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pretraining/mathematica/geometry/solids/10125.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.448 & 0.919 & 0.388 \\
+ 0.412 & 0.324 & 0.252 \\
+ 0.719 & 0.403 & 0.541 \\
+ 0.203 & 0.17 & 0.167 \\
+ 0.165 & 0.558 & 0.1 \\
+ 0.564 & 0.864 & 0.061 \\
+ 0.46 & 0.88 & 0.034 \\
+\end{array}
+\right)$. 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: $1.17$ 
+Surface Area: $0.82$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/10593.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.127 & 0.643 & 0.759 \\
+ 0.848 & 0.922 & 0.691 \\
+ 0.008 & 0.173 & 0.411 \\
+ 0.566 & 0.192 & 0.622 \\
+ 0.61 & 0.722 & 0.475 \\
+ 0.97 & 0.25 & 0.335 \\
+ 0.399 & 0.778 & 0.96 \\
+\end{array}
+\right)$. 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: $1.33$ 
+Surface Area: $1.49$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/11092.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.24 & 0.259 & 0.042 \\
+ 0.978 & 0.075 & 0.048 \\
+ 0.704 & 0.761 & 0.213 \\
+ 0.651 & 0.468 & 0.626 \\
+ 0.003 & 0.12 & 0.243 \\
+ 0.93 & 0.822 & 0.817 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.53$ 
+Volume: $0.1$ 
+Solid Angle: $2.29$

pretraining/mathematica/geometry/solids/13309.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.139 & 0.112 & 0.194 \\
+ 0.939 & 0.167 & 0.458 \\
+ 0.853 & 0.174 & 0.501 \\
+ 0.555 & 0.127 & 0.083 \\
+ 0.1 & 0.423 & 0.401 \\
+ 0.631 & 0.73 & 0.141 \\
+ 0.938 & 0.951 & 0.635 \\
+ 0.933 & 0.843 & 0.991 \\
+ 0.945 & 0.821 & 0.777 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.14$ 
+Surface Area: $1.81$ 
+Solid Angle: $0.85$

pretraining/mathematica/geometry/solids/17042.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.572 & 0.256 & 0.452 \\
+ 0.944 & 0.281 & 0.929 \\
+ 0.7 & 0.735 & 0.514 \\
+ 0.494 & 0.971 & 0.714 \\
+ 0.183 & 0.756 & 0.865 \\
+ 0.135 & 0.194 & 0.586 \\
+ 0.296 & 0.737 & 0.221 \\
+\end{array}
+\right)$. 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: $1.47$ 
+Solid Angle: $2.72$

pretraining/mathematica/geometry/solids/17368.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.938 & 0.541 & 0.078 \\
+ 0.046 & 0.538 & 0.267 \\
+ 1. & 0.333 & 0.122 \\
+ 0.997 & 0.888 & 0.051 \\
+ 0.943 & 0.236 & 0.156 \\
+ 0.979 & 0.43 & 0.145 \\
+ 0.085 & 0.675 & 0.584 \\
+ 0.009 & 0.761 & 0.954 \\
+ 0.36 & 0.597 & 0.891 \\
+ 0.122 & 0.078 & 0.019 \\
+\end{array}
+\right)$. 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: $2.19$ 
+Solid Angle: $5.96$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/17976.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.776 & 0.858 & 0.23 \\
+ 0.189 & 0.959 & 0.321 \\
+ 0.345 & 0.85 & 0.202 \\
+ 0.937 & 0.599 & 0.842 \\
+ 0.748 & 0.914 & 0.379 \\
+ 0.807 & 0.329 & 0.488 \\
+ 0.79 & 0.477 & 0.099 \\
+ 0.494 & 0.867 & 0.573 \\
+ 0.149 & 0.505 & 0.506 \\
+\end{array}
+\right)$. 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: $1.32$ 
+Solid Angle: $2.68$

pretraining/mathematica/geometry/solids/19955.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.817 & 0.898 & 0.43 \\
+ 0.971 & 0.878 & 0.978 \\
+ 0.128 & 0.676 & 0.005 \\
+ 0.013 & 0.752 & 0.663 \\
+ 0.796 & 0.579 & 0.528 \\
+ 0.303 & 0.912 & 0.955 \\
+ 0.042 & 0.213 & 0.281 \\
+ 0.689 & 0.275 & 0.99 \\
+ 0.861 & 0.166 & 0.757 \\
+\end{array}
+\right)$. 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: $2.47$ 
+Solid Angle: $1.89$ 
+Volume: $0.26$

pretraining/mathematica/geometry/solids/20900.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.187 & 0.506 & 0.092 \\
+ 0.091 & 0.343 & 0.261 \\
+ 0.775 & 0.042 & 0.304 \\
+ 0.092 & 0.566 & 0.987 \\
+ 0.083 & 0.743 & 0.189 \\
+ 0.552 & 0.721 & 0.132 \\
+ 0.844 & 0.64 & 0.172 \\
+ 0.788 & 0.873 & 0.865 \\
+\end{array}
+\right)$. 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: $3.25$ 
+Volume: $0.2$ 
+Surface Area: $2.07$

