid Angle: $1.26$

pretraining/mathematica/geometry/solids/30724.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.381 & 0.606 & 0.707 \\
+ 0.548 & 0.245 & 0.24 \\
+ 0.889 & 0.334 & 0.095 \\
+ 0.642 & 0.135 & 0.256 \\
+ 0.555 & 0.061 & 0.987 \\
+ 0.823 & 0.11 & 0.385 \\
+ 0.485 & 0.265 & 0.802 \\
+ 0.434 & 0.419 & 0.876 \\
+\end{array}
+\right)$. 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: $0.47$ 
+Volume: $0.03$ 
+Surface Area: $0.86$

pretraining/mathematica/geometry/solids/32155.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.503 & 0.454 & 0.643 \\
+ 0.019 & 0.916 & 0.221 \\
+ 0.568 & 0.903 & 0.773 \\
+ 0.35 & 0.632 & 0.044 \\
+ 0.236 & 0.901 & 0.236 \\
+ 0.499 & 0.472 & 0.961 \\
+ 0.013 & 0.101 & 0.129 \\
+\end{array}
+\right)$. 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: $0.09$ 
+Surface Area: $1.42$ 
+Solid Angle: $4.06$

pretraining/mathematica/geometry/solids/32338.txt

@@ -0,0 +1,5 @@
+Problem:
+An ellipsoid centered at $\{5.599,-9.511,-2.676\}$ has radii $\{1.59,1.412,3.194\}$. Estimate the ellipsoid's surface area and volume.
+Answer:
+Surface Area: $51.11$ 
+Volume: $30.06$

pretraining/mathematica/geometry/solids/33738.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.828 & 0.516 & 0.843 \\
+ 0.972 & 0.618 & 0.545 \\
+ 0.926 & 0.217 & 0.483 \\
+ 0.376 & 0.749 & 0.542 \\
+ 0.615 & 0.191 & 0.275 \\
+ 0.364 & 0.555 & 0. \\
+\end{array}
+\right)$. 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: $0.99$ 
+Solid Angle: $1.01$

pretraining/mathematica/geometry/solids/36545.txt

@@ -0,0 +1,49 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1 & 0 & 0 \\
+ -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\
+ -\frac{1}{\sqrt{2}} & -\frac{1}{2} & 0 \\
+ -\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\
+ -\frac{1}{\sqrt{2}} & 0 & -\frac{1}{2} \\
+ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\
+ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
+ -\frac{1}{\sqrt{2}} & \frac{1}{2} & 0 \\
+ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
+ -\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 \\
+ -\frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} \\
+ -\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} \\
+ -\frac{1}{2} & \frac{1}{\sqrt{2}} & 0 \\
+ 0 & -1 & 0 \\
+ 0 & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
+ 0 & -\frac{1}{\sqrt{2}} & -\frac{1}{2} \\
+ 0 & -\frac{1}{\sqrt{2}} & \frac{1}{2} \\
+ 0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
+ 0 & -\frac{1}{2} & -\frac{1}{\sqrt{2}} \\
+ 0 & -\frac{1}{2} & \frac{1}{\sqrt{2}} \\
+ 0 & 0 & -1 \\
+ 0 & 0 & 1 \\
+ 0 & \frac{1}{2} & -\frac{1}{\sqrt{2}} \\
+ 0 & \frac{1}{2} & \frac{1}{\sqrt{2}} \\
+ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
+ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{2} \\
+ 0 & \frac{1}{\sqrt{2}} & \frac{1}{2} \\
+ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
+ 0 & 1 & 0 \\
+ \frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 \\
+ \frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} \\
+ \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} \\
+ \frac{1}{2} & \frac{1}{\sqrt{2}} & 0 \\
+ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\
+ \frac{1}{\sqrt{2}} & -\frac{1}{2} & 0 \\
+ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\
+ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{2} \\
+ \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\
+ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
+ \frac{1}{\sqrt{2}} & \frac{1}{2} & 0 \\
+ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
+ 1 & 0 & 0 \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$6 \left(3+2 \sqrt{3}\right)$

