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pretraining/mathematica/geometry/solids/11356.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.275 & 0.231 & 0.996 \\
+ 0.979 & 0.296 & 0.843 \\
+ 0.145 & 0.576 & 0.475 \\
+ 0.354 & 0.118 & 0.551 \\
+ 0.801 & 0.901 & 0.984 \\
+ 0.355 & 0.886 & 0.438 \\
+ 0.584 & 0.718 & 0.11 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.82$ 
+Solid Angle: $1.35$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/11539.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.303 & 0.537 & 0.94 \\
+ 0.869 & 0.107 & 0.174 \\
+ 0.782 & 0.792 & 0.107 \\
+ 0.257 & 0.189 & 0.097 \\
+ 0.187 & 0.6 & 0.286 \\
+ 0.423 & 0.042 & 0.908 \\
+ 0.779 & 0.842 & 0.145 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.78$ 
+Volume: $0.14$ 
+Solid Angle: $0.82$

pretraining/mathematica/geometry/solids/12763.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.362 & 0.157 & 0.256 \\
+ 0.844 & 0.051 & 0.335 \\
+ 0.715 & 0.139 & 0.581 \\
+ 0.999 & 0.664 & 0.051 \\
+ 0.082 & 0.487 & 0.054 \\
+ 0.333 & 0.571 & 0.637 \\
+ 0.876 & 0.137 & 0.176 \\
+ 0.588 & 0.197 & 0.098 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.37$ 
+Volume: $0.1$ 
+Solid Angle: $3.03$

pretraining/mathematica/geometry/solids/13874.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.132 & 0.084 & 0.074 \\
+ 0.028 & 0.309 & 0.984 \\
+ 0.72 & 0.293 & 0.88 \\
+ 0.81 & 0.918 & 0.095 \\
+\end{array}
+\right)$. 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: $0.08$ 
+Solid Angle: $0.4$ 
+Surface Area: $1.63$

pretraining/mathematica/geometry/solids/14895.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.797 & 0.743 & 0.707 \\
+ 0.952 & 0.056 & 0.572 \\
+ 0.961 & 0.654 & 0.103 \\
+ 0.974 & 0.407 & 0.167 \\
+ 0.161 & 0.675 & 0.361 \\
+ 0.43 & 0.997 & 0.336 \\
+ 0.855 & 0.939 & 0.771 \\
+\end{array}
+\right)$. 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: $5.68$ 
+Volume: $0.11$ 
+Surface Area: $1.47$

pretraining/mathematica/geometry/solids/15893.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.829 & 0.267 & 0.804 \\
+ 0.085 & 0.583 & 0.554 \\
+ 0.328 & 0.199 & 0.776 \\
+ 0.26 & 0.221 & 0.79 \\
+ 0.707 & 0.914 & 0.444 \\
+ 0.892 & 0.389 & 0.889 \\
+ 0.258 & 0.382 & 0.878 \\
+ 0.12 & 0.034 & 0.35 \\
+\end{array}
+\right)$. 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: $1.33$ 
+Solid Angle: $1.88$

pretraining/mathematica/geometry/solids/16063.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.972 & 0.524 & 0.523 \\
+ 0.443 & 0.932 & 0.79 \\
+ 0.079 & 0.157 & 0.103 \\
+ 0.74 & 0.439 & 0.21 \\
+ 0.203 & 0.666 & 0.885 \\
+ 0.991 & 0.967 & 0.96 \\
+ 0.513 & 0.059 & 0.231 \\
+ 0.763 & 0.056 & 0.824 \\
+ 0.324 & 0.093 & 0.249 \\
+ 0.523 & 0.995 & 0.108 \\
+\end{array}
+\right)$. 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: $0.32$ 
+Surface Area: $2.7$ 
+Solid Angle: $3.52$

pretraining/mathematica/geometry/solids/17856.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.06 & 0.858 & 0.136 \\
+ 0.696 & 0.674 & 0.992 \\
+ 0.089 & 0.447 & 0.378 \\
+ 0.981 & 0.683 & 0.066 \\
+\end{array}
+\right)$. 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: $1.24$ 
+Solid Angle: $0.56$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/17994.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.325 & 0.364 & 0.31 \\
+ 0.292 & 0.689 & 0.185 \\
+ 0.757 & 0.24 & 0.069 \\
+ 0.431 & 0.066 & 0.004 \\
+ 0.275 & 0.492 & 0.86 \\
+\end{array}
+\right)$. 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: $0.76$ 
+Solid Angle: $5.71$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/18240.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.437 & 0.947 & 0.154 \\
+ 0.786 & 0.738 & 0.868 \\
+ 0.876 & 0.031 & 0.16 \\
+ 0.098 & 0.159 & 0.958 \\
+\end{array}
+\right)$. 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: $1.7$ 
+Solid Angle: $0.51$

pretraining/mathematica/geometry/solids/19053.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.057 & 0.225 & 0.144 \\
+ 0.348 & 0.372 & 0.648 \\
+ 0.39 & 0.22 & 0.769 \\
+ 0.514 & 0.187 & 0.999 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.$ 
+Surface Area: $0.17$ 
+Solid Angle: $0.$

