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pretraining/mathematica/geometry/solids/10358.txt

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+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/13228.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.117 & 0.041 & 0.403 \\
+ 0.364 & 0.968 & 0.553 \\
+ 0.254 & 0.597 & 0.828 \\
+ 0.877 & 0.568 & 0.094 \\
+ 0.268 & 0.731 & 0.934 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.04$ 
+Surface Area: $1.13$ 
+Solid Angle: $0.19$

pretraining/mathematica/geometry/solids/14965.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.394 & 0.806 & 0.616 \\
+ 0.79 & 0.368 & 0.435 \\
+ 0.475 & 0.979 & 0.201 \\
+ 0.167 & 0.61 & 0.552 \\
+ 0.671 & 0.928 & 0.099 \\
+ 0.79 & 0.992 & 0.55 \\
+ 0.09 & 0.056 & 0.873 \\
+\end{array}
+\right)$. 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: $3.68$ 
+Surface Area: $1.32$

pretraining/mathematica/geometry/solids/16444.txt

@@ -0,0 +1,62 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\frac{1}{4} \sqrt{2-\frac{2}{\sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \sqrt{\frac{61}{40}+\frac{131}{40 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{10} \sqrt{85+22 \sqrt{5}} \\
+ \sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\sqrt{\frac{5}{2}+\frac{11}{2 \sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{20} \sqrt{610+262 \sqrt{5}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{10} \sqrt{85+22 \sqrt{5}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ 0 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{7}{10} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} & \frac{1}{10} \sqrt{5+2 \sqrt{5}} \\
+ \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} & \frac{1}{2} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\sqrt{\frac{17}{8}+\frac{31}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{10} \sqrt{130+38 \sqrt{5}} & 0 & \frac{1}{10} \sqrt{145+62 \sqrt{5}} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(5+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\sqrt{\frac{17}{8}+\frac{31}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} \left(2+\sqrt{5}\right) & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(2+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ -\sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{7}{10} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} & \frac{1}{10} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-5-\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & \frac{1}{\sqrt{5}-3} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-5-\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ 0 & \frac{1}{2} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & -1-\frac{\sqrt{5}}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -1-\frac{\sqrt{5}}{2} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$\sqrt{11+4 \sqrt{5}}$

pretraining/mathematica/geometry/solids/17904.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.816 & 0.918 & 0.929 \\
+ 0.044 & 0.732 & 0.221 \\
+ 0.314 & 0.891 & 0.275 \\
+ 0.229 & 0.895 & 0.557 \\
+ 0.773 & 0.29 & 0.318 \\
+ 0.558 & 0.903 & 0.253 \\
+\end{array}
+\right)$. 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: $0.29$ 
+Surface Area: $1.11$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/18426.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.592 & 0.977 & 0.349 \\
+ 0.781 & 0.96 & 0.637 \\
+ 0.351 & 0.784 & 0.956 \\
+ 0.007 & 0.878 & 0.424 \\
+ 0.09 & 0.526 & 0.981 \\
+ 0.832 & 0.143 & 0.314 \\
+ 0.3 & 0.24 & 0.396 \\
+ 0.96 & 0.72 & 0.462 \\
+ 0.7 & 0.638 & 0.278 \\
+ 0.902 & 0.365 & 0.338 \\
+ 0.786 & 0.382 & 0.844 \\
+\end{array}
+\right)$. 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: $2.45$ 
+Surface Area: $2.09$

pretraining/mathematica/geometry/solids/19813.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.606 & 0.728 & 0.128 \\
+ 0.196 & 0.651 & 0.875 \\
+ 0.389 & 0.84 & 0.955 \\
+ 0.934 & 0.503 & 0.053 \\
+ 0.3 & 0.522 & 0.837 \\
+ 0.641 & 0.053 & 0.643 \\
+ 0.863 & 0.491 & 0.901 \\
+ 0.073 & 0.275 & 0.036 \\
+\end{array}
+\right)$. 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: $0.19$ 
+Solid Angle: $2.47$ 
+Surface Area: $2.06$