pretraining/mathematica/geometry/solids/37576.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.233 & 0.554 & 0.886 \\
+ 0.787 & 0.6 & 0.918 \\
+ 0.165 & 0.306 & 0.135 \\
+ 0.869 & 0.676 & 0.036 \\
+ 0.694 & 0.399 & 0.381 \\
+ 0.594 & 0.241 & 0.051 \\
+\end{array}
+\right)$. 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: $0.36$ 
+Volume: $0.06$ 
+Surface Area: $1.32$

pretraining/mathematica/geometry/solids/38504.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 1. & 0.805 & 0.209 \\
+ 0.5 & 0.778 & 0.165 \\
+ 0.485 & 0.26 & 0.078 \\
+ 0.94 & 0.918 & 0.332 \\
+ 0.583 & 0.018 & 0.416 \\
+ 0.464 & 0.374 & 0.148 \\
+ 0.344 & 0.1 & 0.765 \\
+ 0.892 & 0.172 & 0.02 \\
+\end{array}
+\right)$. 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: $0.09$ 
+Solid Angle: $1.33$ 
+Surface Area: $1.36$

pretraining/mathematica/geometry/solids/41440.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.715 & 0.032 & 0.573 \\
+ 0.435 & 0.03 & 0.707 \\
+ 0.387 & 0.919 & 0.994 \\
+ 0.565 & 0.57 & 0.047 \\
+ 0.575 & 0.904 & 0.712 \\
+ 0.982 & 0.786 & 0.15 \\
+ 0.115 & 0.275 & 0.868 \\
+ 0.631 & 0.257 & 0.005 \\
+ 0.791 & 0.3 & 0.046 \\
+ 0.73 & 0.608 & 0.731 \\
+\end{array}
+\right)$. 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: $1.97$ 
+Volume: $0.18$ 
+Surface Area: $2.06$

pretraining/mathematica/geometry/solids/43244.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.706 & 0.451 & 0.207 \\
+ 0.962 & 0.939 & 0.281 \\
+ 0.751 & 0.843 & 0.9 \\
+ 0.189 & 0.816 & 0.57 \\
+ 0.817 & 0.774 & 0.788 \\
+\end{array}
+\right)$. 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: $0.04$ 
+Solid Angle: $0.54$ 
+Surface Area: $0.84$

pretraining/mathematica/geometry/solids/43973.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.144 & 0.183 & 0.413 \\
+ 0.022 & 0.528 & 0.582 \\
+ 0.445 & 0.8 & 0.701 \\
+ 0.651 & 0.052 & 0.752 \\
+ 0.361 & 0.441 & 0.065 \\
+ 0.091 & 0.156 & 0.596 \\
+ 0.113 & 0.4 & 0.329 \\
+ 0.227 & 0.793 & 0.361 \\
+ 0.611 & 0.628 & 0.879 \\
+ 0.953 & 0.237 & 0.299 \\
+ 0.987 & 0.128 & 0.744 \\
+\end{array}
+\right)$. 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: $3.37$ 
+Surface Area: $1.86$

pretraining/mathematica/geometry/solids/46273.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.581 & 0.699 & 0.813 \\
+ 0.906 & 0.269 & 0.634 \\
+ 0.265 & 0.435 & 0.351 \\
+ 0.973 & 0.646 & 0.246 \\
+ 0.795 & 0.065 & 0.953 \\
+ 0.73 & 0.689 & 0.897 \\
+ 0.004 & 0.587 & 0.478 \\
+ 0.968 & 0.728 & 0.082 \\
+ 0.758 & 0.924 & 0.264 \\
+ 0.024 & 0.04 & 0.522 \\
+\end{array}
+\right)$. 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: $3.74$ 
+Surface Area: $2.11$ 
+Volume: $0.18$

