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pretraining/mathematica/geometry/solids/10311.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.825 & 0.138 & 0.963 \\
+ 0.343 & 0.283 & 0.824 \\
+ 0.856 & 0.587 & 0.824 \\
+ 0.201 & 0.916 & 0.999 \\
+ 0.478 & 0.592 & 0.493 \\
+\end{array}
+\right)$. 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: $0.49$ 
+Volume: $0.05$ 
+Surface Area: $0.88$

pretraining/mathematica/geometry/solids/10507.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.213 & 0.629 & 0.087 \\
+ 0.311 & 0.668 & 0.133 \\
+ 0.39 & 0.261 & 0.137 \\
+ 0.743 & 0.276 & 0.934 \\
+ 0.257 & 0.137 & 0.481 \\
+ 0.536 & 0.644 & 0.362 \\
+ 0.833 & 0.809 & 0.141 \\
+ 0.727 & 0.291 & 0.461 \\
+ 0.691 & 0.573 & 0.138 \\
+\end{array}
+\right)$. 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: $1.21$ 
+Volume: $0.07$ 
+Solid Angle: $0.98$

pretraining/mathematica/geometry/solids/10743.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.302 & 0.555 & 0.264 \\
+ 0.131 & 0.691 & 0.627 \\
+ 0.307 & 0.326 & 0.798 \\
+ 0.046 & 0.454 & 0.448 \\
+ 0.416 & 0.364 & 0.967 \\
+ 0.798 & 0.814 & 0.932 \\
+ 0.742 & 0.408 & 0.852 \\
+ 0.07 & 0.61 & 0.352 \\
+ 0.496 & 0.152 & 0.327 \\
+ 0.732 & 0.128 & 0.581 \\
+\end{array}
+\right)$. 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: $2.33$ 
+Volume: $0.1$ 
+Surface Area: $1.35$

pretraining/mathematica/geometry/solids/13610.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.948 & 0.088 & 0.68 \\
+ 0.937 & 0.096 & 0.486 \\
+ 0.228 & 0.784 & 0.853 \\
+ 0.12 & 0.429 & 0.979 \\
+ 0.406 & 0.632 & 0.849 \\
+ 0.274 & 0.3 & 0.57 \\
+ 0.524 & 0.138 & 0.542 \\
+\end{array}
+\right)$. 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: $0.74$ 
+Surface Area: $0.83$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/15102.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-5.297,-3.761,8.497\}$ has radius $0.521$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $0.59$ 
+Surface Area: $3.41$

pretraining/mathematica/geometry/solids/16156.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.186 & 0.361 & 0.222 \\
+ 0.357 & 0.672 & 0.537 \\
+ 0.96 & 0.025 & 0.693 \\
+ 0.048 & 0.15 & 0.753 \\
+\end{array}
+\right)$. 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: $0.88$ 
+Solid Angle: $0.79$

pretraining/mathematica/geometry/solids/18816.txt

@@ -0,0 +1,37 @@
+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}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\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}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \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}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{1+\frac{2}{\sqrt{5}}} & 0 & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{1+\frac{2}{\sqrt{5}}} & 0 & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & -\frac{1}{8} \left(1+\sqrt{5}\right)^2 & 0 \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & 0 \\
+ \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & -\sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & -\frac{1}{8} \left(1+\sqrt{5}\right)^2 & 0 \\
+ -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & 0 \\
+ -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} & 0 \\
+ -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} & 0 \\
+ \frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} & 0 \\
+ \frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} & 0 \\
+ -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & \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}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \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(-3-\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(3+\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{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{1+\frac{2}{\sqrt{5}}} \\
+\end{array}
+\right)$. Determine the Centroid.
+Answer:
+$\{0,0,0\}$