pretraining/mathematica/geometry/solids/21371.txt

@@ -0,0 +1,37 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -\frac{1}{\sqrt{2}} & 0 \\
+ 0 & \frac{1}{\sqrt{2}} & 0 \\
+ \frac{1}{2} \sqrt{\frac{1}{2}-\frac{1}{\sqrt{5}}} & -\frac{1}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2}-\frac{1}{\sqrt{5}}} & \frac{1}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1-\frac{1}{\sqrt{5}}} & -\frac{1}{4} \sqrt{3+\sqrt{5}} & \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1-\frac{1}{\sqrt{5}}} & \frac{\sqrt{3+\sqrt{5}}}{4} & \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1+\frac{1}{\sqrt{5}}} & -\frac{1}{2 \sqrt{3+\sqrt{5}}} & \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ \frac{1}{4} \sqrt{1+\frac{1}{\sqrt{5}}} & \frac{1}{2 \sqrt{3+\sqrt{5}}} & \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & -\frac{1}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & \frac{1}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{1+\frac{1}{\sqrt{5}}} & -\frac{1}{2 \sqrt{3+\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{1+\frac{1}{\sqrt{5}}} & \frac{1}{2 \sqrt{3+\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} & 0 & \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ \frac{\sqrt{5-\sqrt{5}}}{4} & -\frac{1}{4} \sqrt{3+\sqrt{5}} & 0 \\
+ \frac{\sqrt{5-\sqrt{5}}}{4} & \frac{\sqrt{3+\sqrt{5}}}{4} & 0 \\
+ \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{5-\sqrt{5}} & -\frac{1}{4} \sqrt{3+\sqrt{5}} & 0 \\
+ -\frac{1}{4} \sqrt{5-\sqrt{5}} & \frac{\sqrt{3+\sqrt{5}}}{4} & 0 \\
+ -\frac{1}{4} \sqrt{5+\sqrt{5}} & -\frac{1}{2 \sqrt{3+\sqrt{5}}} & 0 \\
+ -\frac{1}{4} \sqrt{5+\sqrt{5}} & \frac{1}{2 \sqrt{3+\sqrt{5}}} & 0 \\
+ \frac{\sqrt{5+\sqrt{5}}}{4} & -\frac{1}{2 \sqrt{3+\sqrt{5}}} & 0 \\
+ \frac{\sqrt{5+\sqrt{5}}}{4} & \frac{1}{2 \sqrt{3+\sqrt{5}}} & 0 \\
+ -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} & 0 & \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & -\frac{1}{2 \sqrt{2}} & \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & \frac{1}{2 \sqrt{2}} & \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{5}}} & -\frac{1}{4} \sqrt{3+\sqrt{5}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{5}}} & \frac{\sqrt{3+\sqrt{5}}}{4} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-2 \sqrt{5}\right)} & -\frac{1}{2 \sqrt{2}} & \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-2 \sqrt{5}\right)} & \frac{1}{2 \sqrt{2}} & \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$10 \sqrt{3}$

pretraining/mathematica/geometry/solids/26046.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.85 & 0.012 & 0.906 \\
+ 0.835 & 0.46 & 0.414 \\
+ 0.944 & 0.897 & 0.632 \\
+ 0.077 & 0.754 & 0.269 \\
+ 0.756 & 0.899 & 0.311 \\
+ 0.253 & 0.752 & 0.749 \\
+ 0.404 & 0.82 & 0.162 \\
+ 0.57 & 0.48 & 0.104 \\
+ 0.569 & 0.156 & 0.825 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Surface Area: $1.8$ 
+Solid Angle: $0.55$

pretraining/mathematica/geometry/solids/26682.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.578 & 0.708 & 0.416 \\
+ 0.247 & 0.492 & 0.827 \\
+ 0.873 & 0.477 & 0.871 \\
+ 0.42 & 0.393 & 0.183 \\
+ 0.706 & 0.414 & 0.127 \\
+ 0.531 & 0.658 & 0.434 \\
+ 0.71 & 0.845 & 0.905 \\
+ 0.157 & 0.024 & 0.595 \\
+ 0.509 & 0.038 & 0.189 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Solid Angle: $2.79$ 
+Surface Area: $1.52$

pretraining/mathematica/geometry/solids/26826.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.696 & 0.407 & 0.392 \\
+ 0.711 & 0.467 & 0.532 \\
+ 0.428 & 0.74 & 0.556 \\
+ 0.008 & 0.255 & 0.828 \\
+ 0.565 & 0.399 & 0.855 \\
+\end{array}
+\right)$. 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: $0.71$ 
+Surface Area: $0.55$