pretraining/mathematica/geometry/solids/21648.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.674 & 0.972 & 0.582 \\
+ 0.872 & 0.13 & 0.737 \\
+ 0.125 & 0.342 & 0.515 \\
+ 0.476 & 0.258 & 0.005 \\
+ 0.227 & 0.856 & 0.294 \\
+ 0.764 & 0.846 & 0.612 \\
+ 0.559 & 0.926 & 0.332 \\
+ 0.364 & 0.247 & 0.855 \\
+ 0.266 & 0.588 & 0.919 \\
+ 0.369 & 0.117 & 0.638 \\
+\end{array}
+\right)$. 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: $1.94$ 
+Surface Area: $1.89$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/21832.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.751 & 0.241 & 0.435 \\
+ 0.007 & 0.312 & 0.024 \\
+ 0.57 & 0.905 & 0.254 \\
+ 0.653 & 0.827 & 0.666 \\
+ 0.814 & 0.645 & 0.964 \\
+ 0.098 & 0.42 & 0.468 \\
+ 0.541 & 0.349 & 0.847 \\
+ 0.332 & 0.13 & 0.73 \\
+ 0.628 & 0.643 & 0.114 \\
+ 0.561 & 0.718 & 0.98 \\
+\end{array}
+\right)$. 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: $1.81$ 
+Solid Angle: $2.25$

pretraining/mathematica/geometry/solids/24116.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.496 & 0.92 & 0.169 \\
+ 0.774 & 0.535 & 0.008 \\
+ 0.646 & 0.752 & 0.428 \\
+ 0.044 & 0.918 & 0.172 \\
+ 0.237 & 0.371 & 0.784 \\
+ 0.73 & 0.147 & 0.086 \\
+ 0.149 & 0.771 & 0.929 \\
+ 0.981 & 0.256 & 0.456 \\
+ 0.418 & 0.657 & 0.972 \\
+ 0.03 & 0.474 & 0.352 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.22$ 
+Solid Angle: $2.56$ 
+Surface Area: $2.17$

pretraining/mathematica/geometry/solids/27103.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.119 & 0.042 & 0.352 \\
+ 0.789 & 0.399 & 0.366 \\
+ 0.669 & 0.605 & 0.206 \\
+ 0.247 & 0.143 & 0.765 \\
+ 0.148 & 0.322 & 0.523 \\
+ 0.633 & 0.133 & 0.42 \\
+ 0.836 & 0.71 & 0.257 \\
+\end{array}
+\right)$. 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: $0.81$ 
+Volume: $0.04$ 
+Solid Angle: $0.9$

pretraining/mathematica/geometry/solids/27291.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & -\frac{1}{2} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & \frac{1}{2} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & -\frac{1}{2} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & \frac{1}{2} \\
+ \frac{1}{\sqrt{3}} & 0 & -\frac{1}{2} \\
+ \frac{1}{\sqrt{3}} & 0 & \frac{1}{2} \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$9$

pretraining/mathematica/geometry/solids/27349.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.54 & 0.637 & 0.619 \\
+ 0.104 & 0.351 & 0.736 \\
+ 0.472 & 0.227 & 0.995 \\
+ 0.195 & 0.42 & 0.333 \\
+ 0.826 & 0.237 & 0.328 \\
+ 0.186 & 0.949 & 0.813 \\
+ 0.807 & 0.053 & 0. \\
+\end{array}
+\right)$. 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:
+Solid Angle: $3.4$ 
+Surface Area: $1.45$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/27702.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.637 & 0.207 & 0.156 \\
+ 0.525 & 0.062 & 0.435 \\
+ 0.23 & 0.838 & 0.613 \\
+ 0.332 & 0.165 & 0.273 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.01$ 
+Surface Area: $0.41$ 
+Solid Angle: $0.5$

pretraining/mathematica/geometry/solids/27743.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.964 & 0.022 & 0.155 \\
+ 0.358 & 0.15 & 0.776 \\
+ 0.451 & 0.858 & 0.153 \\
+ 0.087 & 0.757 & 0.406 \\
+ 0.642 & 0.551 & 0.933 \\
+ 0.328 & 0.843 & 0.855 \\
+ 0.756 & 0.012 & 0.084 \\
+\end{array}
+\right)$. 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: $0.89$ 
+Surface Area: $1.86$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/28185.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.225 & 0.275 & 0.757 \\
+ 0.792 & 0.95 & 0.536 \\
+ 0.585 & 0.656 & 0.646 \\
+ 0.903 & 0.151 & 0.463 \\
+\end{array}
+\right)$. 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: $0.58$ 
+Solid Angle: $0.03$