pretraining/mathematica/geometry/solids/20280.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.623 & 0.019 & 0.423 \\
+ 0.769 & 0.542 & 0.009 \\
+ 0.951 & 0.913 & 0.885 \\
+ 0.478 & 0.986 & 0.105 \\
+ 0.213 & 0.474 & 0.441 \\
+ 0.658 & 0.078 & 0.659 \\
+\end{array}
+\right)$. 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: $1.23$ 
+Surface Area: $1.67$

pretraining/mathematica/geometry/solids/20369.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.764 & 0.972 & 0.91 \\
+ 0.711 & 0.301 & 0.756 \\
+ 0.134 & 0.324 & 0.674 \\
+ 0.882 & 0.009 & 0.651 \\
+ 0.725 & 0.003 & 0.217 \\
+\end{array}
+\right)$. 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: $1.14$ 
+Volume: $0.05$ 
+Solid Angle: $0.18$

pretraining/mathematica/geometry/solids/23112.txt

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+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
+ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
+ 0 & -\sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ 0 & \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
+ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
+ \frac{1}{2} \left(-1-\sqrt{2}\right) & -\frac{1}{2} & 0 \\
+ -\sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} & 0 & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} & 0 & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ -\frac{1}{2} \sqrt{2+\sqrt{2}} & -\frac{1}{2} \sqrt{2+\sqrt{2}} & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ -\frac{1}{2} \sqrt{2+\sqrt{2}} & \frac{\sqrt{2+\sqrt{2}}}{2} & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ \frac{\sqrt{2+\sqrt{2}}}{2} & -\frac{1}{2} \sqrt{2+\sqrt{2}} & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ \frac{\sqrt{2+\sqrt{2}}}{2} & \frac{\sqrt{2+\sqrt{2}}}{2} & -\sqrt{-1-\sqrt{2}+\sqrt{\frac{1}{2} \left(10+7 \sqrt{2}\right)}} \\
+ -\frac{1}{2} & \frac{1}{2} \left(-1-\sqrt{2}\right) & 0 \\
+ \frac{1}{2} & \frac{1}{2} \left(1+\sqrt{2}\right) & 0 \\
+ \frac{1}{2} & \frac{1}{2} \left(-1-\sqrt{2}\right) & 0 \\
+ -\frac{1}{2} & \frac{1}{2} \left(1+\sqrt{2}\right) & 0 \\
+ \frac{1}{2} \left(1+\sqrt{2}\right) & -\frac{1}{2} & 0 \\
+ \frac{1}{2} \left(-1-\sqrt{2}\right) & \frac{1}{2} & 0 \\
+ \frac{1}{2} \left(1+\sqrt{2}\right) & \frac{1}{2} & 0 \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$\text{Root}\left[6561 \text{$\#$1}^8-52488 \text{$\#$1}^7+113724 \text{$\#$1}^6-9720 \text{$\#$1}^5-1616922 \text{$\#$1}^4+396360 \text{$\#$1}^3+1537020 \text{$\#$1}^2-178632 \text{$\#$1}-3391\&,6\right]$

pretraining/mathematica/geometry/solids/23170.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.965 & 0.141 & 0.074 \\
+ 0.589 & 0.516 & 0.891 \\
+ 0.316 & 0.189 & 0.289 \\
+ 0.399 & 0.614 & 0.396 \\
+ 0.59 & 0.917 & 0.453 \\
+ 0.6 & 0.501 & 0.165 \\
+ 0.008 & 0.468 & 0.975 \\
+ 0.476 & 0.034 & 0.339 \\
+ 0.419 & 0.087 & 0.283 \\
+ 0.936 & 0.8 & 0.126 \\
+\end{array}
+\right)$. 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: $1.91$ 
+Solid Angle: $0.78$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/23766.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.16 & 0.548 & 0.703 \\
+ 0.108 & 0.804 & 0.302 \\
+ 0.912 & 0.136 & 0.655 \\
+ 0.583 & 0.536 & 0.011 \\
+ 0.288 & 0.364 & 0.769 \\
+ 0.826 & 0.176 & 0.362 \\
+ 0.974 & 0.921 & 0.691 \\
+ 0.459 & 0.593 & 0.836 \\
+ 0.597 & 0.837 & 0.749 \\
+ 0.157 & 0.404 & 0.746 \\
+\end{array}
+\right)$. 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: $3.89$ 
+Volume: $0.17$ 
+Surface Area: $1.83$

