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pretraining/mathematica/geometry/solids/10317.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.025 & 0.006 & 0.808 \\
+ 0.356 & 0.364 & 0.608 \\
+ 0.858 & 0.673 & 0.277 \\
+ 0.933 & 0.25 & 0.578 \\
+ 0.507 & 0.403 & 0.751 \\
+ 0.322 & 0.953 & 0.878 \\
+\end{array}
+\right)$. 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: $0.2$ 
+Surface Area: $1.31$

pretraining/mathematica/geometry/solids/12068.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.442 & 0.733 & 0.217 \\
+ 0.396 & 0.772 & 0.629 \\
+ 0.357 & 0.038 & 0.556 \\
+ 0.054 & 0.517 & 0.593 \\
+ 0.501 & 0.423 & 0.757 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.03$ 
+Surface Area: $0.66$ 
+Solid Angle: $0.52$

pretraining/mathematica/geometry/solids/13603.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & -1.572 \\
+ -0.577 & 0. & -0.756 \\
+ -0.289 & -0.5 & 0.756 \\
+ -0.289 & 0.5 & 0.756 \\
+ 0.289 & -0.5 & -0.756 \\
+ 0.289 & 0.5 & -0.756 \\
+ 0.577 & 0. & 0.756 \\
+ -0.934 & 0. & 0.178 \\
+ 0.467 & -0.809 & 0.178 \\
+ 0.467 & 0.809 & 0.178 \\
+\end{array}
+\right)$. Determine the Centroid.
+Answer:
+$\{0.,0.,-0.01\}$

pretraining/mathematica/geometry/solids/13952.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.013 & 0.257 & 0.393 \\
+ 0.227 & 0.761 & 0.312 \\
+ 0.627 & 0.158 & 0.927 \\
+ 0.665 & 0.836 & 0.429 \\
+ 0.637 & 0.417 & 0.776 \\
+ 0.133 & 0.637 & 0.328 \\
+ 0.895 & 0.115 & 0.279 \\
+ 0.239 & 0.862 & 0.474 \\
+ 0.889 & 0.278 & 0.099 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $1.05$ 
+Volume: $0.14$ 
+Surface Area: $1.66$

pretraining/mathematica/geometry/solids/14464.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.906 & 0.575 & 0.068 \\
+ 0.693 & 0.671 & 0.893 \\
+ 0.754 & 0.678 & 0.428 \\
+ 0.651 & 0.443 & 0.934 \\
+ 0.813 & 0.374 & 0.902 \\
+ 0.034 & 0.168 & 0.189 \\
+ 0.962 & 0.257 & 0.392 \\
+ 0.171 & 0.768 & 0.464 \\
+ 0.563 & 0.06 & 0.335 \\
+ 0.575 & 0.159 & 0.273 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.82$ 
+Volume: $0.17$ 
+Solid Angle: $1.16$

pretraining/mathematica/geometry/solids/146.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.507 & 0.005 & 0.424 \\
+ 0.314 & 0.287 & 0.561 \\
+ 0.193 & 0.236 & 0.794 \\
+ 0.637 & 0.886 & 0.397 \\
+\end{array}
+\right)$. 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: $0.47$ 
+Volume: $0.01$ 
+Solid Angle: $0.13$

pretraining/mathematica/geometry/solids/1589.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.256 & 0.764 & 0.704 \\
+ 0.205 & 0.594 & 0.398 \\
+ 0.587 & 0.079 & 0.138 \\
+ 0.955 & 0.923 & 0.847 \\
+ 0.615 & 0.714 & 0.124 \\
+ 0.545 & 0.012 & 0.492 \\
+ 0.719 & 0.466 & 0.917 \\
+ 0.305 & 0.912 & 0.409 \\
+ 0.348 & 0.037 & 0.692 \\
+\end{array}
+\right)$. 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: $1.89$ 
+Volume: $0.18$ 
+Solid Angle: $2.68$