pretraining/mathematica/geometry/solids/4836.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.93 & 0.795 & 0.657 \\
+ 0.83 & 0.989 & 0.939 \\
+ 0.125 & 0.658 & 0.587 \\
+ 0.537 & 0.806 & 0.339 \\
+ 0.079 & 0.19 & 0.051 \\
+ 0.059 & 0.855 & 0.92 \\
+ 0.997 & 0.926 & 0.252 \\
+ 0.278 & 0.017 & 0.167 \\
+ 0.067 & 0.154 & 0.208 \\
+ 0.218 & 0.028 & 0.679 \\
+\end{array}
+\right)$. 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: $0.24$ 
+Solid Angle: $3.76$ 
+Surface Area: $2.5$

pretraining/mathematica/geometry/solids/50128.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.027 & 0.019 & 0.144 \\
+ 0.512 & 0.342 & 0.341 \\
+ 0.106 & 0.995 & 0.882 \\
+ 0.06 & 0.073 & 0.848 \\
+\end{array}
+\right)$. 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: $0.41$ 
+Volume: $0.05$ 
+Surface Area: $1.12$

pretraining/mathematica/geometry/solids/50351.txt

@@ -0,0 +1,67 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -0.851 & 2.065 \\
+ 1.618 & 0. & -1.539 \\
+ -0.809 & -0.263 & 2.065 \\
+ 1.309 & 0.951 & -1.539 \\
+ 0.809 & -1.964 & 0.688 \\
+ 1.309 & -1.802 & -0.162 \\
+ -0.5 & 2.065 & 0.688 \\
+ 0. & 2.227 & -0.162 \\
+ -0.5 & -1.539 & 1.539 \\
+ 1.309 & -0.951 & -1.539 \\
+ -1.309 & 0.951 & 1.539 \\
+ 0.5 & 1.539 & -1.539 \\
+ -1.309 & -0.951 & 1.539 \\
+ 0.5 & -0.688 & -2.065 \\
+ -1.618 & 0. & 1.539 \\
+ 0.809 & 0.263 & -2.065 \\
+ 0. & -2.227 & 0.162 \\
+ 0.5 & -2.065 & -0.688 \\
+ -1.309 & 1.802 & 0.162 \\
+ -0.809 & 1.964 & -0.688 \\
+ 0.5 & 0.688 & 2.065 \\
+ 2.118 & 0.688 & -0.162 \\
+ 0.5 & -1.539 & 1.539 \\
+ 1.809 & -1.114 & -0.688 \\
+ -0.5 & 1.539 & 1.539 \\
+ 0.809 & 1.964 & -0.688 \\
+ 1.309 & -0.951 & 1.539 \\
+ 2.118 & -0.688 & 0.162 \\
+ 0.5 & 1.539 & 1.539 \\
+ 1.309 & 1.802 & 0.162 \\
+ 1.618 & 0. & 1.539 \\
+ 2.118 & 0.162 & 0.688 \\
+ 1.309 & 0.951 & 1.539 \\
+ 1.809 & 1.114 & 0.688 \\
+ 1.618 & -1.376 & 0.688 \\
+ 0.809 & -0.263 & 2.065 \\
+ 2.118 & -0.162 & -0.688 \\
+ -0.5 & 0.688 & 2.065 \\
+ 1.618 & 1.376 & -0.688 \\
+ 0.5 & 2.065 & 0.688 \\
+ -0.809 & -1.964 & 0.688 \\
+ 0.5 & -1.539 & -1.539 \\
+ -1.809 & 1.114 & 0.688 \\
+ -0.5 & 1.539 & -1.539 \\
+ -1.309 & -1.802 & -0.162 \\
+ -0.5 & -1.539 & -1.539 \\
+ -2.118 & 0.688 & -0.162 \\
+ -1.309 & 0.951 & -1.539 \\
+ -1.809 & -1.114 & -0.688 \\
+ -1.309 & -0.951 & -1.539 \\
+ -2.118 & -0.162 & -0.688 \\
+ -1.618 & 0. & -1.539 \\
+ -0.5 & -2.065 & -0.688 \\
+ -1.618 & -1.376 & 0.688 \\
+ -0.5 & -0.688 & -2.065 \\
+ -2.118 & 0.162 & 0.688 \\
+ 0. & 0.851 & -2.065 \\
+ -1.618 & 1.376 & -0.688 \\
+ -2.118 & -0.688 & 0.162 \\
+ -0.809 & 0.263 & -2.065 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$120.$