pretraining/mathematica/geometry/solids/20149.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.59 & 0.283 & 0.623 \\
+ 0.517 & 0.704 & 0.219 \\
+ 0.491 & 0.264 & 0.975 \\
+ 0.807 & 0.463 & 0.567 \\
+\end{array}
+\right)$. 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: $0.34$ 
+Solid Angle: $2.02$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/20855.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.065 & 0.293 & 0.193 \\
+ 0.373 & 0.005 & 0.91 \\
+ 0.909 & 0.46 & 0.515 \\
+ 0.443 & 0.537 & 0.026 \\
+ 0.882 & 0.513 & 0.287 \\
+ 0.912 & 0.197 & 0.273 \\
+ 0.183 & 0.89 & 0.854 \\
+ 0.031 & 0.068 & 0.727 \\
+ 0.474 & 0.854 & 0.316 \\
+\end{array}
+\right)$. 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: $0.24$ 
+Surface Area: $2.22$ 
+Solid Angle: $1.76$

pretraining/mathematica/geometry/solids/21614.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.657 & 0.2 & 0.877 \\
+ 0.384 & 0.68 & 0.239 \\
+ 0.503 & 0.032 & 0.627 \\
+ 0.483 & 0.638 & 0.696 \\
+ 0.976 & 0.709 & 0.542 \\
+\end{array}
+\right)$. 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: $0.84$ 
+Volume: $0.04$ 
+Solid Angle: $0.93$

pretraining/mathematica/geometry/solids/22265.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $7.946$ has its base centered at$\{2.824,5.599,2.101\}$ and its tip is at $\{4.487,1.891,3.005\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Centroid: $\{3.24,4.67,2.33\}$ 
+Volume: $275.27$ 
+Surface Area: $422.27$

pretraining/mathematica/geometry/solids/2228.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.776 & 0.918 & 0.22 \\
+ 0.291 & 0.528 & 0.162 \\
+ 0.984 & 0.505 & 0.195 \\
+ 0.843 & 0.497 & 0.479 \\
+ 0.632 & 0.08 & 0.917 \\
+ 0.308 & 0.811 & 0.778 \\
+\end{array}
+\right)$. 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: $1.1$

pretraining/mathematica/geometry/solids/22481.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.126 & 0.975 & 0.736 \\
+ 0.586 & 0.592 & 0.826 \\
+ 0.039 & 0.121 & 0.838 \\
+ 0.989 & 0.239 & 0.689 \\
+\end{array}
+\right)$. 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: $0.1$ 
+Surface Area: $0.88$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/229.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.551 & 0.064 & 0.418 \\
+ 0.049 & 0.532 & 0.383 \\
+ 0.324 & 0.707 & 0.654 \\
+ 0.205 & 0.732 & 0.018 \\
+ 0.792 & 0.104 & 0.927 \\
+ 0.745 & 0.059 & 0.6 \\
+\end{array}
+\right)$. 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: $1.09$ 
+Solid Angle: $1.5$

pretraining/mathematica/geometry/solids/23231.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.626 & 0.415 & 0.953 \\
+ 0.414 & 0.166 & 0.617 \\
+ 0.741 & 0.829 & 0.776 \\
+ 0.925 & 0.368 & 0.115 \\
+ 0.755 & 0.872 & 0.569 \\
+ 0.026 & 0.379 & 0.092 \\
+\end{array}
+\right)$. 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: $0.95$ 
+Surface Area: $1.45$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/23379.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.087 & 0.555 & 0.098 \\
+ 0.854 & 0.588 & 0.089 \\
+ 0.648 & 0.947 & 0.174 \\
+ 0.218 & 0.236 & 0.629 \\
+ 0.401 & 0.535 & 0.567 \\
+ 0.723 & 0.072 & 0.547 \\
+ 0.897 & 0.423 & 0.207 \\
+ 0.103 & 0.475 & 0.722 \\
+ 0.906 & 0.064 & 0.682 \\
+\end{array}
+\right)$. 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: $1.65$ 
+Solid Angle: $1.11$

pretraining/mathematica/geometry/solids/2478.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.698 & 0.767 & 0.293 \\
+ 0.814 & 0.672 & 0.862 \\
+ 0.985 & 0.326 & 0.498 \\
+ 0.647 & 0.331 & 0.256 \\
+\end{array}
+\right)$. 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: $0.42$ 
+Surface Area: $0.47$