pretraining/mathematica/geometry/solids/17403.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.729 & 0.254 & 0.673 \\
+ 0.835 & 0.828 & 0.248 \\
+ 0.445 & 0.435 & 0.825 \\
+ 0.073 & 0.907 & 0.163 \\
+ 0.636 & 0.728 & 0.562 \\
+ 0.506 & 0.125 & 0.572 \\
+ 0.245 & 0.308 & 0.567 \\
+ 0.079 & 0.739 & 0.399 \\
+ 0.824 & 0.425 & 0.844 \\
+ 0.713 & 0.488 & 0.241 \\
+\end{array}
+\right)$. 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: $1.44$ 
+Solid Angle: $3.13$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/17930.txt

@@ -0,0 +1,23 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
+ 0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
+ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
+ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
+ -\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\
+ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
+ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\
+ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
+ \frac{1}{2} & \frac{1}{2} \left(1+\sqrt{2}\right) & 0 \\
+ -\frac{1}{2} & \frac{1}{2} \left(1+\sqrt{2}\right) & 0 \\
+ \frac{1}{2} \left(-1-\sqrt{2}\right) & \frac{1}{2} & 0 \\
+ \frac{1}{2} \left(-1-\sqrt{2}\right) & -\frac{1}{2} & 0 \\
+ -\frac{1}{2} & \frac{1}{2} \left(-1-\sqrt{2}\right) & 0 \\
+ \frac{1}{2} & \frac{1}{2} \left(-1-\sqrt{2}\right) & 0 \\
+ \frac{1}{2} \left(1+\sqrt{2}\right) & -\frac{1}{2} & 0 \\
+ \frac{1}{2} \left(1+\sqrt{2}\right) & \frac{1}{2} & 0 \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$18$

pretraining/mathematica/geometry/solids/18916.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{6.66,-0.053,1.493\}$ has radius $0.176$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $0.39$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/1960.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.586 & 0.747 & 0.22 \\
+ 0.492 & 0.4 & 0.252 \\
+ 0.234 & 0.709 & 0.232 \\
+ 0.294 & 0.481 & 0.261 \\
+ 0.841 & 0.164 & 0.391 \\
+ 0.431 & 0.804 & 0.597 \\
+ 0.01 & 0.803 & 0.644 \\
+ 0.813 & 0.297 & 0.918 \\
+\end{array}
+\right)$. 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: $1.78$ 
+Surface Area: $1.26$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/21985.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.91 & 0.059 & 0.514 \\
+ 0.342 & 0.989 & 0.096 \\
+ 0.27 & 0.49 & 0.668 \\
+ 0.676 & 0.452 & 0.846 \\
+ 0.296 & 0.99 & 0.932 \\
+ 0.492 & 0.217 & 0.273 \\
+ 0.391 & 0.142 & 0.479 \\
+ 0.891 & 0.076 & 0.5 \\
+ 0.527 & 0.951 & 0.683 \\
+ 0.912 & 0.262 & 0.559 \\
+\end{array}
+\right)$. 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: $0.91$ 
+Volume: $0.14$ 
+Surface Area: $1.69$

pretraining/mathematica/geometry/solids/22223.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.623 & 0.113 & 0.225 \\
+ 0.061 & 0.895 & 0.76 \\
+ 0.045 & 0.907 & 0.804 \\
+ 0.466 & 0.988 & 0.59 \\
+ 0.989 & 0.275 & 0.34 \\
+ 0.397 & 0.955 & 0.783 \\
+ 0.442 & 0.472 & 0.367 \\
+ 0.905 & 0.214 & 0.922 \\
+ 0.132 & 0.105 & 0.692 \\
+ 0.819 & 0.202 & 0.184 \\
+\end{array}
+\right)$. 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: $1.99$ 
+Volume: $0.18$ 
+Solid Angle: $1.89$