pretraining/mathematica/geometry/solids/26989.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.041 & 0.352 & 0.584 \\
+ 0.012 & 0.226 & 0.694 \\
+ 0.417 & 0.923 & 0.667 \\
+ 0.003 & 0.931 & 0.937 \\
+ 0.22 & 0.849 & 0.598 \\
+ 0.793 & 0.933 & 0.924 \\
+ 0.445 & 0.217 & 0.424 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.25$ 
+Solid Angle: $3.48$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/27275.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.474 & 0.997 & 0.022 \\
+ 0.39 & 0.852 & 0.67 \\
+ 0.829 & 0.941 & 0.073 \\
+ 0.712 & 0.529 & 0.166 \\
+ 0.35 & 0.871 & 0.012 \\
+ 0.1 & 0.677 & 0.952 \\
+ 0.279 & 0.969 & 0.161 \\
+\end{array}
+\right)$. 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: $1.$ 
+Solid Angle: $2.16$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/29811.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.846 & 0.978 & 0.218 \\
+ 0.019 & 0.795 & 0.327 \\
+ 0.185 & 0.774 & 0.454 \\
+ 0.476 & 0.474 & 0.896 \\
+\end{array}
+\right)$. 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: $0.$ 
+Solid Angle: $0.02$ 
+Surface Area: $0.64$

pretraining/mathematica/geometry/solids/30574.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.059 & 0.767 & 0.416 \\
+ 0.938 & 0.275 & 0.157 \\
+ 0.702 & 0.25 & 0.009 \\
+ 0.683 & 0.951 & 0.252 \\
+ 0.586 & 0.091 & 0.364 \\
+ 0.244 & 0.742 & 0.252 \\
+ 0.916 & 0.461 & 0.776 \\
+ 0.981 & 0.902 & 0.975 \\
+ 0.559 & 0.406 & 0.883 \\
+\end{array}
+\right)$. 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: $2.09$ 
+Solid Angle: $1.13$ 
+Volume: $0.22$

pretraining/mathematica/geometry/solids/30662.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.077 & 0.861 & 0.022 \\
+ 0.701 & 0.635 & 0.077 \\
+ 0.417 & 0.329 & 0.192 \\
+ 0.842 & 0.788 & 0.713 \\
+ 0.052 & 0.178 & 0.803 \\
+ 0.134 & 0.248 & 0.136 \\
+ 0.286 & 0.902 & 0.36 \\
+\end{array}
+\right)$. 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: $1.03$ 
+Volume: $0.14$ 
+Surface Area: $1.72$

pretraining/mathematica/geometry/solids/31097.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.313 & 0.331 & 0.599 \\
+ 0.675 & 0.486 & 0.779 \\
+ 0.548 & 0.935 & 0.687 \\
+ 0.05 & 0.504 & 0.653 \\
+ 0.779 & 0.872 & 0.025 \\
+\end{array}
+\right)$. 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: $0.93$ 
+Solid Angle: $1.69$

pretraining/mathematica/geometry/solids/31519.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.349 & 0.854 & 0.415 \\
+ 0.373 & 0.089 & 0.22 \\
+ 0.734 & 0.256 & 0.768 \\
+ 0.838 & 0.767 & 0.335 \\
+ 0.787 & 0.121 & 0.488 \\
+ 0.857 & 0.197 & 0.413 \\
+ 0.005 & 0.253 & 0.773 \\
+ 0.574 & 0.148 & 0.864 \\
+ 0.885 & 0.859 & 0.739 \\
+\end{array}
+\right)$. 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: $1.63$ 
+Surface Area: $1.81$

pretraining/mathematica/geometry/solids/32057.txt

@@ -0,0 +1,27 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.325 & 0. & 0.425 \\
+ 0.325 & 0. & -0.425 \\
+ -0.526 & 0. & 0.1 \\
+ 0.526 & 0. & -0.1 \\
+ -0.1 & -0.309 & 0.425 \\
+ -0.1 & 0.309 & 0.425 \\
+ 0.1 & -0.309 & -0.425 \\
+ 0.1 & 0.309 & -0.425 \\
+ -0.162 & -0.5 & 0.1 \\
+ -0.162 & 0.5 & 0.1 \\
+ 0.162 & -0.5 & -0.1 \\
+ 0.162 & 0.5 & -0.1 \\
+ -0.425 & -0.309 & -0.1 \\
+ -0.425 & 0.309 & -0.1 \\
+ -0.263 & 0.191 & -0.425 \\
+ -0.263 & -0.191 & -0.425 \\
+ 0.263 & 0.191 & 0.425 \\
+ 0.263 & -0.191 & 0.425 \\
+ 0.425 & -0.309 & 0.1 \\
+ 0.425 & 0.309 & 0.1 \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$1.07$

pretraining/mathematica/geometry/solids/32358.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.669 & 0.95 & 0.304 \\
+ 0.259 & 0.042 & 0.726 \\
+ 0.287 & 0.081 & 0.345 \\
+ 0.623 & 0.412 & 0.287 \\
+ 0.366 & 0.424 & 0.245 \\
+ 0.402 & 0.529 & 0.433 \\
+\end{array}
+\right)$. 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: $0.6$ 
+Volume: $0.02$ 
+Solid Angle: $0.13$