pretraining/mathematica/geometry/solids/25259.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.85 & 0.983 & 0.729 \\
+ 0.397 & 0.153 & 0.502 \\
+ 0.635 & 0.394 & 0.985 \\
+ 0.734 & 0.015 & 0.748 \\
+ 0.221 & 0.895 & 0.88 \\
+ 0.618 & 0.903 & 0.688 \\
+ 0.643 & 0.802 & 0.494 \\
+ 0.203 & 0.63 & 0.835 \\
+\end{array}
+\right)$. 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: $1.28$ 
+Solid Angle: $0.85$

pretraining/mathematica/geometry/solids/25482.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.217 & 0.534 & 0.758 \\
+ 0.632 & 0.955 & 0.423 \\
+ 0.952 & 0.039 & 0.982 \\
+ 0.384 & 0.206 & 0.407 \\
+ 0.721 & 0.183 & 0.952 \\
+\end{array}
+\right)$. 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: $1.07$ 
+Volume: $0.05$ 
+Surface Area: $1.04$

pretraining/mathematica/geometry/solids/28401.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.175 & 0.796 & 0.368 \\
+ 0.863 & 0.648 & 0.277 \\
+ 0.172 & 0.882 & 0.674 \\
+ 0.579 & 0.378 & 0.997 \\
+ 0.597 & 0.937 & 0.991 \\
+ 0.661 & 0.131 & 0.139 \\
+ 0.136 & 0.333 & 0.527 \\
+ 0.828 & 0.519 & 0.428 \\
+ 0.006 & 0.317 & 0.794 \\
+\end{array}
+\right)$. 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: $0.2$ 
+Surface Area: $1.95$ 
+Solid Angle: $2.12$

pretraining/mathematica/geometry/solids/29040.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-3.756,-4.923,0.802\}$ has radius $0.163$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $0.02$ 
+Surface Area: $0.33$

pretraining/mathematica/geometry/solids/32817.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.268 & 0.72 & 0.995 \\
+ 0.823 & 0.749 & 0.127 \\
+ 0.719 & 0.49 & 0.882 \\
+ 0.085 & 0.723 & 0.27 \\
+ 0.413 & 0.339 & 0.365 \\
+ 0.716 & 0.084 & 0.963 \\
+ 0.65 & 0.118 & 0.635 \\
+\end{array}
+\right)$. 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: $0.12$ 
+Surface Area: $1.59$ 
+Solid Angle: $0.83$

pretraining/mathematica/geometry/solids/34397.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & 0 & -\frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{1}{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & 0 & \frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{-1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} & \frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{-1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} & \frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{\sqrt{5}-1}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{\sqrt{5}-1}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{1+\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+ \frac{1+\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} & -\frac{1}{2} \sqrt{1-\frac{1}{4 \left(\frac{5}{8}+\frac{\sqrt{5}}{8}\right)}} \\
+\end{array}
+\right)$. Determine the Circumradius.
+Answer:
+$\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}}$

pretraining/mathematica/geometry/solids/35799.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.262 & 0.284 & 0.966 \\
+ 0.466 & 0.081 & 0.307 \\
+ 0.006 & 0.623 & 0.231 \\
+ 0.767 & 0.723 & 0.514 \\
+ 0.889 & 0.928 & 0.006 \\
+ 0.932 & 0.022 & 0.139 \\
+\end{array}
+\right)$. 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: $0.64$ 
+Volume: $0.17$ 
+Surface Area: $2.02$

pretraining/mathematica/geometry/solids/36234.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.635 & 0.167 & 0.208 \\
+ 0.319 & 0.18 & 0.338 \\
+ 0.019 & 0.417 & 0.598 \\
+ 0.158 & 0.073 & 0.132 \\
+ 0.667 & 0.487 & 0.455 \\
+ 0.087 & 0.554 & 0.683 \\
+ 0.084 & 0.855 & 0.061 \\
+ 0.215 & 0.504 & 0.862 \\
+ 0.293 & 0.816 & 0.987 \\
+\end{array}
+\right)$. 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: $1.6$ 
+Volume: $0.13$ 
+Solid Angle: $1.13$

