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pretraining/mathematica/geometry/solids/10506.txt

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+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1. & 0. & 0. \\
+ -0.707 & -0.707 & 0. \\
+ -0.707 & -0.5 & 0. \\
+ -0.707 & 0. & -0.707 \\
+ -0.707 & 0. & -0.5 \\
+ -0.707 & 0. & 0.5 \\
+ -0.707 & 0. & 0.707 \\
+ -0.707 & 0.5 & 0. \\
+ -0.707 & 0.707 & 0. \\
+ -0.5 & -0.707 & 0. \\
+ -0.5 & 0. & -0.707 \\
+ -0.5 & 0. & 0.707 \\
+ -0.5 & 0.707 & 0. \\
+ 0. & -1. & 0. \\
+ 0. & -0.707 & -0.707 \\
+ 0. & -0.707 & -0.5 \\
+ 0. & -0.707 & 0.5 \\
+ 0. & -0.707 & 0.707 \\
+ 0. & -0.5 & -0.707 \\
+ 0. & -0.5 & 0.707 \\
+ 0. & 0. & -1. \\
+ 0. & 0. & 1. \\
+ 0. & 0.5 & -0.707 \\
+ 0. & 0.5 & 0.707 \\
+ 0. & 0.707 & -0.707 \\
+ 0. & 0.707 & -0.5 \\
+ 0. & 0.707 & 0.5 \\
+ 0. & 0.707 & 0.707 \\
+ 0. & 1. & 0. \\
+ 0.5 & -0.707 & 0. \\
+ 0.5 & 0. & -0.707 \\
+ 0.5 & 0. & 0.707 \\
+ 0.5 & 0.707 & 0. \\
+ 0.707 & -0.707 & 0. \\
+ 0.707 & -0.5 & 0. \\
+ 0.707 & 0. & -0.707 \\
+ 0.707 & 0. & -0.5 \\
+ 0.707 & 0. & 0.5 \\
+ 0.707 & 0. & 0.707 \\
+ 0.707 & 0.5 & 0. \\
+ 0.707 & 0.707 & 0. \\
+ 1. & 0. & 0. \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$38.78$

pretraining/mathematica/geometry/solids/10600.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.487 & 0.997 & 0.718 \\
+ 0.632 & 0.109 & 0.751 \\
+ 0.471 & 0.758 & 0.281 \\
+ 0.044 & 0.403 & 0.418 \\
+ 0.67 & 0.419 & 0.01 \\
+ 0.13 & 0.742 & 0.593 \\
+ 0.996 & 0.172 & 0.166 \\
+ 0.033 & 0.209 & 0.436 \\
+ 0.648 & 0.525 & 0.968 \\
+ 0.041 & 0.558 & 0.352 \\
+\end{array}
+\right)$. 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: $0.2$ 
+Surface Area: $2.03$ 
+Solid Angle: $1.23$

pretraining/mathematica/geometry/solids/11332.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.188 & 0.36 & 0.691 \\
+ 0.499 & 0.322 & 0.83 \\
+ 0.808 & 0.5 & 0.369 \\
+ 0.095 & 0.261 & 0.098 \\
+ 0.586 & 0.856 & 0.607 \\
+\end{array}
+\right)$. 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: $1.51$

pretraining/mathematica/geometry/solids/11801.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.038 & 0.375 & 0.58 \\
+ 0.678 & 0.318 & 0.157 \\
+ 0.119 & 0.46 & 0.395 \\
+ 0.553 & 0.492 & 0.82 \\
+\end{array}
+\right)$. 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: $0.49$

pretraining/mathematica/geometry/solids/13999.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.834 & 0.218 & 0.132 \\
+ 0.589 & 0.263 & 0.794 \\
+ 0.899 & 0.779 & 0.982 \\
+ 0.723 & 0.209 & 0.725 \\
+ 0.525 & 0.276 & 0.963 \\
+ 0.424 & 0.938 & 0.905 \\
+ 0.867 & 0.879 & 0.415 \\
+\end{array}
+\right)$. 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: $1.35$ 
+Solid Angle: $0.34$