pretraining/mathematica/geometry/solids/29358.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.464 & 0.869 & 0.145 \\
+ 0.423 & 0.192 & 0.762 \\
+ 0.362 & 0.97 & 0.197 \\
+ 0.141 & 0.513 & 0.246 \\
+ 0.034 & 0.916 & 0.62 \\
+ 0.621 & 0.735 & 0.765 \\
+ 0.305 & 0.857 & 0.059 \\
+ 0.853 & 0.656 & 0.491 \\
+ 0.921 & 0.215 & 0.765 \\
+ 0.34 & 0.734 & 0.745 \\
+ 0.153 & 0.409 & 0.719 \\
+\end{array}
+\right)$. 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: $3.13$ 
+Volume: $0.14$ 
+Surface Area: $1.69$

pretraining/mathematica/geometry/solids/29801.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.598 & 0.928 & 0.966 \\
+ 0.116 & 0.045 & 0.087 \\
+ 0.198 & 0.061 & 0.302 \\
+ 0.868 & 0.268 & 0.115 \\
+ 0.131 & 0.26 & 0.963 \\
+ 0.821 & 0.588 & 0.005 \\
+ 0.458 & 0.666 & 0.982 \\
+ 0.134 & 0.604 & 0.059 \\
+ 0.841 & 0.013 & 0.921 \\
+\end{array}
+\right)$. 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: $0.34$ 
+Surface Area: $2.83$ 
+Solid Angle: $1.02$

pretraining/mathematica/geometry/solids/29873.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 Circumradius.
+Answer:
+$2.23$

pretraining/mathematica/geometry/solids/30320.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.786 & 0.792 & 0.418 \\
+ 0.609 & 0.133 & 0.626 \\
+ 0.816 & 0.21 & 0.821 \\
+ 0.921 & 0.219 & 0.731 \\
+ 0.846 & 0.179 & 0.747 \\
+\end{array}
+\right)$. 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: $0.04$ 
+Surface Area: $0.28$

pretraining/mathematica/geometry/solids/3146.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.081 & 0.28 & 0.965 \\
+ 0.764 & 0.746 & 0.186 \\
+ 0.145 & 0.375 & 0.228 \\
+ 0.161 & 0.799 & 0.493 \\
+ 0.157 & 0.645 & 0.963 \\
+ 0.583 & 0.374 & 0.131 \\
+ 0.993 & 0.238 & 0.483 \\
+ 0.998 & 0.713 & 0.603 \\
+ 0.191 & 0.471 & 0.037 \\
+\end{array}
+\right)$. 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: $0.96$ 
+Surface Area: $2.05$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/34863.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.12 & 0.954 & 0.873 \\
+ 0.872 & 0.389 & 0.573 \\
+ 0.571 & 0.725 & 0.344 \\
+ 0.009 & 0.51 & 0.104 \\
+\end{array}
+\right)$. 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: $0.95$ 
+Solid Angle: $0.21$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/35227.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.625 & 0.932 & 0.84 \\
+ 0.778 & 0.911 & 0.242 \\
+ 0.091 & 0.126 & 0.303 \\
+ 0.273 & 0.735 & 0.227 \\
+ 0.528 & 0.345 & 0.623 \\
+\end{array}
+\right)$. 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: $1.06$ 
+Volume: $0.06$ 
+Solid Angle: $0.51$

pretraining/mathematica/geometry/solids/35977.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.492 & 0.063 & 0.875 \\
+ 0.128 & 0.641 & 0.378 \\
+ 0.057 & 0.995 & 0.244 \\
+ 0.125 & 0.807 & 0.679 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $0.02$ 
+Surface Area: $0.44$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/36250.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.239 & 0.426 & 0.003 \\
+ 0.095 & 0.61 & 0.842 \\
+ 0.833 & 0.602 & 0.975 \\
+ 0.723 & 0.74 & 0.388 \\
+ 0.404 & 0.179 & 0.255 \\
+ 0.434 & 0.266 & 0.763 \\
+ 0.897 & 0.212 & 0.847 \\
+ 0.815 & 0.04 & 0.888 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.64$ 
+Volume: $0.14$ 
+Surface Area: $1.71$

pretraining/mathematica/geometry/solids/38411.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.95 & 0.415 & 0.1 \\
+ 0.723 & 0.564 & 0.002 \\
+ 0.71 & 0.245 & 0.391 \\
+ 0.43 & 0.843 & 0.221 \\
+ 0.044 & 0.141 & 0.235 \\
+ 0.723 & 0.308 & 0.847 \\
+ 0.244 & 0.166 & 0.718 \\
+ 0.108 & 0.09 & 0.679 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.56$ 
+Volume: $0.12$ 
+Solid Angle: $1.01$

pretraining/mathematica/geometry/solids/3881.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.201 & 0.158 & 0.786 \\
+ 0.699 & 0.538 & 0.424 \\
+ 0.209 & 0.617 & 0.392 \\
+ 0.952 & 0.757 & 0.716 \\
+ 0.217 & 0.297 & 0.819 \\
+\end{array}
+\right)$. 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: $0.58$ 
+Volume: $0.03$ 
+Surface Area: $0.7$