pretraining/mathematica/geometry/solids/32518.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.984 & 0.706 & 0.303 \\
+ 0.323 & 0.591 & 0.295 \\
+ 0.659 & 0.03 & 0.284 \\
+ 0.182 & 0.406 & 0.438 \\
+ 0.227 & 0.223 & 0.588 \\
+ 0.159 & 0.874 & 0.969 \\
+ 0.047 & 0.745 & 0.714 \\
+ 0.776 & 0.834 & 0.608 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.62$ 
+Volume: $0.09$ 
+Surface Area: $1.55$

pretraining/mathematica/geometry/solids/32638.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.638 & 0.93 & 0.283 \\
+ 0.86 & 0.469 & 0.166 \\
+ 0.019 & 0.842 & 0.881 \\
+ 0.36 & 0.957 & 0.786 \\
+ 0.395 & 0.72 & 0.36 \\
+ 0.809 & 0.594 & 0.899 \\
+ 0.224 & 0.152 & 0.502 \\
+ 0.832 & 0.887 & 0.159 \\
+ 0.003 & 0.108 & 0.673 \\
+\end{array}
+\right)$. 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: $0.17$ 
+Surface Area: $1.97$ 
+Solid Angle: $2.99$

pretraining/mathematica/geometry/solids/3315.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.87 & 0.491 & 0.258 \\
+ 0.954 & 0.589 & 0.617 \\
+ 0.525 & 0.661 & 0.722 \\
+ 0.364 & 0.16 & 0.273 \\
+ 0.695 & 0.749 & 0.691 \\
+ 0.586 & 0.849 & 0.865 \\
+ 0.696 & 0.294 & 0.662 \\
+ 0.439 & 0.835 & 0.662 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.05$ 
+Solid Angle: $1.08$ 
+Surface Area: $0.9$

pretraining/mathematica/geometry/solids/33375.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.33 & 0.985 & 0.362 \\
+ 0.364 & 0.131 & 0.81 \\
+ 0.563 & 0.066 & 0.799 \\
+ 0.857 & 0.803 & 0.966 \\
+ 0.604 & 0.664 & 0.515 \\
+ 0.104 & 0.944 & 0.215 \\
+ 0.27 & 0.901 & 0.832 \\
+ 0.072 & 0.987 & 0.704 \\
+ 0.919 & 0.994 & 0.59 \\
+ 0.171 & 0.446 & 0.437 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Solid Angle: $4.99$ 
+Surface Area: $1.86$

pretraining/mathematica/geometry/solids/3452.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.958 & 0.324 & 0.504 \\
+ 0.182 & 0.923 & 0.016 \\
+ 0.499 & 0.364 & 0.937 \\
+ 0.35 & 0.808 & 0.415 \\
+ 0.58 & 0.882 & 0.097 \\
+\end{array}
+\right)$. 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: $0.03$ 
+Solid Angle: $0.18$ 
+Surface Area: $0.96$

pretraining/mathematica/geometry/solids/3814.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -0.5 & 0. \\
+ 0. & 0.5 & 0. \\
+ -0.853 & 0.5 & 0.522 \\
+ -0.5 & 0. & 1.313 \\
+ -0.853 & -0.5 & 0.522 \\
+ 0.853 & 0.5 & 0.522 \\
+ 0.5 & 0. & 1.313 \\
+ 0.853 & -0.5 & 0.522 \\
+ 0. & 0.789 & 0.957 \\
+ 0. & -0.789 & 0.957 \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$1.98$

pretraining/mathematica/geometry/solids/42845.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.909 & 0.502 & 0.564 \\
+ 0.904 & 0.351 & 0.212 \\
+ 0.21 & 0.317 & 0.76 \\
+ 0.417 & 0.311 & 0.101 \\
+ 0.662 & 0.909 & 0.845 \\
+ 0.821 & 0.15 & 0.475 \\
+ 0.21 & 0.221 & 0.292 \\
+\end{array}
+\right)$. 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: $1.3$ 
+Solid Angle: $2.58$

pretraining/mathematica/geometry/solids/43738.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\sqrt{1+\frac{2}{\sqrt{5}}} & 0 & 0 \\
+ -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & 0 \\
+ -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \sqrt{\frac{2}{5+\sqrt{5}}} & 0 \\
+ 0 & -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ 0 & -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ 0 & \sqrt{\frac{2}{5+\sqrt{5}}} & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ 0 & \sqrt{\frac{2}{5+\sqrt{5}}} & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{2}{5+\sqrt{5}}} & -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & 0 \\
+ \sqrt{\frac{2}{5+\sqrt{5}}} & \sqrt{\frac{2}{5+\sqrt{5}}} & 0 \\
+ \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{1+\frac{2}{\sqrt{5}}} & 0 & 0 \\
+\end{array}
+\right)$. Determine the Centroid.
+Answer:
+$\{0,0,0\}$