pretraining/mathematica/geometry/solids/23758.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.289 & 0.216 & 0.598 \\
+ 0.196 & 0.925 & 0.217 \\
+ 0.128 & 0.598 & 0.866 \\
+ 0.063 & 0.262 & 0.571 \\
+\end{array}
+\right)$. 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: $0.48$ 
+Solid Angle: $0.69$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/24418.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.009 & 0.259 & 0.154 \\
+ 0.049 & 0.398 & 0.999 \\
+ 0.713 & 0.481 & 0.746 \\
+ 0.378 & 0.625 & 0.803 \\
+ 0.926 & 0.683 & 0.56 \\
+ 0.205 & 0.17 & 0.042 \\
+ 0.971 & 0.717 & 0.713 \\
+ 0.371 & 0.659 & 0.642 \\
+\end{array}
+\right)$. 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: $1.37$ 
+Solid Angle: $0.85$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/25281.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.751 & 0.557 & 0.727 \\
+ 0.043 & 0.246 & 0.511 \\
+ 0.206 & 0.018 & 0.401 \\
+ 0.34 & 0.128 & 0.661 \\
+ 0.727 & 0.394 & 0.72 \\
+ 0.836 & 0.452 & 0.955 \\
+ 0.259 & 0.357 & 0.137 \\
+ 0.167 & 0.811 & 0.966 \\
+ 0.272 & 0.961 & 0.437 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.14$ 
+Surface Area: $1.63$ 
+Solid Angle: $3.42$

pretraining/mathematica/geometry/solids/25975.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.759 & 0.977 & 0.711 \\
+ 0.565 & 0.425 & 0.037 \\
+ 0.723 & 0.966 & 0.137 \\
+ 0.858 & 0.807 & 0.135 \\
+ 0.743 & 0.293 & 0.397 \\
+ 0.849 & 0.214 & 0.312 \\
+ 0.123 & 0.665 & 0.233 \\
+ 0.432 & 0.383 & 0.152 \\
+ 0.865 & 0.982 & 0.119 \\
+ 0.199 & 0.912 & 0.914 \\
+ 0.814 & 0.547 & 0.004 \\
+\end{array}
+\right)$. 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: $1.78$ 
+Solid Angle: $1.52$

pretraining/mathematica/geometry/solids/26829.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.399 & 0.89 & 0.928 \\
+ 0.747 & 0.006 & 0.211 \\
+ 0.959 & 0.825 & 0.935 \\
+ 0.536 & 0.595 & 0.6 \\
+ 0.04 & 0.627 & 0.767 \\
+ 0.731 & 0.07 & 0.174 \\
+ 0. & 0.196 & 0.306 \\
+ 0.155 & 0.1 & 0.969 \\
+ 0.617 & 0.362 & 0.775 \\
+\end{array}
+\right)$. 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: $1.29$ 
+Volume: $0.18$ 
+Surface Area: $2.14$

pretraining/mathematica/geometry/solids/27015.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.438 & 0.853 & 0.015 \\
+ 0.011 & 0.8 & 0.241 \\
+ 0.768 & 0.701 & 0.611 \\
+ 0.596 & 0.457 & 0.174 \\
+ 0.839 & 0.474 & 0.24 \\
+ 0.685 & 0.753 & 0.777 \\
+ 0.314 & 0.32 & 0.795 \\
+ 0.074 & 0.227 & 0.926 \\
+ 0.448 & 0.818 & 0.673 \\
+ 0.47 & 0.095 & 0.374 \\
+\end{array}
+\right)$. 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: $1.26$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/3113.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.871 & 0.576 & 0.417 \\
+ 0.965 & 0.04 & 0.939 \\
+ 0.656 & 0.541 & 0.793 \\
+ 0.33 & 0.714 & 0.47 \\
+ 0.934 & 0.966 & 0.1 \\
+ 0.405 & 0.061 & 0.696 \\
+\end{array}
+\right)$. 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: $1.3$ 
+Solid Angle: $6.11$