pretraining/mathematica/geometry/solids/50356.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.966 & 0.775 & 0.301 \\
+ 0.693 & 0.007 & 0.521 \\
+ 0.287 & 0.797 & 0.205 \\
+ 0.147 & 0.189 & 0.04 \\
+ 0.776 & 0.486 & 0.65 \\
+ 0.914 & 0.189 & 0.867 \\
+ 0.589 & 0.706 & 0.448 \\
+\end{array}
+\right)$. 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: $0.85$ 
+Surface Area: $1.47$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/51903.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.69 & 0.047 & 0.484 \\
+ 0.508 & 0.495 & 0.11 \\
+ 0.669 & 0.799 & 0.804 \\
+ 0.004 & 0.584 & 0.475 \\
+ 0.64 & 0.47 & 0.897 \\
+ 0.716 & 0.402 & 0.189 \\
+ 0.682 & 0.77 & 0.587 \\
+ 0.384 & 0.392 & 0.316 \\
+ 0.969 & 0.586 & 0.845 \\
+\end{array}
+\right)$. 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: $1.23$ 
+Surface Area: $1.38$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/52289.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.739 & 0.267 & 0.238 \\
+ 0.562 & 0.041 & 0.831 \\
+ 0.177 & 0.22 & 0.597 \\
+ 0.23 & 0.853 & 0.636 \\
+ 0.202 & 0.528 & 0.748 \\
+ 0.569 & 0.087 & 0.571 \\
+ 0.086 & 0.503 & 0.286 \\
+ 0.869 & 0.875 & 0.615 \\
+ 0.685 & 0.167 & 0.854 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.59$ 
+Volume: $0.13$ 
+Solid Angle: $1.4$

pretraining/mathematica/geometry/solids/52741.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.972 & 0.636 & 0.443 \\
+ 0.045 & 0.986 & 0.102 \\
+ 0.909 & 0.49 & 0.393 \\
+ 0.373 & 0.542 & 0.057 \\
+ 0.186 & 0.989 & 0.237 \\
+ 0.35 & 0.148 & 0.581 \\
+ 0.918 & 0.208 & 0.77 \\
+ 0.287 & 0.065 & 0.144 \\
+ 0.002 & 0.168 & 0.66 \\
+ 0.309 & 0.404 & 0.649 \\
+ 0.608 & 0.499 & 0.664 \\
+\end{array}
+\right)$. 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: $0.2$ 
+Surface Area: $2.07$ 
+Solid Angle: $1.42$

pretraining/mathematica/geometry/solids/54338.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.021 & 0.584 & 0.452 \\
+ 0.742 & 0.946 & 0.312 \\
+ 0.676 & 0.031 & 0.652 \\
+ 0.805 & 0.401 & 0.089 \\
+\end{array}
+\right)$. 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: $0.3$ 
+Volume: $0.04$ 
+Surface Area: $1.05$

pretraining/mathematica/geometry/solids/54538.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.739 & 0.857 & 0.527 \\
+ 0.275 & 0.036 & 0.732 \\
+ 0.927 & 0.885 & 0.223 \\
+ 0.989 & 0.883 & 0.579 \\
+ 0.716 & 0.597 & 0.107 \\
+ 0.859 & 0.766 & 0.002 \\
+\end{array}
+\right)$. 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: $2.63$ 
+Volume: $0.02$ 
+Surface Area: $0.87$