pretraining/mathematica/geometry/solids/44057.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.579 & 0.344 & 0.247 \\
+ 0.42 & 0.416 & 0.155 \\
+ 0.709 & 0.439 & 0.307 \\
+ 0.064 & 0.645 & 0.194 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.$ 
+Surface Area: $0.14$ 
+Solid Angle: $0.63$

pretraining/mathematica/geometry/solids/44068.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.071 & 0.947 & 0.227 \\
+ 0.272 & 0.96 & 0.577 \\
+ 0.689 & 0.451 & 0.297 \\
+ 0.481 & 0.199 & 0.267 \\
+ 0.298 & 0.57 & 0.727 \\
+ 0.605 & 0.031 & 0.492 \\
+ 0.754 & 0.989 & 0.166 \\
+ 0.309 & 0.992 & 0.542 \\
+ 0.919 & 0.635 & 0.372 \\
+ 0.166 & 0.303 & 0.797 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.73$ 
+Solid Angle: $1.18$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/48421.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.158 & 0.328 & 0.322 \\
+ 0.675 & 0.486 & 0.236 \\
+ 0.791 & 0.835 & 0.745 \\
+ 0.059 & 0.701 & 0.259 \\
+ 0.321 & 0.206 & 0.546 \\
+ 0.607 & 0.881 & 0.493 \\
+ 0.035 & 0.908 & 0.683 \\
+ 0.502 & 0.623 & 0.207 \\
+ 0.568 & 0.872 & 0.223 \\
+ 0.625 & 0.635 & 0.964 \\
+ 0.753 & 0.875 & 0.175 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.64$ 
+Volume: $0.15$ 
+Solid Angle: $2.06$

pretraining/mathematica/geometry/solids/48630.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.718 & 0.834 & 0.173 \\
+ 0.862 & 0.715 & 0.063 \\
+ 0.246 & 0.592 & 0.966 \\
+ 0.682 & 0.226 & 0.763 \\
+ 0.617 & 0.542 & 0.358 \\
+ 0.438 & 0.859 & 0.718 \\
+\end{array}
+\right)$. 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: $0.89$ 
+Volume: $0.03$ 
+Solid Angle: $0.78$

pretraining/mathematica/geometry/solids/49674.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.828 & 0.934 & 0.681 \\
+ 0.086 & 0.047 & 0.262 \\
+ 0.944 & 0.83 & 0.396 \\
+ 0.708 & 0.986 & 0.117 \\
+\end{array}
+\right)$. 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: $0.8$ 
+Volume: $0.02$ 
+Solid Angle: $0.48$

pretraining/mathematica/geometry/solids/49908.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.656 & 0.503 & 0.214 \\
+ 0.482 & 0.337 & 0.991 \\
+ 0.174 & 0.216 & 0.965 \\
+ 0.691 & 0.221 & 0.875 \\
+ 0.143 & 0.943 & 0.676 \\
+ 0.957 & 0.249 & 0.45 \\
+ 0.664 & 0.841 & 0.054 \\
+ 0.793 & 0.224 & 0.789 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.64$ 
+Volume: $0.08$ 
+Solid Angle: $3.5$

pretraining/mathematica/geometry/solids/50552.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & \frac{1}{3} \left(3+\sqrt{6}\right) \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & 0 \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & 1 \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & 0 \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & 1 \\
+ \frac{1}{\sqrt{3}} & 0 & 0 \\
+ \frac{1}{\sqrt{3}} & 0 & 1 \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$7$

pretraining/mathematica/geometry/solids/53004.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.731 & 0.804 & 0.599 \\
+ 0.557 & 0.088 & 0.383 \\
+ 0.064 & 0.083 & 0.47 \\
+ 0.314 & 0.577 & 0.55 \\
+ 0.237 & 0.885 & 0.21 \\
+\end{array}
+\right)$. 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: $0.94$ 
+Volume: $0.04$ 
+Solid Angle: $0.4$

pretraining/mathematica/geometry/solids/5357.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1 & 0 & 0 \\
+ -\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\
+ -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
+ -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\
+ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
+ 0 & -1 & 0 \\
+ 0 & 0 & -1 \\
+ 0 & 0 & 1 \\
+ 0 & 1 & 0 \\
+ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\
+ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
+ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\
+ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
+ 1 & 0 & 0 \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$14$

pretraining/mathematica/geometry/solids/54706.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.835 & 0. & 0.537 \\
+ 0.901 & 0.059 & 0.931 \\
+ 0.927 & 0.587 & 0.435 \\
+ 0.526 & 0.869 & 0.427 \\
+ 0.493 & 0.244 & 0.923 \\
+ 0.125 & 0.416 & 0.151 \\
+ 0.296 & 0.832 & 0.047 \\
+ 0.284 & 0.613 & 0.04 \\
+ 0.188 & 0.676 & 0.739 \\
+ 0.995 & 0.661 & 0.147 \\
+\end{array}
+\right)$. 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: $0.21$ 
+Solid Angle: $1.97$ 
+Surface Area: $2.16$