pretraining/mathematica/geometry/solids/42195.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.103 & 0.924 & 0.787 \\
+ 0.286 & 0.737 & 0.183 \\
+ 0.896 & 0.229 & 0.525 \\
+ 0.889 & 0.481 & 0.978 \\
+ 0.306 & 0.146 & 0.307 \\
+ 0.18 & 0.049 & 0.556 \\
+ 0.686 & 0.819 & 0.907 \\
+\end{array}
+\right)$. 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: $0.93$ 
+Volume: $0.16$ 
+Surface Area: $1.84$

pretraining/mathematica/geometry/solids/42619.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.004 & 0.268 & 0.354 \\
+ 0.118 & 0.915 & 0.943 \\
+ 0.066 & 0.799 & 0.931 \\
+ 0.893 & 0.332 & 0.255 \\
+ 0.136 & 0.634 & 0.957 \\
+\end{array}
+\right)$. 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: $0.96$ 
+Solid Angle: $0.27$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/44264.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.275 & 0.934 & 0.977 \\
+ 0.113 & 0.738 & 0.683 \\
+ 0.043 & 0.39 & 0.369 \\
+ 0.579 & 0.284 & 0.001 \\
+\end{array}
+\right)$. 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: $0.05$ 
+Surface Area: $0.59$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/46488.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.662 & 0.048 & 0.303 \\
+ 0.89 & 0.442 & 0.84 \\
+ 0.38 & 0.587 & 0.871 \\
+ 0.064 & 0.018 & 0.622 \\
+ 0.462 & 0.388 & 0.806 \\
+ 0.207 & 0.837 & 0.926 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.37$ 
+Surface Area: $1.15$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/48299.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.307 & 0.007 & 0.102 \\
+ 0.101 & 0.972 & 0.177 \\
+ 0.37 & 0.544 & 0.081 \\
+ 0.446 & 0.127 & 0.815 \\
+ 0.901 & 0.41 & 0.742 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.38$ 
+Volume: $0.09$ 
+Solid Angle: $0.64$

pretraining/mathematica/geometry/solids/48735.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.193 & 0.562 & 0.357 \\
+ 0.44 & 0.122 & 0.607 \\
+ 0.066 & 0.817 & 0.489 \\
+ 0.929 & 0.485 & 0.078 \\
+ 0.605 & 0.143 & 0.557 \\
+ 0.475 & 0.988 & 0.159 \\
+ 0.148 & 0.059 & 0.475 \\
+ 0.583 & 0.265 & 0.771 \\
+ 0.82 & 0.556 & 0.631 \\
+\end{array}
+\right)$. 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: $1.66$ 
+Solid Angle: $5.1$

pretraining/mathematica/geometry/solids/49228.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.083 & 0.454 & 0.672 \\
+ 0.531 & 0.473 & 0.884 \\
+ 0.679 & 0.03 & 0.598 \\
+ 0.893 & 0.679 & 0.954 \\
+ 0.235 & 0.678 & 0.404 \\
+ 0.777 & 0.709 & 0.087 \\
+ 0.868 & 0.618 & 0.419 \\
+\end{array}
+\right)$. 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: $0.88$ 
+Surface Area: $1.41$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/49310.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.731 & 0.849 & 0.322 \\
+ 0.265 & 0.907 & 0.488 \\
+ 0.416 & 0.842 & 0.345 \\
+ 0.26 & 0.199 & 0.99 \\
+ 0.277 & 0.588 & 0.724 \\
+ 0.59 & 0.685 & 0.117 \\
+ 0.26 & 0.247 & 0.719 \\
+ 0.636 & 0.395 & 0.935 \\
+ 0.745 & 0.355 & 0.092 \\
+\end{array}
+\right)$. 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:
+Volume: $0.09$ 
+Solid Angle: $1.84$ 
+Surface Area: $1.34$

pretraining/mathematica/geometry/solids/54110.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.777 & 0.658 & 0.253 \\
+ 0.618 & 0.145 & 0.877 \\
+ 0.711 & 0.553 & 0.965 \\
+ 0.65 & 0.211 & 0.57 \\
+ 0.646 & 0.861 & 0.747 \\
+ 0.087 & 0.64 & 0.412 \\
+ 0.207 & 0.198 & 0.358 \\
+ 0.319 & 0.456 & 0.29 \\
+ 0.156 & 0.817 & 0.628 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Solid Angle: $1.22$ 
+Surface Area: $1.45$

pretraining/mathematica/geometry/solids/56293.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.244 & 0.748 & 0.131 \\
+ 0.891 & 0.251 & 0.143 \\
+ 0.114 & 0.82 & 0.93 \\
+ 0.991 & 0.346 & 0.508 \\
+\end{array}
+\right)$. 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: $1.06$ 
+Solid Angle: $0.16$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/5653.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.859 & 0.126 & 0.313 \\
+ 0.169 & 0.631 & 0.524 \\
+ 0.505 & 0.741 & 0.318 \\
+ 0.874 & 0.315 & 0.86 \\
+ 0.771 & 0.377 & 0.985 \\
+ 0.418 & 0.593 & 0.906 \\
+ 0.751 & 0.942 & 0.653 \\
+\end{array}
+\right)$. 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: $1.26$ 
+Volume: $0.09$ 
+Solid Angle: $0.54$