pretraining/mathematica/geometry/solids/14407.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.504 & 0.552 & 0.596 \\
+ 0.084 & 0.34 & 0.409 \\
+ 0.619 & 0.937 & 0.905 \\
+ 0.254 & 0.557 & 0.297 \\
+\end{array}
+\right)$. 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: $1.4$ 
+Surface Area: $0.34$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/14737.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.931 & 0.811 & 0.3 \\
+ 0.85 & 0.989 & 0.939 \\
+ 0.856 & 0.189 & 0.842 \\
+ 0.513 & 0.034 & 0.245 \\
+ 0.913 & 0.521 & 0.405 \\
+ 0.301 & 0.91 & 0.685 \\
+\end{array}
+\right)$. 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: $1.28$ 
+Surface Area: $1.61$

pretraining/mathematica/geometry/solids/16999.txt

@@ -0,0 +1,52 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.5 & 0.289 & -2.157 \\
+ -0.5 & -1.223 & 1.8 \\
+ -0.5 & 1.223 & -1.8 \\
+ -0.5 & -0.289 & 2.157 \\
+ -0.5 & -1.8 & -1.223 \\
+ -0.5 & 2.157 & 0.289 \\
+ -0.5 & 1.8 & 1.223 \\
+ 0. & -0.577 & -2.157 \\
+ 0. & 1.868 & -1.223 \\
+ 0. & 0.577 & 2.157 \\
+ 0.5 & 0.289 & -2.157 \\
+ 0.5 & -1.223 & 1.8 \\
+ 0.5 & 1.223 & -1.8 \\
+ 0.5 & -0.289 & 2.157 \\
+ 0.5 & -1.8 & -1.223 \\
+ 0.5 & 2.157 & 0.289 \\
+ 0.5 & 1.8 & 1.223 \\
+ -1.809 & 0.467 & -1.223 \\
+ -1.809 & -0.467 & 1.223 \\
+ -0.809 & -1.044 & -1.8 \\
+ -0.809 & 1.979 & -0.645 \\
+ -0.809 & 1.044 & 1.8 \\
+ -1.618 & -0.934 & -1.223 \\
+ -1.618 & -1.512 & 0.289 \\
+ -2.118 & -0.289 & -0.645 \\
+ -2.118 & -0.645 & 0.289 \\
+ -1.309 & -1.69 & -0.645 \\
+ -1.309 & -0.178 & -1.8 \\
+ -1.309 & -1.333 & 1.223 \\
+ -1.309 & 1.333 & -1.223 \\
+ -1.309 & 0.178 & 1.8 \\
+ 0.809 & -1.044 & -1.8 \\
+ 0.809 & 1.979 & -0.645 \\
+ 0.809 & 1.044 & 1.8 \\
+ 1.618 & -0.934 & -1.223 \\
+ 1.618 & -1.512 & 0.289 \\
+ 2.118 & -0.289 & -0.645 \\
+ 2.118 & -0.645 & 0.289 \\
+ 1.309 & -1.69 & -0.645 \\
+ 1.309 & -0.178 & -1.8 \\
+ 1.309 & -1.333 & 1.223 \\
+ 1.309 & 1.333 & -1.223 \\
+ 1.309 & 0.178 & 1.8 \\
+ 1.809 & 0.467 & -1.223 \\
+ 1.809 & -0.467 & 1.223 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$75.$