pretraining/mathematica/geometry/solids/31552.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.252 & 0.836 & 0.084 \\
+ 0.84 & 0.93 & 0.2 \\
+ 0.213 & 0.03 & 0.741 \\
+ 0.419 & 0.627 & 0.219 \\
+ 0.443 & 0.705 & 0.703 \\
+ 0.62 & 0.5 & 0.328 \\
+\end{array}
+\right)$. 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: $0.67$ 
+Volume: $0.05$ 
+Surface Area: $1.02$

pretraining/mathematica/geometry/solids/31660.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.079 & 0.239 & 0.427 \\
+ 0.262 & 0.867 & 0.889 \\
+ 0.592 & 0.832 & 0.47 \\
+ 0.589 & 0.689 & 0.187 \\
+ 0.4 & 0.065 & 0.379 \\
+ 0.11 & 0.949 & 0.574 \\
+ 0.535 & 0.299 & 0.974 \\
+ 0.054 & 0.25 & 0.803 \\
+\end{array}
+\right)$. 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: $1.61$ 
+Solid Angle: $2.47$

pretraining/mathematica/geometry/solids/34546.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.595 & 0.363 & 0.184 \\
+ 0.855 & 0.398 & 0.864 \\
+ 0.199 & 0.094 & 0.146 \\
+ 0.659 & 0.975 & 0.242 \\
+ 0.971 & 0.575 & 0.255 \\
+ 0.411 & 0.249 & 0.939 \\
+ 0.133 & 0.841 & 0.365 \\
+ 0.694 & 0.95 & 0.158 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Surface Area: $1.89$ 
+Solid Angle: $4.94$

pretraining/mathematica/geometry/solids/34739.txt

@@ -0,0 +1,45 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & -\frac{1}{2 \sqrt{\frac{3}{21+\sqrt[3]{3 \left(3123-29 \sqrt{33}\right)}+\sqrt[3]{3 \left(3123+29 \sqrt{33}\right)}}}} \\
+ 0 & 0 & \frac{1}{2 \sqrt{\frac{3}{21+\sqrt[3]{3 \left(3123-29 \sqrt{33}\right)}+\sqrt[3]{3 \left(3123+29 \sqrt{33}\right)}}}} \\
+ 0 & -\frac{1}{2 \sqrt{\frac{3}{21+\sqrt[3]{3 \left(3123-29 \sqrt{33}\right)}+\sqrt[3]{3 \left(3123+29 \sqrt{33}\right)}}}} & 0 \\
+ 0 & \frac{1}{2 \sqrt{\frac{3}{21+\sqrt[3]{3 \left(3123-29 \sqrt{33}\right)}+\sqrt[3]{3 \left(3123+29 \sqrt{33}\right)}}}} & 0 \\
+ -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{21+\sqrt[3]{3 \left(3123-29 \sqrt{33}\right)}+\sqrt[3]{3 \left(3123+29 \sqrt{33}\right)}}}} & 0 & 0 \\
+ \frac{1}{2 \sqrt{\frac{3}{21+\sqrt[3]{3 \left(3123-29 \sqrt{33}\right)}+\sqrt[3]{3 \left(3123+29 \sqrt{33}\right)}}}} & 0 & 0 \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ -\frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & -\frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+ \frac{1}{2 \sqrt{\frac{3}{-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{7+\sqrt[3]{199-3 \sqrt{33}}+\sqrt[3]{199+3 \sqrt{33}}}}} & \frac{1}{2 \sqrt{\frac{3}{15+\sqrt[3]{3 \left(423-43 \sqrt{33}\right)}+\sqrt[3]{3 \left(423+43 \sqrt{33}\right)}}}} \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$60$