pretraining/mathematica/geometry/solids/24700.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.209 & 0.486 & 0.518 \\
+ 0.141 & 0.457 & 0.815 \\
+ 0.599 & 0.255 & 0.095 \\
+ 0.028 & 0.998 & 0.379 \\
+ 0.969 & 0.254 & 0.395 \\
+ 0.684 & 0.941 & 0.783 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.11$ 
+Solid Angle: $5.17$ 
+Surface Area: $1.59$

pretraining/mathematica/geometry/solids/25168.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.417 & 0.069 & 0.112 \\
+ 0.115 & 0.414 & 0.885 \\
+ 0.094 & 0.638 & 0.382 \\
+ 0.142 & 0.369 & 0.648 \\
+ 0.292 & 0.115 & 0.949 \\
+ 0.587 & 0.051 & 0.844 \\
+ 0.4 & 0.775 & 0.354 \\
+ 0.495 & 0.718 & 0.614 \\
+ 0.575 & 0.682 & 0.036 \\
+ 0.582 & 0.338 & 0.168 \\
+\end{array}
+\right)$. 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: $1.21$ 
+Volume: $0.13$ 
+Surface Area: $1.58$

pretraining/mathematica/geometry/solids/2613.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.06 & 0.863 & 0.229 \\
+ 0.615 & 0.335 & 0.5 \\
+ 0.012 & 0.763 & 0.28 \\
+ 0.131 & 0.642 & 0.863 \\
+ 0.289 & 0.417 & 0.113 \\
+ 0.684 & 0.94 & 0.403 \\
+ 0.968 & 0.58 & 0.394 \\
+ 0.842 & 0.24 & 0.069 \\
+\end{array}
+\right)$. 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: $1.67$ 
+Volume: $0.13$ 
+Surface Area: $1.58$

pretraining/mathematica/geometry/solids/26747.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.584 & 0.666 & 0.013 \\
+ 0.893 & 0.084 & 0.61 \\
+ 0.701 & 0.562 & 0.045 \\
+ 0.964 & 0.639 & 0.015 \\
+ 0.24 & 0.9 & 0.005 \\
+ 0.187 & 0.182 & 0.882 \\
+ 0.807 & 0.639 & 0.468 \\
+ 0.235 & 0.696 & 0.915 \\
+\end{array}
+\right)$. 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: $2.$ 
+Solid Angle: $4.38$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/27191.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & -\frac{1}{2} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & \frac{1}{2} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & -\frac{1}{2} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & \frac{1}{2} \\
+ \frac{1}{\sqrt{3}} & 0 & -\frac{1}{2} \\
+ \frac{1}{\sqrt{3}} & 0 & \frac{1}{2} \\
+ -\sqrt{\frac{7}{12}+\frac{1}{\sqrt{6}}} & 0 & 0 \\
+ \sqrt{\frac{7}{48}+\frac{1}{4 \sqrt{6}}} & \frac{1}{4} \left(-1-\sqrt{6}\right) & 0 \\
+ \sqrt{\frac{7}{48}+\frac{1}{4 \sqrt{6}}} & \frac{1}{4} \left(1+\sqrt{6}\right) & 0 \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$\frac{7 \sqrt{3}}{2}$

pretraining/mathematica/geometry/solids/27839.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.4 & 0.208 & 0.939 \\
+ 0.844 & 0.444 & 0.944 \\
+ 0.21 & 0.28 & 0.786 \\
+ 0.793 & 0.031 & 0.973 \\
+ 0.644 & 0.219 & 0.041 \\
+\end{array}
+\right)$. 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: $0.92$ 
+Volume: $0.04$ 
+Solid Angle: $2.14$