pretraining/mathematica/geometry/solids/17262.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1. & 0. & -0.5 \\
+ -1. & 0. & 0.5 \\
+ -0.5 & -0.866 & -0.5 \\
+ -0.5 & -0.866 & 0.5 \\
+ -0.5 & 0.866 & -0.5 \\
+ -0.5 & 0.866 & 0.5 \\
+ 0.5 & -0.866 & -0.5 \\
+ 0.5 & -0.866 & 0.5 \\
+ 0.5 & 0.866 & -0.5 \\
+ 0.5 & 0.866 & 0.5 \\
+ 1. & 0. & -0.5 \\
+ 1. & 0. & 0.5 \\
+ 1.362 & 0.787 & 0. \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$11.93$

pretraining/mathematica/geometry/solids/17450.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.884 & 0.577 & 0.074 \\
+ 0.066 & 0.521 & 0.609 \\
+ 0.167 & 0.099 & 0.324 \\
+ 0.846 & 0.398 & 0.037 \\
+\end{array}
+\right)$. 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:
+Surface Area: $0.59$ 
+Volume: $0.$ 
+Solid Angle: $0.05$

pretraining/mathematica/geometry/solids/18810.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.419 & 0.305 & 0.093 \\
+ 0.864 & 0.026 & 0.945 \\
+ 0.448 & 0.521 & 0.088 \\
+ 0.267 & 0.694 & 0.484 \\
+ 0.303 & 0.098 & 0.991 \\
+ 0.803 & 0.707 & 0.088 \\
+ 0.674 & 0.036 & 0.421 \\
+ 0.76 & 0.423 & 0.927 \\
+ 0.272 & 0.039 & 0.97 \\
+ 0.289 & 0.95 & 0.891 \\
+\end{array}
+\right)$. 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: $2.22$ 
+Volume: $0.23$ 
+Solid Angle: $2.07$

pretraining/mathematica/geometry/solids/20678.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.435 & 0.835 & 0.088 \\
+ 0.284 & 0.19 & 0.106 \\
+ 0.139 & 0.845 & 0.066 \\
+ 0.489 & 0.856 & 0.494 \\
+ 0.708 & 0.365 & 0.275 \\
+ 0.308 & 0.103 & 0.52 \\
+ 0.298 & 0.888 & 0.173 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.06$ 
+Solid Angle: $2.43$ 
+Surface Area: $1.01$

pretraining/mathematica/geometry/solids/20902.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.929 & 0.982 & 0.491 \\
+ 0.026 & 0.478 & 0.919 \\
+ 0.849 & 0.94 & 0.456 \\
+ 0.459 & 0.351 & 0.411 \\
+ 0.136 & 0.56 & 0.439 \\
+ 0.642 & 0.322 & 0.243 \\
+ 0.942 & 0.442 & 0.84 \\
+ 0.041 & 0.64 & 0.966 \\
+ 0.955 & 0.709 & 0.91 \\
+\end{array}
+\right)$. 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: $1.65$ 
+Solid Angle: $1.09$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/23641.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.833 & 0.692 & 0.572 \\
+ 0.363 & 0.469 & 0.165 \\
+ 0.217 & 0.228 & 0.061 \\
+ 0.237 & 0.673 & 0.381 \\
+ 0.205 & 0.723 & 0.717 \\
+ 0.84 & 0.228 & 0.366 \\
+ 0.041 & 0.099 & 0.797 \\
+ 0.981 & 0.661 & 0.713 \\
+\end{array}
+\right)$. 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: $3.35$ 
+Surface Area: $1.65$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/23942.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.366 & 0.256 & 0.906 \\
+ 0.822 & 0.895 & 0.148 \\
+ 0.072 & 0.902 & 0.433 \\
+ 0.703 & 0.235 & 0.144 \\
+ 0.403 & 0.061 & 0.223 \\
+ 0.235 & 0.403 & 0.019 \\
+\end{array}
+\right)$. 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: $1.65$ 
+Solid Angle: $0.6$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/26034.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.024 & 0.799 & 0.562 \\
+ 0.152 & 0.184 & 0.691 \\
+ 0.829 & 0.3 & 0.347 \\
+ 0.027 & 0.076 & 0.273 \\
+ 0.222 & 0.312 & 0.888 \\
+ 0.709 & 0.965 & 0.83 \\
+ 0.617 & 0.102 & 0.356 \\
+ 0.563 & 0.708 & 0.581 \\
+\end{array}
+\right)$. 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: $1.06$ 
+Volume: $0.12$ 
+Surface Area: $1.69$