pretraining/mathematica/geometry/solids/41758.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.016 & 0.337 & 0.96 \\
+ 0.616 & 0.368 & 0.51 \\
+ 0.05 & 0.826 & 0.185 \\
+ 0.89 & 0.481 & 0.548 \\
+ 0.495 & 0.816 & 0.822 \\
+ 0.581 & 0.233 & 0.94 \\
+\end{array}
+\right)$. 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: $0.67$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/43873.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.085 & 0.842 & 0.821 \\
+ 0.492 & 0.776 & 0.294 \\
+ 0.868 & 0.138 & 0.139 \\
+ 0.831 & 0.034 & 0.228 \\
+ 0.412 & 0.263 & 0.609 \\
+ 0.543 & 0.059 & 0.122 \\
+ 0.277 & 0.957 & 0.701 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.13$ 
+Volume: $0.05$ 
+Solid Angle: $0.36$

pretraining/mathematica/geometry/solids/44889.txt

@@ -0,0 +1,62 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0.688 & -0.5 & 2.065 \\
+ 0. & 1.618 & -1.539 \\
+ -0.263 & -0.809 & 2.065 \\
+ -0.951 & 1.309 & -1.539 \\
+ 1.729 & 0.809 & 1.158 \\
+ 1.802 & 1.309 & -0.162 \\
+ -2.065 & -0.5 & 0.688 \\
+ -2.227 & 0. & -0.162 \\
+ 1.729 & -0.809 & 1.158 \\
+ 0.951 & 1.309 & -1.539 \\
+ -0.951 & -1.309 & 1.539 \\
+ -1.539 & 0.5 & -1.539 \\
+ 0.951 & -1.309 & 1.539 \\
+ 0. & -1.618 & 1.539 \\
+ 2.154 & 0.5 & 0.308 \\
+ 2.065 & 0.5 & -0.688 \\
+ -1.802 & -1.309 & 0.162 \\
+ -1.964 & -0.809 & -0.688 \\
+ -0.263 & 0.809 & 2.065 \\
+ -0.688 & 2.118 & -0.162 \\
+ 1.466 & 0. & 1.684 \\
+ 1.114 & 1.809 & -0.688 \\
+ -1.539 & -0.5 & 1.539 \\
+ -1.964 & 0.809 & -0.688 \\
+ 0.951 & 1.309 & 1.539 \\
+ 0.688 & 2.118 & 0.162 \\
+ -1.539 & 0.5 & 1.539 \\
+ -1.802 & 1.309 & 0.162 \\
+ 0. & 1.618 & 1.539 \\
+ -0.162 & 2.118 & 0.688 \\
+ -0.951 & 1.309 & 1.539 \\
+ -1.114 & 1.809 & 0.688 \\
+ 1.376 & 1.618 & 0.688 \\
+ 0.688 & 0.5 & 2.065 \\
+ 0.162 & 2.118 & -0.688 \\
+ -0.851 & 0. & 2.065 \\
+ -1.376 & 1.618 & -0.688 \\
+ -2.065 & 0.5 & 0.688 \\
+ 2.154 & -0.5 & 0.308 \\
+ 1.539 & 0.5 & -1.539 \\
+ -1.114 & -1.809 & 0.688 \\
+ -1.539 & -0.5 & -1.539 \\
+ 1.802 & -1.309 & -0.162 \\
+ 1.539 & -0.5 & -1.539 \\
+ -0.688 & -2.118 & -0.162 \\
+ -0.951 & -1.309 & -1.539 \\
+ 1.114 & -1.809 & -0.688 \\
+ 0.951 & -1.309 & -1.539 \\
+ 0.162 & -2.118 & -0.688 \\
+ 0. & -1.618 & -1.539 \\
+ 2.065 & -0.5 & -0.688 \\
+ 1.376 & -1.618 & 0.688 \\
+ -0.162 & -2.118 & 0.688 \\
+ -1.376 & -1.618 & -0.688 \\
+ 0.688 & -2.118 & 0.162 \\
+\end{array}
+\right)$. Determine the Centroid.
+Answer:
+$\{0.,0.,0.1\}$