pretraining/mathematica/geometry/solids/56705.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.935 & 0.073 & 0.009 \\
+ 0.533 & 0.775 & 0.458 \\
+ 0.883 & 0.204 & 0.135 \\
+ 0.009 & 0.489 & 0.167 \\
+\end{array}
+\right)$. 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: $0.$ 
+Solid Angle: $0.06$

pretraining/mathematica/geometry/solids/59027.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.702 & 0.057 & 0.079 \\
+ 0.255 & 0.289 & 0.937 \\
+ 0.97 & 0.299 & 0.174 \\
+ 0.708 & 0.945 & 0.306 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.06$ 
+Solid Angle: $0.53$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/59226.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & -\frac{1}{2} & \sqrt{\text{Root}\left[722534400 \text{$\#$1}^{10}-4601020416 \text{$\#$1}^9+12682706944 \text{$\#$1}^8-19788308480 \text{$\#$1}^7+19238761984 \text{$\#$1}^6-12174590464 \text{$\#$1}^5+5171166400 \text{$\#$1}^4-1551068256 \text{$\#$1}^3+354890305 \text{$\#$1}^2-60378624 \text{$\#$1}+5308416\&,3\right]} \\
+ -\frac{1}{2} & 0 & -\sqrt{\text{Root}\left[2822400 \text{$\#$1}^{10}-10916736 \text{$\#$1}^9+17041168 \text{$\#$1}^8-13361088 \text{$\#$1}^7+5304000 \text{$\#$1}^6-1085888 \text{$\#$1}^5+415496 \text{$\#$1}^4-303984 \text{$\#$1}^3+105616 \text{$\#$1}^2-15032 \text{$\#$1}+529\&,3\right]} \\
+ -\frac{1}{2} & \frac{1}{2} & \sqrt{\text{Root}\left[722534400 \text{$\#$1}^{10}-4601020416 \text{$\#$1}^9+12682706944 \text{$\#$1}^8-19788308480 \text{$\#$1}^7+19238761984 \text{$\#$1}^6-12174590464 \text{$\#$1}^5+5171166400 \text{$\#$1}^4-1551068256 \text{$\#$1}^3+354890305 \text{$\#$1}^2-60378624 \text{$\#$1}+5308416\&,3\right]} \\
+ 0 & -\text{Root}\left[65536 \text{$\#$1}^{10}-270336 \text{$\#$1}^9+387328 \text{$\#$1}^8-207872 \text{$\#$1}^7-3328 \text{$\#$1}^6+68352 \text{$\#$1}^5-47008 \text{$\#$1}^4-10560 \text{$\#$1}^3+12976 \text{$\#$1}^2+4464 \text{$\#$1}+345\&,4\right] & \sqrt{\text{Root}\left[184968806400 \text{$\#$1}^{10}-298144628736 \text{$\#$1}^9+200126890240 \text{$\#$1}^8-78744037376 \text{$\#$1}^7+21900300032 \text{$\#$1}^6-4494628864 \text{$\#$1}^5+690379360 \text{$\#$1}^4-81591424 \text{$\#$1}^3+6973168 \text{$\#$1}^2-347328 \text{$\#$1}+6561\&,2\right]} \\
+ 0 & \text{Root}\left[65536 \text{$\#$1}^{10}-270336 \text{$\#$1}^9+387328 \text{$\#$1}^8-207872 \text{$\#$1}^7-3328 \text{$\#$1}^6+68352 \text{$\#$1}^5-47008 \text{$\#$1}^4-10560 \text{$\#$1}^3+12976 \text{$\#$1}^2+4464 \text{$\#$1}+345\&,4\right] & \sqrt{\text{Root}\left[184968806400 \text{$\#$1}^{10}-298144628736 \text{$\#$1}^9+200126890240 \text{$\#$1}^8-78744037376 \text{$\#$1}^7+21900300032 \text{$\#$1}^6-4494628864 \text{$\#$1}^5+690379360 \text{$\#$1}^4-81591424 \text{$\#$1}^3+6973168 \text{$\#$1}^2-347328 \text{$\#$1}+6561\&,2\right]} \\