pretraining/mathematica/geometry/solids/26209.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 FaceCount.
+Answer:
+$120$

pretraining/mathematica/geometry/solids/27271.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.846 & 0.464 & 0.947 \\
+ 0.202 & 0.557 & 0.794 \\
+ 0.235 & 0.373 & 0.609 \\
+ 0.827 & 0.847 & 0.775 \\
+ 0.617 & 0.637 & 0.08 \\
+ 0.334 & 0.481 & 0.214 \\
+ 0.328 & 0.115 & 0.279 \\
+ 0.37 & 0.623 & 0.438 \\
+\end{array}
+\right)$. 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: $0.94$ 
+Volume: $0.09$ 
+Surface Area: $1.32$

pretraining/mathematica/geometry/solids/28998.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.963 & 0.241 & 0.791 \\
+ 0.782 & 0.849 & 0.561 \\
+ 0.939 & 0.257 & 0.37 \\
+ 0.449 & 0.252 & 0.049 \\
+ 0.967 & 0.479 & 0.439 \\
+ 0.217 & 0.546 & 0.019 \\
+ 0.645 & 0.877 & 0.125 \\
+ 0.465 & 0.647 & 0.338 \\
+ 0.803 & 0.983 & 0.193 \\
+\end{array}
+\right)$. 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: $0.09$ 
+Surface Area: $1.34$

pretraining/mathematica/geometry/solids/31345.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.707 & 0.214 & 0.023 \\
+ 0.819 & 0.44 & 0.486 \\
+ 0.695 & 0.768 & 0.422 \\
+ 0.818 & 0.139 & 0.463 \\
+ 0.149 & 0.988 & 0.362 \\
+ 0.358 & 0.668 & 0.128 \\
+ 0.184 & 0.377 & 0.303 \\
+ 0.171 & 0.651 & 0.888 \\
+ 0.839 & 0.184 & 0.434 \\
+\end{array}
+\right)$. 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: $1.54$ 
+Volume: $0.13$ 
+Solid Angle: $1.04$

pretraining/mathematica/geometry/solids/32281.txt

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

pretraining/mathematica/geometry/solids/36.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.737 & 0.604 & 0.183 \\
+ 0.966 & 0.85 & 0.939 \\
+ 0.894 & 0.537 & 0.087 \\
+ 0.609 & 0.21 & 0.131 \\
+ 0.803 & 0.989 & 0.531 \\
+ 0.829 & 0.095 & 0.562 \\
+ 0.947 & 0.188 & 0.266 \\
+ 0.331 & 0.274 & 0.661 \\
+ 0.494 & 0.563 & 0.685 \\
+\end{array}
+\right)$. 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: $4.21$ 
+Surface Area: $1.54$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/3690.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.757 & 0.655 & 0.934 \\
+ 0.259 & 0.795 & 0.778 \\
+ 0.658 & 0.053 & 0.856 \\
+ 0.173 & 0.055 & 0.562 \\
+ 0.728 & 0.896 & 0.17 \\
+ 0.25 & 0.942 & 0.16 \\
+\end{array}
+\right)$. 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: $1.75$ 
+Volume: $0.13$ 
+Solid Angle: $1.33$