pretraining/mathematica/geometry/solids/45592.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.962 & 0.884 & 0.2 \\
+ 0.376 & 0.516 & 0.255 \\
+ 0.166 & 0.83 & 0.661 \\
+ 0.548 & 0.523 & 0.161 \\
+ 0.751 & 0.64 & 0.543 \\
+ 0.742 & 0.811 & 0.254 \\
+\end{array}
+\right)$. 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: $0.33$ 
+Surface Area: $0.64$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/45739.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.827 & 0.969 & 0.822 \\
+ 0.514 & 0.694 & 0.935 \\
+ 0.807 & 0.191 & 0.547 \\
+ 0.227 & 0.12 & 0.52 \\
+ 0.1 & 0.952 & 0.52 \\
+ 0.809 & 0.013 & 0.832 \\
+ 0.677 & 0.958 & 0.464 \\
+\end{array}
+\right)$. 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: $1.72$ 
+Solid Angle: $1.31$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/47211.txt

@@ -0,0 +1,39 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1.466 & 0. & -0.733 \\
+ -1.376 & 0. & 0.263 \\
+ -1.186 & -0.862 & 0.733 \\
+ -1.186 & 0.862 & 0.733 \\
+ -1.114 & -0.809 & -0.263 \\
+ -1.114 & 0.809 & -0.263 \\
+ -0.851 & 0. & 1.114 \\
+ -0.688 & -0.5 & -1.114 \\
+ -0.688 & 0.5 & -1.114 \\
+ -0.453 & -1.394 & -0.733 \\
+ -0.453 & 1.394 & -0.733 \\
+ -0.425 & -1.309 & 0.263 \\
+ -0.425 & 1.309 & 0.263 \\
+ -0.263 & -0.809 & 1.114 \\
+ -0.263 & 0.809 & 1.114 \\
+ 0. & 0. & -1.639 \\
+ 0. & 0. & 1.639 \\
+ 0.263 & -0.809 & -1.114 \\
+ 0.263 & 0.809 & -1.114 \\
+ 0.425 & -1.309 & -0.263 \\
+ 0.425 & 1.309 & -0.263 \\
+ 0.453 & -1.394 & 0.733 \\
+ 0.453 & 1.394 & 0.733 \\
+ 0.688 & -0.5 & 1.114 \\
+ 0.688 & 0.5 & 1.114 \\
+ 0.851 & 0. & -1.114 \\
+ 1.114 & -0.809 & 0.263 \\
+ 1.114 & 0.809 & 0.263 \\
+ 1.186 & -0.862 & -0.733 \\
+ 1.186 & 0.862 & -0.733 \\
+ 1.376 & 0. & -0.263 \\
+ 1.466 & 0. & 0.733 \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$3.28$

pretraining/mathematica/geometry/solids/48684.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.984 & 0.246 & 0.357 \\
+ 0.283 & 0.826 & 0.897 \\
+ 0.291 & 0.71 & 0.264 \\
+ 0.299 & 0.093 & 0.257 \\
+ 0.155 & 0.381 & 0.171 \\
+ 0.096 & 0.295 & 0.766 \\
+ 0.554 & 0.396 & 0.995 \\
+ 0.83 & 0.668 & 0.129 \\
+ 0.724 & 0.622 & 0.643 \\
+ 0.347 & 0.912 & 0.774 \\
+\end{array}
+\right)$. 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: $2.01$ 
+Volume: $0.21$ 
+Solid Angle: $1.25$

pretraining/mathematica/geometry/solids/48984.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.738 & 0.755 & 0.775 \\
+ 0.269 & 0.186 & 0.158 \\
+ 0.967 & 0.226 & 0.404 \\
+ 0.117 & 0.139 & 0.711 \\
+ 0.274 & 0.364 & 0.415 \\
+ 0.914 & 0.496 & 0.493 \\
+ 0.804 & 0.254 & 0.938 \\
+ 0.668 & 0.523 & 0.248 \\
+ 0.768 & 0.096 & 0.189 \\
+ 0.192 & 0.311 & 0.846 \\
+\end{array}
+\right)$. 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: $1.32$ 
+Surface Area: $1.64$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/50079.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.197 & 0.352 & 0.557 \\
+ 0.025 & 0.066 & 0.98 \\
+ 0.514 & 0.893 & 0.329 \\
+ 0.653 & 0.741 & 0.367 \\
+\end{array}
+\right)$. 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: $1.$ 
+Volume: $0.01$ 
+Surface Area: $0.38$