pretraining/mathematica/geometry/solids/29641.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.121 & 0.602 & 0.904 \\
+ 0.146 & 0.49 & 0.771 \\
+ 0.39 & 0.089 & 0.845 \\
+ 0.099 & 0.94 & 0.239 \\
+ 0.432 & 0.434 & 0.101 \\
+ 0.957 & 0.471 & 0.509 \\
+ 0.134 & 0.644 & 0.82 \\
+ 0.457 & 0.688 & 0.236 \\
+\end{array}
+\right)$. 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: $1.1$ 
+Surface Area: $1.47$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/29780.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.822 & 0.92 & 0.712 \\
+ 0.608 & 0.083 & 0.367 \\
+ 0.593 & 0.922 & 0.037 \\
+ 0.104 & 0.738 & 0.854 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.83$ 
+Surface Area: $1.28$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/34068.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.574 & 0.857 & 0.724 \\
+ 0.779 & 0.221 & 0.125 \\
+ 0.467 & 0.936 & 0.279 \\
+ 0.064 & 0.753 & 0.741 \\
+ 0.006 & 0.933 & 0.399 \\
+ 0.028 & 0.978 & 0.682 \\
+ 0.335 & 0.463 & 0.804 \\
+ 0.801 & 0.102 & 0.564 \\
+\end{array}
+\right)$. 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: $1.57$ 
+Solid Angle: $2.17$

pretraining/mathematica/geometry/solids/34621.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.133 & 0.482 & 0.234 \\
+ 0.396 & 0.993 & 0.426 \\
+ 0.888 & 0.849 & 0.928 \\
+ 0.385 & 0.873 & 0.456 \\
+ 0.135 & 0.436 & 0.263 \\
+ 0.769 & 0.879 & 0.824 \\
+ 0.784 & 0.539 & 0.407 \\
+ 0.773 & 0.602 & 0.676 \\
+\end{array}
+\right)$. 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: $0.82$ 
+Solid Angle: $0.93$

pretraining/mathematica/geometry/solids/34626.txt

@@ -0,0 +1,5 @@
+Problem:
+An ellipsoid centered at $\{3.66,2.09,0.68\}$ has radii $\{4.295,4.738,1.593\}$. Estimate the ellipsoid's surface area and volume.
+Answer:
+Surface Area: $156.94$ 
+Volume: $135.79$

pretraining/mathematica/geometry/solids/34930.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.962 & 0.249 & 0.929 \\
+ 0.785 & 0.089 & 0.926 \\
+ 0.668 & 0.525 & 0.68 \\
+ 0.348 & 0.565 & 0.354 \\
+ 0.521 & 0.9 & 0.938 \\
+ 0.056 & 0.46 & 0.658 \\
+ 0.521 & 0.761 & 0.637 \\
+ 0.041 & 0.055 & 0.554 \\
+\end{array}
+\right)$. 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: $0.73$ 
+Surface Area: $1.41$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/37107.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.514 & 0.706 & 0.725 \\
+ 0.653 & 0.216 & 0.347 \\
+ 0.196 & 0.42 & 0.236 \\
+ 0.715 & 0.739 & 0.013 \\
+\end{array}
+\right)$. 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: $0.69$ 
+Solid Angle: $0.39$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/43991.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.431 & 0.808 & 0.203 \\
+ 0.038 & 0.355 & 0.957 \\
+ 0.565 & 0.413 & 0.206 \\
+ 0.189 & 0.676 & 0.814 \\
+ 0.053 & 0.571 & 0.099 \\
+ 0.968 & 0.635 & 0.842 \\
+ 0.722 & 0.779 & 0.956 \\
+ 0.173 & 0.126 & 0.564 \\
+\end{array}
+\right)$. 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: $2.$ 
+Volume: $0.17$ 
+Surface Area: $1.83$