pretraining/mathematica/geometry/solids/37631.txt

@@ -0,0 +1,27 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -1.618 & 0. \\
+ 0. & 1.618 & 0. \\
+ 0.851 & 0. & -0.526 \\
+ 0.851 & 0. & 0.526 \\
+ 0.263 & -0.809 & -0.526 \\
+ 0.263 & -0.809 & 0.526 \\
+ 0.263 & 0.809 & -0.526 \\
+ 0.263 & 0.809 & 0.526 \\
+ -0.951 & -1.309 & 0. \\
+ -0.951 & 1.309 & 0. \\
+ 0.951 & -1.309 & 0. \\
+ 0.951 & 1.309 & 0. \\
+ -0.688 & -0.5 & -0.526 \\
+ -0.688 & -0.5 & 0.526 \\
+ -0.688 & 0.5 & -0.526 \\
+ -0.688 & 0.5 & 0.526 \\
+ -1.539 & -0.5 & 0. \\
+ -1.539 & 0.5 & 0. \\
+ 1.539 & -0.5 & 0. \\
+ 1.539 & 0.5 & 0. \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$4.65$

pretraining/mathematica/geometry/solids/3793.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.496 & 0.737 & 0.309 \\
+ 0.284 & 0.167 & 0.477 \\
+ 0.633 & 0.678 & 0.901 \\
+ 0.715 & 0.947 & 0.534 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $0.55$ 
+Volume: $0.$ 
+Solid Angle: $0.2$

pretraining/mathematica/geometry/solids/38435.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.288 & 0.977 & 0.916 \\
+ 0.869 & 0.047 & 0.938 \\
+ 0.167 & 0.452 & 0.073 \\
+ 0.851 & 0.737 & 0.845 \\
+ 0.584 & 0.325 & 0.236 \\
+ 0.934 & 0.174 & 0.837 \\
+ 0.591 & 0.376 & 0.207 \\
+ 0.916 & 0.528 & 0.909 \\
+ 0.442 & 0.72 & 0.049 \\
+\end{array}
+\right)$. 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: $0.72$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/38653.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ \frac{1}{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & 0 & -\frac{1}{2} \\
+ \frac{1}{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & 0 & \frac{1}{2} \\
+ \frac{-1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} & -\frac{1}{2} \\
+ \frac{-1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} & \frac{1}{2} \\
+ \frac{-1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} & -\frac{1}{2} \\
+ \frac{-1-\sqrt{5}}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} & \frac{1}{2} \\
+ \frac{\sqrt{5}-1}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \\
+ \frac{\sqrt{5}-1}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \\
+ \frac{\sqrt{5}-1}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & -\frac{1}{2} \\
+ \frac{\sqrt{5}-1}{8 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \sqrt{\frac{\frac{5}{8}+\frac{\sqrt{5}}{8}}{\frac{5}{8}-\frac{\sqrt{5}}{8}}} & \frac{1}{2} \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$7$

pretraining/mathematica/geometry/solids/41348.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.582 & 0.928 & 0.566 \\
+ 0.309 & 0.599 & 0.939 \\
+ 0.655 & 0.06 & 0.203 \\
+ 0.832 & 0.836 & 0.235 \\
+ 0.201 & 0.961 & 0.018 \\
+ 0.919 & 0.619 & 0.687 \\
+ 0.799 & 0.833 & 0.773 \\
+ 0.794 & 0.217 & 0.016 \\
+ 0.405 & 0.141 & 0.514 \\
+ 0.928 & 0.204 & 0.708 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $2.27$ 
+Solid Angle: $4.05$ 
+Volume: $0.26$

pretraining/mathematica/geometry/solids/41936.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1.225 & -1.225 & 0.408 \\
+ -1.225 & 1.225 & 0. \\
+ -0.408 & -0.816 & 0.816 \\
+ 0. & 0. & 6.124 \\
+ 0.408 & 0.816 & 0.816 \\
+ 1.225 & -1.225 & 0. \\
+ 1.225 & 1.225 & 0.408 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$21.$