pretraining/mathematica/geometry/solids/55790.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.115 & 0.792 & 0.71 \\
+ 0.2 & 0.483 & 0.473 \\
+ 0.91 & 0.826 & 0.356 \\
+ 0.611 & 0.15 & 0.286 \\
+ 0.122 & 0.674 & 0.574 \\
+ 0.568 & 0.173 & 0.641 \\
+ 0.038 & 0.621 & 0.775 \\
+ 0.246 & 0.264 & 0.823 \\
+ 0.696 & 0.26 & 0.09 \\
+\end{array}
+\right)$. 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: $1.35$ 
+Solid Angle: $1.23$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/56128.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.552 & 0.096 & 0.537 \\
+ 0.943 & 0.991 & 0.426 \\
+ 0.653 & 0.857 & 0.272 \\
+ 0.531 & 0.126 & 0.857 \\
+ 0.565 & 0.92 & 0.307 \\
+ 0.525 & 0.074 & 0.176 \\
+ 0.426 & 0.832 & 0.175 \\
+ 0.688 & 0.167 & 0.423 \\
+ 0.611 & 0.196 & 0.088 \\
+\end{array}
+\right)$. 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: $1.26$ 
+Solid Angle: $4.64$

pretraining/mathematica/geometry/solids/5977.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.56 & 0.939 & 0.256 \\
+ 0.754 & 0.161 & 0.714 \\
+ 0.353 & 0.311 & 0.922 \\
+ 0.795 & 0.671 & 0.509 \\
+ 0.028 & 0.731 & 0.138 \\
+ 0.838 & 0.274 & 0.13 \\
+\end{array}
+\right)$. 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: $1.51$ 
+Volume: $0.1$ 
+Solid Angle: $1.32$

pretraining/mathematica/geometry/solids/63724.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.906 & 0.025 & 0.844 \\
+ 0.676 & 0.435 & 0.823 \\
+ 0.246 & 0.784 & 0.572 \\
+ 0.878 & 0.801 & 0.857 \\
+ 0.024 & 0.17 & 0.472 \\
+ 0.55 & 0.83 & 0.461 \\
+ 0.007 & 0.721 & 0.422 \\
+ 0.685 & 0.928 & 0.63 \\
+\end{array}
+\right)$. 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: $0.28$ 
+Volume: $0.07$ 
+Surface Area: $1.55$

pretraining/mathematica/geometry/solids/65918.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.492 & 0.272 & 0.662 \\
+ 0.647 & 0.219 & 0.933 \\
+ 0.133 & 0.454 & 0.02 \\
+ 0.83 & 0.057 & 0.548 \\
+ 0.815 & 0.372 & 0.388 \\
+\end{array}
+\right)$. 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: $3.88$ 
+Surface Area: $0.66$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/69281.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.198 & 0.494 & 0.866 \\
+ 0.242 & 0.236 & 0.984 \\
+ 0.516 & 0.486 & 0.558 \\
+ 0.653 & 0.896 & 0.285 \\
+\end{array}
+\right)$. 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: $0.32$ 
+Solid Angle: $0.17$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/70232.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.36 & 0.892 & 0.421 \\
+ 0.458 & 0.691 & 0.205 \\
+ 0.211 & 0.218 & 0.635 \\
+ 0.56 & 0.129 & 0.311 \\
+ 0.989 & 0.506 & 0.996 \\
+ 0.295 & 0.654 & 0.856 \\
+ 0.192 & 0.159 & 0.396 \\
+\end{array}
+\right)$. 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: $0.11$ 
+Solid Angle: $1.36$ 
+Surface Area: $1.43$

pretraining/mathematica/geometry/solids/70395.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.15 & 0.718 & 0.594 \\
+ 0.632 & 0.605 & 0.682 \\
+ 0.907 & 0.44 & 0.891 \\
+ 0.766 & 0.103 & 0.371 \\
+ 0.279 & 0.091 & 0.004 \\
+ 0.159 & 0.047 & 0.238 \\
+ 0.061 & 0.406 & 0.204 \\
+ 0.057 & 0.739 & 0.216 \\
+ 0.262 & 0.558 & 0.798 \\
+ 0.389 & 0.152 & 0.797 \\
+\end{array}
+\right)$. 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: $2.04$ 
+Surface Area: $1.77$ 
+Volume: $0.14$