pretraining/mathematica/geometry/solids/45621.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.223 & 0.232 & 0.85 \\
+ 0.111 & 0.752 & 0.804 \\
+ 0.151 & 1. & 0.002 \\
+ 0.215 & 0.088 & 0.728 \\
+ 0.6 & 0.636 & 0.607 \\
+ 0.061 & 0.755 & 0.83 \\
+ 0.664 & 0.539 & 0.196 \\
+ 0.078 & 0.943 & 0.367 \\
+ 0.234 & 0.648 & 0.242 \\
+\end{array}
+\right)$. 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: $1.82$ 
+Volume: $0.09$ 
+Surface Area: $1.42$

pretraining/mathematica/geometry/solids/46029.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.6 & 0.08 & 0.746 \\
+ 0.926 & 0.865 & 0.44 \\
+ 0.097 & 0.984 & 0.154 \\
+ 0.736 & 0.087 & 0.313 \\
+ 0.92 & 0.252 & 0.708 \\
+ 0.238 & 0.79 & 0.614 \\
+ 0.535 & 0.298 & 0.818 \\
+ 0.301 & 0.104 & 0.039 \\
+\end{array}
+\right)$. 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: $0.18$ 
+Solid Angle: $2.09$ 
+Surface Area: $2.06$

pretraining/mathematica/geometry/solids/4659.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.351 & 0.218 & 0.986 \\
+ 0.977 & 0.562 & 0.548 \\
+ 0.124 & 0.658 & 0.592 \\
+ 0.548 & 0.082 & 0.017 \\
+ 0.575 & 0.923 & 0.5 \\
+ 0.755 & 0.365 & 0.764 \\
+ 0.114 & 0.418 & 0.816 \\
+ 0.112 & 0.107 & 0.195 \\
+ 0.581 & 0.47 & 0.002 \\
+ 0.417 & 0.622 & 0.026 \\
+ 0.306 & 0.818 & 0.225 \\
+\end{array}
+\right)$. 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: $2.13$ 
+Volume: $0.23$ 
+Solid Angle: $1.26$

pretraining/mathematica/geometry/solids/4664.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.41 & 0.236 & 0.713 \\
+ 0.106 & 0.478 & 0.888 \\
+ 0.752 & 0.822 & 0.018 \\
+ 0.934 & 0.48 & 0.98 \\
+ 0.496 & 0.733 & 0.086 \\
+ 0.082 & 0.374 & 0.067 \\
+ 0.41 & 0.978 & 0.757 \\
+ 0.906 & 0.232 & 0.044 \\
+\end{array}
+\right)$. 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: $0.27$ 
+Solid Angle: $3.6$ 
+Surface Area: $2.45$

pretraining/mathematica/geometry/solids/47560.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.134 & 0.986 & 0.814 \\
+ 0.011 & 0.97 & 0.188 \\
+ 0.257 & 0.163 & 0.543 \\
+ 0.721 & 0.415 & 0.704 \\
+ 0.379 & 0.657 & 0.338 \\
+\end{array}
+\right)$. 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: $0.05$ 
+Solid Angle: $0.5$

pretraining/mathematica/geometry/solids/48275.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.964 & 0.97 & 0.882 \\
+ 0.412 & 0.127 & 0.553 \\
+ 0.914 & 0.675 & 0.42 \\
+ 0.978 & 0.916 & 0.49 \\
+ 0.612 & 0.035 & 0.83 \\
+ 0.647 & 0.418 & 0.425 \\
+ 0.359 & 0.908 & 0.583 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.08$ 
+Surface Area: $1.29$ 
+Solid Angle: $0.81$

pretraining/mathematica/geometry/solids/48397.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.46 & 0.648 & 0.328 \\
+ 0.26 & 0.104 & 0.752 \\
+ 0.203 & 0.98 & 0.729 \\
+ 0.774 & 0.94 & 0.861 \\
+ 0.891 & 0.999 & 0.992 \\
+ 0.052 & 0.592 & 0.564 \\
+ 0.868 & 0.39 & 0.398 \\
+ 0.989 & 0.046 & 0.329 \\
+ 0.667 & 0.06 & 0.956 \\
+\end{array}
+\right)$. 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: $2.71$ 
+Surface Area: $2.23$