pretraining/mathematica/geometry/solids/42118.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.343 & 0.545 & 0.548 \\
+ 0.187 & 0.149 & 0.592 \\
+ 0.242 & 0.232 & 0.812 \\
+ 0.006 & 0.281 & 0.615 \\
+ 0.158 & 0.951 & 0.382 \\
+ 0.24 & 0.747 & 0.09 \\
+ 0.129 & 0.055 & 0.804 \\
+ 0.194 & 0.234 & 0.906 \\
+\end{array}
+\right)$. 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: $2.96$

pretraining/mathematica/geometry/solids/42178.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.181 & 0.15 & 0.782 \\
+ 0.536 & 0.018 & 0.015 \\
+ 0.528 & 0.088 & 0.038 \\
+ 0.526 & 0.825 & 0.954 \\
+ 0.071 & 0.266 & 0.012 \\
+ 0.652 & 0.06 & 0.187 \\
+ 0.94 & 0.5 & 0.459 \\
+ 0.825 & 0.763 & 0.472 \\
+\end{array}
+\right)$. 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: $0.15$ 
+Surface Area: $1.82$ 
+Solid Angle: $1.08$

pretraining/mathematica/geometry/solids/4351.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.507 & 0.543 & 0.919 \\
+ 0.535 & 0.24 & 0.32 \\
+ 0.075 & 0.968 & 0.529 \\
+ 0.626 & 0.929 & 0.465 \\
+\end{array}
+\right)$. 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: $0.03$ 
+Surface Area: $0.79$ 
+Solid Angle: $0.59$

pretraining/mathematica/geometry/solids/48546.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.503 & 0.481 & 0.153 \\
+ 0.908 & 0.545 & 0.892 \\
+ 0.892 & 0.068 & 0.379 \\
+ 0.729 & 0.814 & 0.118 \\
+ 0.266 & 0.272 & 0.794 \\
+ 0.135 & 0.967 & 0.145 \\
+ 0.458 & 0.256 & 0.382 \\
+\end{array}
+\right)$. 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: $3.96$ 
+Surface Area: $1.76$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/49275.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.336 & 0.004 & 0.879 \\
+ 0.128 & 0.179 & 0.224 \\
+ 0.392 & 0.203 & 0.089 \\
+ 0.623 & 0.891 & 0.436 \\
+ 0.223 & 0.367 & 0.196 \\
+ 0.718 & 0.634 & 0.796 \\
+ 0.067 & 0.942 & 0.626 \\
+ 0.148 & 0.408 & 0.542 \\
+ 0.222 & 0.936 & 0.96 \\
+ 0.409 & 0.443 & 0.043 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.86$ 
+Surface Area: $1.97$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/49338.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.914 & 0.365 & 0.09 \\
+ 0.186 & 0.359 & 0.601 \\
+ 0.617 & 0.989 & 0.314 \\
+ 0.573 & 0.702 & 0.563 \\
+ 0.824 & 0.483 & 0.003 \\
+ 0.585 & 0.143 & 0.381 \\
+ 0.68 & 0.341 & 0.516 \\
+\end{array}
+\right)$. 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: $1.2$ 
+Surface Area: $0.97$

pretraining/mathematica/geometry/solids/50676.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.876 & 0.207 & 0.152 \\
+ 0.129 & 0.898 & 0.544 \\
+ 0.444 & 0.377 & 0.933 \\
+ 0.941 & 0.929 & 0.949 \\
+ 0.106 & 0.193 & 0.158 \\
+ 0.322 & 0.656 & 0.993 \\
+ 0.116 & 0.348 & 0.792 \\
+\end{array}
+\right)$. 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: $0.19$ 
+Surface Area: $2.15$ 
+Solid Angle: $0.61$

pretraining/mathematica/geometry/solids/53090.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.48 & 0.696 & 0.497 \\
+ 0.717 & 0.161 & 0.774 \\
+ 0.174 & 0.089 & 0.661 \\
+ 0.94 & 0.905 & 0.539 \\
+ 0.792 & 0.08 & 0.947 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.19$ 
+Volume: $0.01$ 
+Surface Area: $0.89$