pretraining/mathematica/geometry/solids/55675.txt

@@ -0,0 +1,69 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & -\frac{5}{\sqrt{5+\sqrt{5}}} \\
+ 0 & 0 & \frac{5}{\sqrt{5+\sqrt{5}}} \\
+ 0 & -\frac{5+\sqrt{5}}{2 \sqrt{2}} & 0 \\
+ 0 & \frac{5+\sqrt{5}}{2 \sqrt{2}} & 0 \\
+ -\sqrt{5-\sqrt{5}} & 0 & \frac{\sqrt{5-\sqrt{5}}}{2} \\
+ \sqrt{5-\sqrt{5}} & 0 & -\frac{1}{2} \sqrt{5-\sqrt{5}} \\
+ -\sqrt{\frac{5}{2}-\sqrt{5}} & -\sqrt{\frac{5}{2}} & \frac{\sqrt{5-\sqrt{5}}}{2} \\
+ -\sqrt{\frac{5}{2}-\sqrt{5}} & \sqrt{\frac{5}{2}} & \frac{\sqrt{5-\sqrt{5}}}{2} \\
+ \sqrt{\frac{5}{2}-\sqrt{5}} & -\sqrt{\frac{5}{2}} & -\frac{1}{2} \sqrt{5-\sqrt{5}} \\
+ \sqrt{\frac{5}{2}-\sqrt{5}} & \sqrt{\frac{5}{2}} & -\frac{1}{2} \sqrt{5-\sqrt{5}} \\
+ -\sqrt{\frac{5}{2}+\sqrt{5}} & 0 & -\frac{1}{2} \sqrt{5+\sqrt{5}} \\
+ -\frac{1}{2} \sqrt{5+\sqrt{5}} & 0 & \sqrt{\frac{5}{2}+\sqrt{5}} \\
+ -\frac{1}{2} \sqrt{5+\sqrt{5}} & \frac{\sqrt{5}-5}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{5-\sqrt{5}} \\
+ -\frac{1}{2} \sqrt{5+\sqrt{5}} & -\frac{\sqrt{5}-5}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{5-\sqrt{5}} \\
+ \frac{\sqrt{5+\sqrt{5}}}{2} & 0 & -\sqrt{\frac{5}{2}+\sqrt{5}} \\
+ \frac{\sqrt{5+\sqrt{5}}}{2} & \frac{\sqrt{5}-5}{2 \sqrt{2}} & \frac{\sqrt{5-\sqrt{5}}}{2} \\
+ \frac{\sqrt{5+\sqrt{5}}}{2} & -\frac{\sqrt{5}-5}{2 \sqrt{2}} & \frac{\sqrt{5-\sqrt{5}}}{2} \\
+ \sqrt{\frac{5}{2}+\sqrt{5}} & 0 & \frac{\sqrt{5+\sqrt{5}}}{2} \\
+ -\frac{1}{2} \sqrt{\frac{25}{2}+5 \sqrt{5}} & -\frac{\sqrt{\frac{5}{2}}}{2} & 0 \\
+ -\frac{1}{2} \sqrt{\frac{25}{2}+5 \sqrt{5}} & \frac{\sqrt{\frac{5}{2}}}{2} & 0 \\
+ -\frac{1}{4} \sqrt{5 \left(5+\sqrt{5}\right)} & -\frac{5+3 \sqrt{5}}{4 \sqrt{2}} & 0 \\
+ -\frac{1}{4} \sqrt{5 \left(5+\sqrt{5}\right)} & \frac{5+3 \sqrt{5}}{4 \sqrt{2}} & 0 \\
+ \frac{1}{4} \sqrt{5 \left(5+\sqrt{5}\right)} & -\frac{5+3 \sqrt{5}}{4 \sqrt{2}} & 0 \\
+ \frac{1}{4} \sqrt{5 \left(5+\sqrt{5}\right)} & \frac{5+3 \sqrt{5}}{4 \sqrt{2}} & 0 \\
+ \sqrt{\frac{25}{8}+\frac{5 \sqrt{5}}{4}} & -\frac{\sqrt{\frac{5}{2}}}{2} & 0 \\
+ \sqrt{\frac{25}{8}+\frac{5 \sqrt{5}}{4}} & \frac{\sqrt{\frac{5}{2}}}{2} & 0 \\
+ -\sqrt{2+\frac{4}{\sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\sqrt{1+\frac{1}{\sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ \sqrt{1+\frac{1}{\sqrt{5}}} & 0 & \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ \sqrt{2+\frac{4}{\sqrt{5}}} & 0 & \frac{1}{\sqrt{5+\sqrt{5}}} \\
+ -\frac{1}{4} \sqrt{25+11 \sqrt{5}} & -\frac{5+\sqrt{5}}{4 \sqrt{2}} & \frac{\sqrt{5+\sqrt{5}}}{2} \\
+ -\frac{1}{4} \sqrt{25+11 \sqrt{5}} & \frac{5+\sqrt{5}}{4 \sqrt{2}} & \frac{\sqrt{5+\sqrt{5}}}{2} \\
+ -\frac{1}{4} \sqrt{5-\sqrt{5}} & -\frac{5+\sqrt{5}}{4 \sqrt{2}} & \sqrt{\frac{5}{2}+\sqrt{5}} \\
+ -\frac{1}{4} \sqrt{5-\sqrt{5}} & \frac{5+\sqrt{5}}{4 \sqrt{2}} & \sqrt{\frac{5}{2}+\sqrt{5}} \\
+ \frac{\sqrt{5-\sqrt{5}}}{4} & -\frac{5+\sqrt{5}}{4 \sqrt{2}} & -\sqrt{\frac{5}{2}+\sqrt{5}} \\