pretraining/mathematica/geometry/solids/50805.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.381 & 0.548 & 0.607 \\
+ 0.528 & 0.297 & 0.597 \\
+ 0.064 & 0.294 & 0.47 \\
+ 0.714 & 0.9 & 0.912 \\
+\end{array}
+\right)$. 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: $3.72$ 
+Surface Area: $0.33$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/51157.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.102 & 0.045 & 0.238 \\
+ 0.255 & 0.255 & 0.012 \\
+ 0.823 & 0.304 & 0.478 \\
+ 0.523 & 0.166 & 0.767 \\
+\end{array}
+\right)$. 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: $0.19$ 
+Surface Area: $0.55$

pretraining/mathematica/geometry/solids/51983.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.872 & 0.53 & 0.585 \\
+ 0.896 & 0.531 & 0.917 \\
+ 0.269 & 0.469 & 0.308 \\
+ 0.927 & 0.454 & 0.495 \\
+ 0.038 & 0.348 & 0.415 \\
+ 0.089 & 0.104 & 0.476 \\
+ 0.101 & 0.227 & 0.914 \\
+\end{array}
+\right)$. 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: $1.12$ 
+Solid Angle: $3.79$

pretraining/mathematica/geometry/solids/54409.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.087 & 0.569 & 0.99 \\
+ 0.45 & 0.849 & 0.09 \\
+ 0.035 & 0.646 & 0.855 \\
+ 0.061 & 0.873 & 0.849 \\
+ 0.13 & 0.775 & 0.529 \\
+ 0.497 & 0.477 & 0.485 \\
+ 0.571 & 0.558 & 0.958 \\
+ 0.153 & 0.115 & 0.737 \\
+ 0.276 & 0.243 & 0.849 \\
+ 0.174 & 0.164 & 0.407 \\
+\end{array}
+\right)$. 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: $1.43$ 
+Solid Angle: $2.5$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/55446.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.675 & 0.035 & 0.211 \\
+ 0.745 & 0.433 & 0.089 \\
+ 0.304 & 0.61 & 0.78 \\
+ 0.371 & 0.229 & 0.121 \\
+ 0.18 & 0.18 & 0.556 \\
+\end{array}
+\right)$. 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: $0.77$ 
+Volume: $0.03$ 
+Solid Angle: $0.92$

pretraining/mathematica/geometry/solids/56413.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.319 & 0.343 & 0.235 \\
+ 0.932 & 0.92 & 0.488 \\
+ 0.98 & 0.604 & 0.462 \\
+ 0.59 & 0.797 & 0.853 \\
+ 0.555 & 0.323 & 0.345 \\
+ 0.295 & 0.611 & 0.119 \\
+\end{array}
+\right)$. 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: $0.9$ 
+Solid Angle: $0.88$