pretraining/mathematica/geometry/solids/51032.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.808 & 0.941 & 0.566 \\
+ 0.39 & 0.891 & 0.501 \\
+ 0.956 & 0.602 & 0.518 \\
+ 0.343 & 0.476 & 0.099 \\
+\end{array}
+\right)$. 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: $0.4$ 
+Surface Area: $0.51$

pretraining/mathematica/geometry/solids/52299.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.413 & 0.503 & 0.804 \\
+ 0.347 & 0.251 & 0.623 \\
+ 0.513 & 0.156 & 0.349 \\
+ 0.134 & 0.516 & 0.951 \\
+\end{array}
+\right)$. 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: $0.22$ 
+Volume: $0.$ 
+Surface Area: $0.19$

pretraining/mathematica/geometry/solids/5231.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.494 & 0.3 & 0.166 \\
+ 0.951 & 0.307 & 0.731 \\
+ 0.314 & 0.051 & 0.838 \\
+ 0.041 & 0.778 & 0.517 \\
+ 0.213 & 0.47 & 0.151 \\
+\end{array}
+\right)$. 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: $1.14$ 
+Surface Area: $1.18$

pretraining/mathematica/geometry/solids/52629.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.783 & 0.138 & 0.728 \\
+ 0.517 & 0.158 & 0.155 \\
+ 0.079 & 0.501 & 0.165 \\
+ 0.256 & 0.937 & 0.974 \\
+ 0.382 & 0.938 & 0.886 \\
+ 0.264 & 0.684 & 0.205 \\
+ 0.298 & 0.15 & 0.015 \\
+ 0.779 & 0.695 & 0.04 \\
+\end{array}
+\right)$. 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: $1.89$ 
+Solid Angle: $0.78$

pretraining/mathematica/geometry/solids/53513.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.003 & 0.795 & 0.124 \\
+ 0.677 & 0.429 & 0.79 \\
+ 0.071 & 0.398 & 0.186 \\
+ 0.515 & 0.229 & 0.602 \\
+ 0.264 & 0.669 & 0.062 \\
+ 0.119 & 0.43 & 0.098 \\
+ 0.764 & 0.147 & 0.949 \\
+\end{array}
+\right)$. 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:
+Volume: $0.03$ 
+Surface Area: $0.8$ 
+Solid Angle: $0.57$

pretraining/mathematica/geometry/solids/55617.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.439 & 0.21 & 0.379 \\
+ 0.364 & 0.635 & 0.529 \\
+ 0.931 & 0.292 & 0.253 \\
+ 0.58 & 0.228 & 0.142 \\
+ 0.945 & 0.195 & 0.914 \\
+ 0.527 & 0.612 & 0.44 \\
+\end{array}
+\right)$. 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: $0.82$ 
+Volume: $0.04$ 
+Solid Angle: $2.1$

pretraining/mathematica/geometry/solids/56110.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.88 & 0.133 & 0.015 \\
+ 0.806 & 0.311 & 0.595 \\
+ 0.78 & 0.636 & 0.557 \\
+ 0.374 & 0.458 & 0.405 \\
+ 0.634 & 0.883 & 0.448 \\
+ 0.139 & 0.542 & 0.179 \\
+ 0.252 & 0.613 & 0.878 \\
+ 0.575 & 0.81 & 0.64 \\
+\end{array}
+\right)$. 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: $0.37$ 
+Volume: $0.09$ 
+Surface Area: $1.35$

pretraining/mathematica/geometry/solids/56371.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.247 & 0.175 & 0.191 \\
+ 0.478 & 0.735 & 0.003 \\
+ 0.431 & 0.024 & 0.923 \\
+ 0.052 & 0.604 & 0.758 \\
+ 0.288 & 0.907 & 0.395 \\
+ 0.047 & 0.768 & 0.112 \\
+ 0.697 & 0.855 & 0.448 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.7$ 
+Volume: $0.13$ 
+Solid Angle: $1.31$