+ \frac{\sqrt{5-\sqrt{5}}}{4} & \frac{5+\sqrt{5}}{4 \sqrt{2}} & -\sqrt{\frac{5}{2}+\sqrt{5}} \\
+ \frac{1}{4} \sqrt{25+11 \sqrt{5}} & -\frac{5+\sqrt{5}}{4 \sqrt{2}} & -\frac{1}{2} \sqrt{5+\sqrt{5}} \\
+ \frac{1}{4} \sqrt{25+11 \sqrt{5}} & \frac{5+\sqrt{5}}{4 \sqrt{2}} & -\frac{1}{2} \sqrt{5+\sqrt{5}} \\
+ -\frac{1}{2} \sqrt{\frac{5}{2}+\sqrt{5}} & -\frac{\sqrt{\frac{5}{2}}}{2} & -\sqrt{\frac{5}{2}+\sqrt{5}} \\
+ -\frac{1}{2} \sqrt{\frac{5}{2}+\sqrt{5}} & \frac{\sqrt{\frac{5}{2}}}{2} & -\sqrt{\frac{5}{2}+\sqrt{5}} \\
+ -\frac{1}{4} \sqrt{5+\sqrt{5}} & -\frac{5+3 \sqrt{5}}{4 \sqrt{2}} & -\frac{1}{2} \sqrt{5+\sqrt{5}} \\
+ -\frac{1}{4} \sqrt{5+\sqrt{5}} & \frac{5+3 \sqrt{5}}{4 \sqrt{2}} & -\frac{1}{2} \sqrt{5+\sqrt{5}} \\
+ \frac{\sqrt{5+\sqrt{5}}}{4} & -\frac{5+3 \sqrt{5}}{4 \sqrt{2}} & \frac{\sqrt{5+\sqrt{5}}}{2} \\
+ \frac{\sqrt{5+\sqrt{5}}}{4} & \frac{5+3 \sqrt{5}}{4 \sqrt{2}} & \frac{\sqrt{5+\sqrt{5}}}{2} \\
+ \frac{1}{2} \sqrt{\frac{5}{2}+\sqrt{5}} & -\frac{\sqrt{\frac{5}{2}}}{2} & \sqrt{\frac{5}{2}+\sqrt{5}} \\
+ \frac{1}{2} \sqrt{\frac{5}{2}+\sqrt{5}} & \frac{\sqrt{\frac{5}{2}}}{2} & \sqrt{\frac{5}{2}+\sqrt{5}} \\
+ -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} & -\frac{1}{2} \sqrt{3+\sqrt{5}} & \frac{1}{\sqrt{5+\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} & \frac{\sqrt{3+\sqrt{5}}}{2} & \frac{1}{\sqrt{5+\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} & -\frac{1}{2} \sqrt{3+\sqrt{5}} & -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} & \frac{\sqrt{3+\sqrt{5}}}{2} & -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ \frac{1}{\sqrt{5+\sqrt{5}}} & -\frac{1}{2} \sqrt{3+\sqrt{5}} & \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ \frac{1}{\sqrt{5+\sqrt{5}}} & \frac{\sqrt{3+\sqrt{5}}}{2} & \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} & -\frac{1}{2} \sqrt{3+\sqrt{5}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} & \frac{\sqrt{3+\sqrt{5}}}{2} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & -\frac{1}{\sqrt{2}} & \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ -\sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & \frac{1}{\sqrt{2}} & \frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} & -\frac{3+\sqrt{5}}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{5}}} & \frac{3+\sqrt{5}}{2 \sqrt{2}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{5}}} \\
+ \frac{1}{\sqrt{5-\sqrt{5}}} & -\frac{3+\sqrt{5}}{2 \sqrt{2}} & \frac{1}{\sqrt{5+\sqrt{5}}} \\
+ \frac{1}{\sqrt{5-\sqrt{5}}} & \frac{3+\sqrt{5}}{2 \sqrt{2}} & \frac{1}{\sqrt{5+\sqrt{5}}} \\
+ \sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & -\frac{1}{\sqrt{2}} & -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{2}+\frac{1}{\sqrt{5}}} & \frac{1}{\sqrt{2}} & -\frac{1}{2} \sqrt{5+\frac{11}{\sqrt{5}}} \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$180$