+ 0 & -\text{Root}\left[196608 \text{$\#$1}^{10}-860160 \text{$\#$1}^9+1514240 \text{$\#$1}^8-1201152 \text{$\#$1}^7+70912 \text{$\#$1}^6+643840 \text{$\#$1}^5-426592 \text{$\#$1}^4-9920 \text{$\#$1}^3+99216 \text{$\#$1}^2-26384 \text{$\#$1}+67\&,3\right] & -\sqrt{\text{Root}\left[6502809600 \text{$\#$1}^{10}-21590613504 \text{$\#$1}^9+31258427913 \text{$\#$1}^8-26107499520 \text{$\#$1}^7+13891942264 \text{$\#$1}^6-4874307760 \text{$\#$1}^5+1108772352 \text{$\#$1}^4-150640320 \text{$\#$1}^3+10362496 \text{$\#$1}^2-309888 \text{$\#$1}+1600\&,2\right]} \\
+ 0 & \text{Root}\left[196608 \text{$\#$1}^{10}-860160 \text{$\#$1}^9+1514240 \text{$\#$1}^8-1201152 \text{$\#$1}^7+70912 \text{$\#$1}^6+643840 \text{$\#$1}^5-426592 \text{$\#$1}^4-9920 \text{$\#$1}^3+99216 \text{$\#$1}^2-26384 \text{$\#$1}+67\&,3\right] & -\sqrt{\text{Root}\left[6502809600 \text{$\#$1}^{10}-21590613504 \text{$\#$1}^9+31258427913 \text{$\#$1}^8-26107499520 \text{$\#$1}^7+13891942264 \text{$\#$1}^6-4874307760 \text{$\#$1}^5+1108772352 \text{$\#$1}^4-150640320 \text{$\#$1}^3+10362496 \text{$\#$1}^2-309888 \text{$\#$1}+1600\&,2\right]} \\
+ \frac{1}{2} & -\frac{1}{2} & \sqrt{\text{Root}\left[722534400 \text{$\#$1}^{10}-4601020416 \text{$\#$1}^9+12682706944 \text{$\#$1}^8-19788308480 \text{$\#$1}^7+19238761984 \text{$\#$1}^6-12174590464 \text{$\#$1}^5+5171166400 \text{$\#$1}^4-1551068256 \text{$\#$1}^3+354890305 \text{$\#$1}^2-60378624 \text{$\#$1}+5308416\&,3\right]} \\
+ \frac{1}{2} & 0 & -\sqrt{\text{Root}\left[2822400 \text{$\#$1}^{10}-10916736 \text{$\#$1}^9+17041168 \text{$\#$1}^8-13361088 \text{$\#$1}^7+5304000 \text{$\#$1}^6-1085888 \text{$\#$1}^5+415496 \text{$\#$1}^4-303984 \text{$\#$1}^3+105616 \text{$\#$1}^2-15032 \text{$\#$1}+529\&,3\right]} \\
+ \frac{1}{2} & \frac{1}{2} & \sqrt{\text{Root}\left[722534400 \text{$\#$1}^{10}-4601020416 \text{$\#$1}^9+12682706944 \text{$\#$1}^8-19788308480 \text{$\#$1}^7+19238761984 \text{$\#$1}^6-12174590464 \text{$\#$1}^5+5171166400 \text{$\#$1}^4-1551068256 \text{$\#$1}^3+354890305 \text{$\#$1}^2-60378624 \text{$\#$1}+5308416\&,3\right]} \\
+ -\text{Root}\left[6720 \text{$\#$1}^{10}-24768 \text{$\#$1}^9+29456 \text{$\#$1}^8-5632 \text{$\#$1}^7-13088 \text{$\#$1}^6+8192 \text{$\#$1}^5+376 \text{$\#$1}^4-1984 \text{$\#$1}^3+756 \text{$\#$1}^2+76 \text{$\#$1}-95\&,3\right] & -\frac{1}{2} & 0 \\
+ -\text{Root}\left[6720 \text{$\#$1}^{10}-24768 \text{$\#$1}^9+29456 \text{$\#$1}^8-5632 \text{$\#$1}^7-13088 \text{$\#$1}^6+8192 \text{$\#$1}^5+376 \text{$\#$1}^4-1984 \text{$\#$1}^3+756 \text{$\#$1}^2+76 \text{$\#$1}-95\&,3\right] & \frac{1}{2} & 0 \\
+ \text{Root}\left[6720 \text{$\#$1}^{10}-24768 \text{$\#$1}^9+29456 \text{$\#$1}^8-5632 \text{$\#$1}^7-13088 \text{$\#$1}^6+8192 \text{$\#$1}^5+376 \text{$\#$1}^4-1984 \text{$\#$1}^3+756 \text{$\#$1}^2+76 \text{$\#$1}-95\&,3\right] & -\frac{1}{2} & 0 \\
+ \text{Root}\left[6720 \text{$\#$1}^{10}-24768 \text{$\#$1}^9+29456 \text{$\#$1}^8-5632 \text{$\#$1}^7-13088 \text{$\#$1}^6+8192 \text{$\#$1}^5+376 \text{$\#$1}^4-1984 \text{$\#$1}^3+756 \text{$\#$1}^2+76 \text{$\#$1}-95\&,3\right] & \frac{1}{2} & 0 \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$21$