pretraining/mathematica/geometry/solids/36626.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.28 & 0.819 & 0.54 \\
+ 0.049 & 0.822 & 0.108 \\
+ 0.28 & 0.584 & 0.973 \\
+ 0.716 & 0.991 & 0.794 \\
+ 0.678 & 0.722 & 0.098 \\
+ 0.365 & 0.157 & 0.519 \\
+ 0.617 & 0.997 & 0.592 \\
+ 0.429 & 0.317 & 0.716 \\
+ 0.776 & 0.915 & 0.081 \\
+\end{array}
+\right)$. 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: $1.69$ 
+Solid Angle: $5.68$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/38765.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.428 & 0.378 & 0.246 \\
+ 0.083 & 0.626 & 0.396 \\
+ 0.598 & 0.917 & 0.99 \\
+ 0.595 & 0.077 & 0.822 \\
+ 0.821 & 0.594 & 0.62 \\
+ 0.113 & 0.082 & 0.927 \\
+ 0.533 & 0.811 & 0.164 \\
+ 0.824 & 0.242 & 0.654 \\
+ 0.318 & 0.159 & 0.313 \\
+\end{array}
+\right)$. 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: $3.81$ 
+Surface Area: $1.95$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/39008.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.679 & 0.215 & 0.874 \\
+ 0.001 & 0.114 & 0.528 \\
+ 0.168 & 0.156 & 0.812 \\
+ 0.587 & 0.611 & 0.216 \\
+ 0.753 & 0.826 & 0.082 \\
+ 0.743 & 0.911 & 0.689 \\
+ 0.932 & 0.614 & 0.702 \\
+ 0.653 & 0.144 & 0.615 \\
+ 0.061 & 0.247 & 0.453 \\
+\end{array}
+\right)$. 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: $1.91$ 
+Surface Area: $1.6$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/39105.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-4.011,-2.697,-9.839\}$ has radius $3.898$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $248.07$ 
+Surface Area: $190.93$

pretraining/mathematica/geometry/solids/39451.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.251 & 0.959 & 0.968 \\
+ 0.347 & 0.716 & 0.603 \\
+ 0.211 & 0.081 & 0.702 \\
+ 0.124 & 0.391 & 0.984 \\
+\end{array}
+\right)$. 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: $0.47$ 
+Solid Angle: $0.02$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/41917.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.334 & 0.033 & 0.232 \\
+ 0.9 & 0.488 & 0.241 \\
+ 0.23 & 0.451 & 0.091 \\
+ 0.764 & 0.306 & 0.746 \\
+ 0.061 & 0.145 & 0.665 \\
+\end{array}
+\right)$. 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: $1.71$ 
+Surface Area: $1.06$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/43085.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.223 & 0.942 & 0.313 \\
+ 0.146 & 0.601 & 0.842 \\
+ 0.912 & 0.29 & 0.583 \\
+ 0.085 & 0.658 & 0.637 \\
+ 0.815 & 0.144 & 0.559 \\
+ 0.412 & 0.241 & 0.171 \\
+ 0.586 & 0.774 & 0.948 \\
+\end{array}
+\right)$. 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: $0.1$ 
+Solid Angle: $0.89$ 
+Surface Area: $1.42$

pretraining/mathematica/geometry/solids/4531.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.112 & 0.56 & 0.771 \\
+ 0.95 & 0.542 & 0.782 \\
+ 0.659 & 0.314 & 0.517 \\
+ 0.486 & 0.147 & 0.848 \\
+ 0.509 & 0.925 & 0.455 \\
+ 0.524 & 0.425 & 0.505 \\
+ 0.185 & 0.926 & 0.91 \\
+ 0.154 & 0.239 & 0.625 \\
+\end{array}
+\right)$. 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: $1.3$ 
+Solid Angle: $2.32$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/47819.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.25 & 0.194 & 0.198 \\
+ 0.81 & 0.516 & 0.143 \\
+ 0.176 & 0.031 & 0.632 \\
+ 0.512 & 0.656 & 0.16 \\
+ 0.072 & 0.101 & 0.72 \\
+ 0.074 & 0.783 & 0.536 \\
+ 0.755 & 0.546 & 0.784 \\
+ 0.151 & 0.199 & 0.036 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $6.05$ 
+Surface Area: $1.63$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/4808.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.258 & 0.077 & 0.718 \\
+ 0.094 & 0.19 & 0.973 \\
+ 0.137 & 0.885 & 0.229 \\
+ 0.553 & 0.583 & 0.946 \\
+ 0.812 & 0.544 & 0.931 \\
+ 0.484 & 0.178 & 0.086 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Surface Area: $1.59$ 
+Solid Angle: $2.1$