pretraining/mathematica/geometry/solids/56972.txt

@@ -0,0 +1,5 @@
+Problem:
+An ellipsoid centered at $\{3.357,-9.861,-3.373\}$ has radii $\{3.997,3.673,3.524\}$. Estimate the ellipsoid's surface area and volume.
+Answer:
+Volume: $216.69$ 
+Surface Area: $174.84$

pretraining/mathematica/geometry/solids/58630.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.436 & 0.473 & 0.028 \\
+ 0.43 & 0.825 & 0.503 \\
+ 0.196 & 0.607 & 0.307 \\
+ 0.005 & 0.149 & 0.7 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.27$ 
+Surface Area: $0.53$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/64144.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.637 & 0.346 & 0.376 \\
+ 0.439 & 0.374 & 0.894 \\
+ 0.067 & 0.743 & 0.522 \\
+ 0.54 & 0.483 & 0.354 \\
+ 0.565 & 0.365 & 0.321 \\
+ 0.381 & 0.12 & 0.059 \\
+ 0.309 & 0.061 & 0.905 \\
+\end{array}
+\right)$. 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: $0.06$ 
+Surface Area: $1.03$ 
+Solid Angle: $2.24$

pretraining/mathematica/geometry/solids/64789.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.355 & 0.337 & 0.468 \\
+ 0.477 & 0.09 & 0.157 \\
+ 0.441 & 0.132 & 0.787 \\
+ 0.717 & 0.073 & 0.594 \\
+\end{array}
+\right)$. 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: $0.32$ 
+Solid Angle: $0.5$

pretraining/mathematica/geometry/solids/66810.txt

@@ -0,0 +1,6 @@
+Problem:
+A cylinder with radius $5.857$ is around the line from $\{8.658,-4.438,7.692\}$ to $\{5.723,-9.579,-0.16\}$. Estimate the cylinder's surface area, volume, and centroid.
+Answer:
+Centroid: $\{7.19,-7.01,3.77\}$ 
+Surface Area: $577.38$ 
+Volume: $1059.65$

pretraining/mathematica/geometry/solids/67661.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.088 & 0.202 & 0.015 \\
+ 0.505 & 0.06 & 0.026 \\
+ 0.7 & 0.398 & 0.939 \\
+ 0.876 & 0.254 & 0.074 \\
+ 0.611 & 0.447 & 0.226 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.04$ 
+Surface Area: $0.91$ 
+Solid Angle: $0.33$

pretraining/mathematica/geometry/solids/7026.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.052 & 0.676 & 0.411 \\
+ 0.869 & 0.413 & 0.927 \\
+ 0.849 & 0.326 & 0.265 \\
+ 0.587 & 0.826 & 0.23 \\
+ 0.023 & 0.666 & 0.016 \\
+ 0.495 & 0.187 & 0.539 \\
+\end{array}
+\right)$. 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: $1.47$ 
+Volume: $0.1$ 
+Surface Area: $1.38$

pretraining/mathematica/geometry/solids/70330.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.172 & 0.392 & 0.32 \\
+ 0.295 & 0.785 & 0.133 \\
+ 0.734 & 0.894 & 0.926 \\
+ 0.45 & 0.289 & 0.13 \\
+ 0.031 & 0.519 & 0.562 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.04$ 
+Surface Area: $0.93$ 
+Solid Angle: $2.29$

pretraining/mathematica/geometry/solids/70624.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.208 & 0.345 & 0.279 \\
+ 0.705 & 0.908 & 0.881 \\
+ 0.272 & 0.086 & 0.082 \\
+ 0.387 & 0.973 & 0.626 \\
+ 0.697 & 0.888 & 0.153 \\
+ 0.818 & 0.946 & 0.899 \\
+ 0.022 & 0.353 & 0.489 \\
+\end{array}
+\right)$. 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: $5.85$ 
+Surface Area: $1.4$