pretraining/mathematica/geometry/solids/59549.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.973 & 0.787 & 0.38 \\
+ 0.51 & 0.015 & 0.021 \\
+ 0.782 & 0.803 & 0.816 \\
+ 0.487 & 0.966 & 0.233 \\
+ 0.746 & 0.399 & 0.119 \\
+ 0.823 & 0.351 & 0.949 \\
+\end{array}
+\right)$. 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: $1.92$ 
+Surface Area: $1.44$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/6007.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.445 & 0.312 & 0.189 \\
+ 0.067 & 0.817 & 0.756 \\
+ 0.205 & 0.124 & 0.284 \\
+ 0.975 & 0.607 & 0.821 \\
+\end{array}
+\right)$. 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: $1.13$ 
+Surface Area: $0.97$

pretraining/mathematica/geometry/solids/60954.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.044 & 0.668 & 0.355 \\
+ 0.681 & 0.999 & 0.776 \\
+ 0.224 & 0.44 & 0.444 \\
+ 0.323 & 0.09 & 0.507 \\
+ 0.057 & 0.618 & 0.841 \\
+\end{array}
+\right)$. 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: $0.85$ 
+Solid Angle: $0.86$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/63162.txt

@@ -0,0 +1,82 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ \sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{2} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{3+\frac{4 \sqrt{5}}{3}} & \frac{1}{2} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & \frac{1}{4} \left(-5-3 \sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & \frac{1}{4} \left(5+3 \sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{7}{4}+\frac{2 \sqrt{5}}{3}} & -1-\frac{\sqrt{5}}{2} & \frac{5+\sqrt{5}}{4 \sqrt{3}} \\
+ \sqrt{\frac{7}{4}+\frac{2 \sqrt{5}}{3}} & \frac{1}{2} \left(2+\sqrt{5}\right) & \frac{5+\sqrt{5}}{4 \sqrt{3}} \\
+ -\frac{1}{\sqrt{3}} & 0 & -\frac{9+5 \sqrt{5}}{4 \sqrt{3}} \\
+ \frac{\sqrt{\frac{5}{3}}}{2} & -1-\frac{\sqrt{5}}{2} & -\sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ \frac{\sqrt{\frac{5}{3}}}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) & -\sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ 0 & \frac{1}{2} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ 0 & \frac{1}{2} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ -\frac{4+\sqrt{5}}{2 \sqrt{3}} & -1-\frac{\sqrt{5}}{2} & -\frac{5+\sqrt{5}}{4 \sqrt{3}} \\
+ -\frac{4+\sqrt{5}}{2 \sqrt{3}} & \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{5+\sqrt{5}}{4 \sqrt{3}} \\
+ 0 & \frac{1}{2} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ 0 & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ \frac{1}{\sqrt{3}} & 0 & \frac{9+5 \sqrt{5}}{4 \sqrt{3}} \\
+ -\frac{1}{4} \sqrt{3} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{4} \left(\sqrt{3}+\sqrt{15}\right) \\
+ -\frac{1}{4} \sqrt{3} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(\sqrt{3}+\sqrt{15}\right) \\
+ -\frac{\sqrt{\frac{5}{3}}}{2} & -1-\frac{\sqrt{5}}{2} & \sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ -\frac{\sqrt{\frac{5}{3}}}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) & \sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & 0 & \sqrt{\frac{83}{24}+\frac{11 \sqrt{5}}{8}} \\
+ -\frac{3+\sqrt{5}}{2 \sqrt{3}} & \frac{1}{2} \left(-1-\sqrt{5}\right) & \sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ -\frac{3+\sqrt{5}}{2 \sqrt{3}} & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ \frac{3+\sqrt{5}}{4 \sqrt{3}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\sqrt{\frac{83}{24}+\frac{11 \sqrt{5}}{8}} \\
+ \frac{3+\sqrt{5}}{4 \sqrt{3}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\sqrt{\frac{83}{24}+\frac{11 \sqrt{5}}{8}} \\
+ -\frac{3+\sqrt{5}}{4 \sqrt{3}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \sqrt{\frac{83}{24}+\frac{11 \sqrt{5}}{8}} \\
+ -\frac{3+\sqrt{5}}{4 \sqrt{3}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{83}{24}+\frac{11 \sqrt{5}}{8}} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & \frac{9+5 \sqrt{5}}{4 \sqrt{3}} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & \frac{9+5 \sqrt{5}}{4 \sqrt{3}} \\
+ \frac{1}{2 \sqrt{3}} & -\frac{1}{2} & -\sqrt{\frac{103}{24}+\frac{15 \sqrt{5}}{8}} \\
+ \frac{1}{2 \sqrt{3}} & \frac{1}{2} & -\sqrt{\frac{103}{24}+\frac{15 \sqrt{5}}{8}} \\