pretraining/mathematica/geometry/solids/54383.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.652 & 0.615 & 0.272 \\
+ 0.804 & 0.346 & 0.217 \\
+ 0.159 & 0.967 & 0.652 \\
+ 0.878 & 0.361 & 0.198 \\
+ 0.865 & 0.396 & 0.836 \\
+ 0.131 & 0.333 & 0.592 \\
+ 0.029 & 0.026 & 0.68 \\
+ 0.886 & 0.833 & 0.612 \\
+\end{array}
+\right)$. 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: $3.38$ 
+Surface Area: $1.73$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/54673.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.12 & 0.549 & 0.99 \\
+ 0.061 & 0.47 & 0.782 \\
+ 0.127 & 0.186 & 0.769 \\
+ 0.683 & 0.944 & 0.85 \\
+ 0.547 & 0.586 & 0.134 \\
+ 0.836 & 0.579 & 0.017 \\
+ 0.225 & 0.148 & 0.136 \\
+ 0.162 & 0.191 & 0.437 \\
+ 0.039 & 0.434 & 0.481 \\
+ 0.355 & 0.694 & 0.326 \\
+\end{array}
+\right)$. 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: $1.77$ 
+Volume: $0.11$ 
+Solid Angle: $1.05$

pretraining/mathematica/geometry/solids/55540.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.662 & 0.396 & 0.217 \\
+ 0.77 & 0.617 & 0.63 \\
+ 0.047 & 0.475 & 0.804 \\
+ 0.935 & 0.344 & 0.9 \\
+ 0.754 & 0.254 & 0.553 \\
+ 0.286 & 0.836 & 0.489 \\
+ 0.498 & 0.874 & 0.595 \\
+ 0.348 & 0.953 & 0.821 \\
+ 0.649 & 0.115 & 0.874 \\
+ 0.271 & 0.167 & 0.725 \\
+\end{array}
+\right)$. 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: $1.54$ 
+Solid Angle: $1.18$

pretraining/mathematica/geometry/solids/5728.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.153 & 0.614 & 0.955 \\
+ 0.061 & 0.473 & 0.492 \\
+ 0.297 & 0.558 & 0.958 \\
+ 0.837 & 0.871 & 0.578 \\
+ 0.966 & 0.337 & 0.538 \\
+ 0.344 & 0.851 & 0.42 \\
+ 0.567 & 0.847 & 0.194 \\
+ 0.096 & 0.055 & 0.798 \\
+ 0.447 & 0.498 & 0.88 \\
+ 0.731 & 0.748 & 0.669 \\
+ 0.835 & 0.015 & 0.298 \\
+\end{array}
+\right)$. 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: $2.04$ 
+Solid Angle: $1.91$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/577.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.027 & 0.518 & 0.405 \\
+ 0.815 & 0.259 & 0.305 \\
+ 0.693 & 0.996 & 0.864 \\
+ 0.532 & 0.584 & 0.835 \\
+ 0.376 & 0.738 & 0.112 \\
+ 0.372 & 0.222 & 0.496 \\
+ 0.326 & 0.624 & 0.069 \\
+ 0.12 & 0.423 & 0.499 \\
+ 0.395 & 0.315 & 0.031 \\
+ 0.287 & 0.013 & 0.254 \\
+ 0.611 & 0.742 & 0.248 \\
+\end{array}
+\right)$. 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: $1.65$ 
+Solid Angle: $1.99$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/58367.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.212 & 0.867 & 0.198 \\
+ 0.751 & 0.09 & 0.661 \\
+ 0.497 & 0.511 & 0.654 \\
+ 0.526 & 0.134 & 0.7 \\
+ 0.963 & 0.016 & 0.027 \\
+ 0.356 & 0.458 & 0.259 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.04$ 
+Solid Angle: $0.21$ 
+Surface Area: $1.08$