pretraining/mathematica/geometry/solids/57010.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.219 & 0.234 & 0.988 \\
+ 0.935 & 0.355 & 0.469 \\
+ 0.466 & 0.368 & 0.384 \\
+ 0.298 & 0.215 & 0.907 \\
+ 0.445 & 0.518 & 0.613 \\
+ 0.688 & 0.077 & 0.141 \\
+ 0.595 & 0.993 & 0.968 \\
+\end{array}
+\right)$. 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: $0.5$ 
+Surface Area: $1.23$ 
+Volume: $0.06$

pretraining/mathematica/geometry/solids/57812.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.077 & 0.279 & 0.392 \\
+ 0.923 & 0.772 & 0.547 \\
+ 0.544 & 0.562 & 0.931 \\
+ 0.878 & 0.443 & 0.038 \\
+ 0.507 & 0.032 & 0.168 \\
+ 0.746 & 0.027 & 0.133 \\
+ 0.339 & 0.791 & 0.266 \\
+ 0.723 & 0.834 & 0.326 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.7$ 
+Solid Angle: $1.07$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/61837.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.311 & 0.17 & 0.559 \\
+ 0.836 & 0.544 & 0.275 \\
+ 0.28 & 0.412 & 0.135 \\
+ 0.174 & 0.107 & 0.631 \\
+ 0.944 & 0.321 & 0.581 \\
+ 0.616 & 0.723 & 0.457 \\
+ 0.125 & 0.904 & 0.818 \\
+ 0.686 & 0.829 & 0.042 \\
+ 0.015 & 0.739 & 0.33 \\
+ 0.4 & 0.454 & 0.713 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $5.85$ 
+Surface Area: $1.8$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/62226.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.642 & 0.558 & 0.026 \\
+ 0.001 & 0.918 & 0.194 \\
+ 0.627 & 0.109 & 0.413 \\
+ 0.905 & 0.401 & 0.073 \\
+ 0.008 & 0.93 & 0.318 \\
+ 0.464 & 0.581 & 0.913 \\
+ 0.097 & 0.243 & 0.897 \\
+ 0.786 & 0.316 & 0.089 \\
+ 0.263 & 0.103 & 0.263 \\
+ 0.595 & 0.321 & 0.97 \\
+\end{array}
+\right)$. 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: $2.12$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/6241.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.206 & 0.301 & 0.66 \\
+ 0.735 & 0.473 & 0.101 \\
+ 0.41 & 0.274 & 0.871 \\
+ 0.49 & 0.027 & 0.434 \\
+ 0.257 & 0.291 & 0.291 \\
+ 0.805 & 0.076 & 0.708 \\
+ 0.299 & 0.405 & 0.878 \\
+ 0.666 & 0.377 & 0.078 \\
+ 0.854 & 0.943 & 0.073 \\
+ 0.181 & 0.3 & 0.47 \\
+\end{array}
+\right)$. 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: $2.96$ 
+Volume: $0.11$ 
+Surface Area: $1.44$

pretraining/mathematica/geometry/solids/64770.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.678 & 0.58 & 0.458 \\
+ 0.125 & 0.756 & 0.733 \\
+ 0.706 & 0.133 & 0.282 \\
+ 0.033 & 0.378 & 0.942 \\
+ 0.584 & 0.991 & 0.593 \\
+ 0.283 & 0.013 & 0.483 \\
+ 0.112 & 0.408 & 0.112 \\
+ 0.556 & 0.8 & 0.271 \\
+\end{array}
+\right)$. 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: $3.47$ 
+Surface Area: $1.67$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/64809.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.403 & 0.349 & 0.078 \\
+ 0.518 & 0.485 & 0.548 \\
+ 0.466 & 0.115 & 0.611 \\
+ 0.097 & 0.86 & 0.308 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.39$ 
+Surface Area: $0.52$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/65893.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.107 & 0.937 & 0.256 \\
+ 0.068 & 0.113 & 0.586 \\
+ 0.291 & 0.954 & 0.172 \\
+ 0.462 & 0.768 & 0.881 \\
+ 0.549 & 0.284 & 0.164 \\
+ 0.449 & 0.098 & 0.411 \\
+ 0.347 & 0.991 & 0.933 \\
+ 0.994 & 0.339 & 0.17 \\
+ 0.579 & 0.83 & 0.838 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.18$ 
+Surface Area: $2.$ 
+Solid Angle: $1.75$

pretraining/mathematica/geometry/solids/66305.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.806 & 0.309 & 0.59 \\
+ 0.039 & 0.394 & 0.175 \\
+ 0.343 & 0.774 & 0.181 \\
+ 0.052 & 0.201 & 0.013 \\
+ 0.797 & 0.099 & 0.647 \\
+ 0.235 & 0.584 & 0.001 \\
+ 0.598 & 0.725 & 0.749 \\
+\end{array}
+\right)$. 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: $2.73$ 
+Surface Area: $1.22$ 
+Volume: $0.06$