+ \sqrt{\frac{21}{8}+\frac{9 \sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{29}{8}+\frac{35 \sqrt{5}}{24}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{21}{8}+\frac{9 \sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{29}{8}+\frac{35 \sqrt{5}}{24}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{2} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{29}{8}+\frac{35 \sqrt{5}}{24}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{21}{8}+\frac{9 \sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{29}{8}+\frac{35 \sqrt{5}}{24}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{21}{8}+\frac{9 \sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{15}{4}+\frac{5 \sqrt{5}}{3}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ -\frac{3+2 \sqrt{5}}{2 \sqrt{3}} & -\frac{1}{2} & -\sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ \sqrt{\frac{15}{4}+\frac{5 \sqrt{5}}{3}} & \frac{1}{2} & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ -\frac{3+2 \sqrt{5}}{2 \sqrt{3}} & \frac{1}{2} & -\sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ \frac{1}{2} \sqrt{\frac{29}{3}+4 \sqrt{5}} & -\frac{1}{2} & \sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ -\frac{5+2 \sqrt{5}}{2 \sqrt{3}} & -\frac{1}{2} & \frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{29}{3}+4 \sqrt{5}} & \frac{1}{2} & \sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ -\frac{5+2 \sqrt{5}}{2 \sqrt{3}} & \frac{1}{2} & \frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & 0 & -\frac{11+3 \sqrt{5}}{4 \sqrt{3}} \\
+ \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{1}{2} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{\frac{47}{24}+\frac{7 \sqrt{5}}{8}} \\
+ -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{2} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{2} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{21}{8}+\frac{9 \sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{21}{8}+\frac{9 \sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ \frac{\sqrt{5}-5}{4 \sqrt{15}} & \frac{1}{4} \left(-5-3 \sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \frac{\sqrt{5}-5}{4 \sqrt{15}} & \frac{1}{4} \left(5+3 \sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{49}{10}+\frac{13 \sqrt{5}}{6}} & 0 & \frac{1}{2} \sqrt{\frac{29}{10}-\frac{7 \sqrt{5}}{6}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{30} \left(603+217 \sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{\sqrt{5}-7}{4 \sqrt{15}} \\
+ -\frac{13+4 \sqrt{5}}{2 \sqrt{15}} & \frac{1}{2} & -\frac{5+17 \sqrt{5}}{20 \sqrt{3}} \\
+ -\frac{13+4 \sqrt{5}}{2 \sqrt{15}} & -\frac{1}{2} & -\frac{5+17 \sqrt{5}}{20 \sqrt{3}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{30} \left(603+217 \sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{\sqrt{5}-7}{4 \sqrt{15}} \\
+ \sqrt{\frac{49}{40}+\frac{13 \sqrt{5}}{24}} & -\frac{5}{4}-\frac{13}{4 \sqrt{5}} & \frac{1}{2} \sqrt{\frac{29}{10}-\frac{7 \sqrt{5}}{6}} \\
+ \sqrt{\frac{7}{20}+\frac{2}{3 \sqrt{5}}} & -1-\frac{9}{2 \sqrt{5}} & \frac{\sqrt{5}-7}{4 \sqrt{15}} \\
+ \frac{5+13 \sqrt{5}}{20 \sqrt{3}} & -\frac{5}{4}-\frac{13}{4 \sqrt{5}} & -\frac{5+17 \sqrt{5}}{20 \sqrt{3}} \\
+ \sqrt{\frac{69}{40}+\frac{91}{24 \sqrt{5}}} & -\frac{3}{4}-\frac{13}{4 \sqrt{5}} & -\frac{5+17 \sqrt{5}}{20 \sqrt{3}} \\
+ \sqrt{\frac{109}{40}+\frac{23 \sqrt{5}}{24}} & -\frac{3}{4}-\frac{13}{4 \sqrt{5}} & \frac{\sqrt{5}-7}{4 \sqrt{15}} \\
+ \sqrt{\frac{49}{40}+\frac{13 \sqrt{5}}{24}} & \frac{5}{4}+\frac{13}{4 \sqrt{5}} & \frac{1}{2} \sqrt{\frac{29}{10}-\frac{7 \sqrt{5}}{6}} \\
+ \sqrt{\frac{109}{40}+\frac{23 \sqrt{5}}{24}} & \frac{3}{4}+\frac{13}{4 \sqrt{5}} & \frac{\sqrt{5}-7}{4 \sqrt{15}} \\
+ \sqrt{\frac{69}{40}+\frac{91}{24 \sqrt{5}}} & \frac{3}{4}+\frac{13}{4 \sqrt{5}} & -\frac{5+17 \sqrt{5}}{20 \sqrt{3}} \\
+ \frac{5+13 \sqrt{5}}{20 \sqrt{3}} & \frac{5}{4}+\frac{13}{4 \sqrt{5}} & -\frac{5+17 \sqrt{5}}{20 \sqrt{3}} \\
+ \sqrt{\frac{7}{20}+\frac{2}{3 \sqrt{5}}} & 1+\frac{9}{2 \sqrt{5}} & \frac{\sqrt{5}-7}{4 \sqrt{15}} \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$\frac{1}{4} \left(60+35 \sqrt{3}+90 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)$

pretraining/mathematica/geometry/solids/63590.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.002 & 0.703 & 0.87 \\
+ 0.013 & 0.018 & 0.917 \\
+ 0.299 & 0.371 & 0.843 \\
+ 0.527 & 0.984 & 0.686 \\
+ 0.005 & 0.987 & 0.183 \\
+ 0.289 & 0.991 & 0.964 \\
+ 0.553 & 0.762 & 0.033 \\
+ 0.148 & 0.37 & 0.221 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $3.28$ 
+Surface Area: $1.99$ 
+Volume: $0.18$