pretraining/mathematica/geometry/solids/58701.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.297 & 0.419 & 0.065 \\
+ 0.978 & 0.516 & 0.615 \\
+ 0.566 & 0.85 & 0.551 \\
+ 0.349 & 0.437 & 0.821 \\
+ 0.175 & 0.132 & 0.763 \\
+ 0.614 & 0.056 & 0.838 \\
+ 0.12 & 0.584 & 0.321 \\
+ 0.104 & 0.35 & 0.815 \\
+ 0.874 & 0.151 & 0.118 \\
+\end{array}
+\right)$. 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: $1.86$ 
+Solid Angle: $1.58$

pretraining/mathematica/geometry/solids/59209.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.866 & 0.395 & 0.635 \\
+ 0.408 & 0.105 & 0.162 \\
+ 0.531 & 0.905 & 0.786 \\
+ 0.221 & 0.675 & 0.092 \\
+ 0.014 & 0.233 & 0.406 \\
+ 0.616 & 0.036 & 0.047 \\
+ 0.284 & 0.814 & 0.76 \\
+\end{array}
+\right)$. 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: $0.76$ 
+Volume: $0.11$ 
+Surface Area: $1.58$

pretraining/mathematica/geometry/solids/6150.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.707 & -0.408 & -0.289 \\
+ -0.707 & -0.408 & 0.289 \\
+ -0.707 & 0.408 & -0.289 \\
+ -0.707 & 0.408 & 0.289 \\
+ 0. & -0.816 & -0.289 \\
+ 0. & -0.816 & 0.289 \\
+ 0. & 0. & -0.866 \\
+ 0. & 0. & 0.866 \\
+ 0. & 0.816 & -0.289 \\
+ 0. & 0.816 & 0.289 \\
+ 0.707 & -0.408 & -0.289 \\
+ 0.707 & -0.408 & 0.289 \\
+ 0.707 & 0.408 & -0.289 \\
+ 0.707 & 0.408 & 0.289 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$24.$

pretraining/mathematica/geometry/solids/65133.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.313 & 0.997 & 0.118 \\
+ 0.32 & 0.759 & 0.448 \\
+ 0.22 & 0.701 & 0.683 \\
+ 0.723 & 0.255 & 0.339 \\
+ 0.597 & 0.491 & 0.391 \\
+\end{array}
+\right)$. 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: $0.49$ 
+Volume: $0.$ 
+Solid Angle: $0.04$

pretraining/mathematica/geometry/solids/6563.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.079 & 0.046 & 0.683 \\
+ 0.729 & 0.71 & 0.54 \\
+ 0.141 & 0.721 & 0.785 \\
+ 0.153 & 0.588 & 0.041 \\
+ 0.31 & 0.948 & 0.672 \\
+ 0.392 & 0.816 & 0.322 \\
+ 0.86 & 0.573 & 0.068 \\
+ 0.786 & 0.158 & 0.048 \\
+ 0.731 & 0.19 & 0.192 \\
+ 0.04 & 0.279 & 0.864 \\
+\end{array}
+\right)$. 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: $0.19$ 
+Surface Area: $2.02$ 
+Solid Angle: $1.02$

pretraining/mathematica/geometry/solids/65895.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.7 & 0.98 & 0.356 \\
+ 0.567 & 0.531 & 0.808 \\
+ 0.798 & 0.297 & 0.394 \\
+ 0.322 & 0.511 & 0.545 \\
+ 0.677 & 0.333 & 0.812 \\
+\end{array}
+\right)$. 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: $0.27$ 
+Surface Area: $0.63$

pretraining/mathematica/geometry/solids/66016.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.723 & 0.925 & 0.281 \\
+ 0.033 & 0.515 & 0.021 \\
+ 0.499 & 0.398 & 0.232 \\
+ 0.049 & 0.554 & 0.03 \\
+\end{array}
+\right)$. 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: $0.32$ 
+Solid Angle: $0.$