pretraining/mathematica/geometry/solids/56858.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.161 & 0.446 & 0.323 \\
+ 0.266 & 0.487 & 0.244 \\
+ 0.958 & 0.034 & 0.412 \\
+ 0.171 & 0.812 & 0.731 \\
+ 0.853 & 0.733 & 0.563 \\
+\end{array}
+\right)$. 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: $1.01$ 
+Solid Angle: $1.35$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/56888.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.292 & 0.327 & 0.668 \\
+ 0.48 & 0.677 & 0.435 \\
+ 0.358 & 0.921 & 0.62 \\
+ 0.277 & 0.37 & 0.929 \\
+ 0.311 & 0.849 & 0.402 \\
+ 0.001 & 0.265 & 0.943 \\
+ 0.88 & 0.945 & 0.354 \\
+ 0.75 & 0.647 & 0.671 \\
+ 0.795 & 0.696 & 0.867 \\
+\end{array}
+\right)$. 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: $2.49$ 
+Surface Area: $1.14$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/57669.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.609 & 0.684 & 0.853 \\
+ 0.373 & 0.883 & 0.658 \\
+ 0.084 & 0.816 & 0.051 \\
+ 0.207 & 0.78 & 0.891 \\
+ 0.762 & 0.143 & 0.768 \\
+\end{array}
+\right)$. 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: $1.69$ 
+Surface Area: $0.97$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/5926.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.215 & 0.427 & 0.278 \\
+ 0.67 & 0.631 & 0.233 \\
+ 0.764 & 0.749 & 0.512 \\
+ 0.785 & 0.066 & 0.506 \\
+ 0.957 & 0.622 & 0.318 \\
+\end{array}
+\right)$. 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: $0.26$ 
+Surface Area: $0.66$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/59631.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.355 & 0.016 & 0.61 \\
+ 0.388 & 0.371 & 0.792 \\
+ 0.899 & 0.055 & 0.663 \\
+ 0.043 & 0.546 & 0.199 \\
+ 0.996 & 0.545 & 0.474 \\
+ 0.613 & 0.377 & 0.072 \\
+ 0.808 & 0.123 & 0.744 \\
+ 0.899 & 0.796 & 0.398 \\
+ 0.029 & 0.269 & 0.194 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.71$ 
+Volume: $0.15$ 
+Solid Angle: $1.98$

pretraining/mathematica/geometry/solids/59972.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.875 & 0.101 & 0.075 \\
+ 0.025 & 0.323 & 0.42 \\
+ 0.889 & 0.402 & 0.386 \\
+ 0.14 & 0.338 & 0.336 \\
+ 0.76 & 0.717 & 0.051 \\
+ 0.481 & 0.237 & 0.459 \\
+ 0.25 & 0.716 & 0.819 \\
+ 0.905 & 0.499 & 0.923 \\
+ 0.062 & 0.49 & 0.726 \\
+\end{array}
+\right)$. 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: $0.15$ 
+Solid Angle: $0.86$ 
+Surface Area: $1.81$

pretraining/mathematica/geometry/solids/62056.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.351 & 0.689 & 0.942 \\
+ 0.027 & 0.474 & 0.839 \\
+ 0.719 & 0.751 & 0.397 \\
+ 0.313 & 0.565 & 0.572 \\
+ 0.669 & 0.235 & 0.057 \\
+ 0.156 & 0.157 & 0.92 \\
+ 0.325 & 0.787 & 0.933 \\
+\end{array}
+\right)$. 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: $3.26$ 
+Surface Area: $1.19$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/62957.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.507 & 0.478 & 0.121 \\
+ 0.361 & 0.751 & 0.211 \\
+ 0.961 & 0.523 & 0.311 \\
+ 0.815 & 0.287 & 0.056 \\
+ 0.195 & 0.439 & 0.754 \\
+ 0.217 & 0.375 & 0.998 \\
+ 0.847 & 0.882 & 0.624 \\
+ 0.084 & 0.785 & 0.228 \\
+ 0.653 & 0.82 & 0.677 \\
+ 0.907 & 0.075 & 0.431 \\
+\end{array}
+\right)$. 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: $1.86$ 
+Solid Angle: $3.24$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/63221.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.697 & 0.504 & 0.63 \\
+ 0.676 & 0.654 & 0.513 \\
+ 0.156 & 0.4 & 0.246 \\
+ 0.081 & 0.923 & 0.54 \\
+\end{array}
+\right)$. 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: $0.01$ 
+Surface Area: $0.48$

pretraining/mathematica/geometry/solids/63287.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.703 & 0.041 & 0.915 \\
+ 0.67 & 0.475 & 0.047 \\
+ 0.849 & 0.265 & 0.379 \\
+ 0.078 & 0.186 & 0.51 \\
+ 0.053 & 0.385 & 0.587 \\
+ 0.789 & 0.205 & 0.508 \\
+ 0.975 & 0.791 & 0.603 \\
+ 0.039 & 0.546 & 0.448 \\
+ 0.535 & 0.308 & 0.139 \\
+\end{array}
+\right)$. 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: $1.66$ 
+Volume: $0.14$ 
+Solid Angle: $0.76$