pretraining/mathematica/geometry/solids/50614.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.587 & 0.852 & 0.92 \\
+ 0.074 & 0.103 & 0.773 \\
+ 0.933 & 0.925 & 0.287 \\
+ 0.189 & 0.985 & 0.789 \\
+ 0.407 & 0.849 & 0.902 \\
+ 0.374 & 0.379 & 0.371 \\
+\end{array}
+\right)$. 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: $1.32$ 
+Solid Angle: $1.68$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/50903.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.226 & 0.791 & 0.145 \\
+ 0.418 & 0.162 & 0.715 \\
+ 0.958 & 0.662 & 0.065 \\
+ 0.31 & 0.408 & 0.84 \\
+ 0.558 & 0.161 & 0.608 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.29$ 
+Surface Area: $0.96$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/51124.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1.309 & 0. & -0.5 \\
+ -1.309 & 0. & 0.5 \\
+ -0.5 & -0.5 & -0.809 \\
+ -0.5 & -0.5 & 0.809 \\
+ -0.5 & 0.5 & -0.809 \\
+ -0.5 & 0.5 & 0.809 \\
+ 0. & -0.809 & 0. \\
+ 0. & 0.809 & 0. \\
+ 0.5 & -0.5 & -0.809 \\
+ 0.5 & -0.5 & 0.809 \\
+ 0.5 & 0.5 & -0.809 \\
+ 0.5 & 0.5 & 0.809 \\
+ 1.309 & 0. & -0.5 \\
+ 1.309 & 0. & 0.5 \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$2.8$

pretraining/mathematica/geometry/solids/5320.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.351 & 0.21 & 0.918 \\
+ 0.532 & 0.953 & 0.493 \\
+ 0.998 & 0.194 & 0.129 \\
+ 0.64 & 0.155 & 0.365 \\
+ 0.09 & 0.365 & 0.281 \\
+ 0.386 & 0.196 & 0.301 \\
+ 0.859 & 0.99 & 0.199 \\
+ 0.065 & 0.32 & 0.095 \\
+ 0.161 & 0.566 & 0.537 \\
+ 0.34 & 0.2 & 0.904 \\
+ 0.24 & 0.951 & 0.21 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.2$ 
+Solid Angle: $1.31$ 
+Surface Area: $2.13$

pretraining/mathematica/geometry/solids/53341.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.602 & 0.178 & 0.48 \\
+ 0.665 & 0.348 & 0.042 \\
+ 0.211 & 0.044 & 0.195 \\
+ 0.768 & 0.801 & 0.152 \\
+ 0.139 & 0.593 & 0.74 \\
+ 0.35 & 0.558 & 0.971 \\
+ 0.081 & 0.115 & 0.922 \\
+ 0.807 & 0.524 & 0.17 \\
+ 0.756 & 0.85 & 0.583 \\
+\end{array}
+\right)$. 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: $0.15$ 
+Solid Angle: $3.38$ 
+Surface Area: $1.85$

pretraining/mathematica/geometry/solids/55685.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.744 & 0.677 & 0.942 \\
+ 0.372 & 0.671 & 0.631 \\
+ 0.462 & 0.813 & 0.184 \\
+ 0.799 & 0.236 & 0.906 \\
+ 0.307 & 0.119 & 0.834 \\
+ 0.77 & 0.229 & 0.979 \\
+ 0.505 & 0.514 & 0.887 \\
+ 0.981 & 0.759 & 0.847 \\
+\end{array}
+\right)$. 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: $0.07$ 
+Surface Area: $1.16$ 
+Solid Angle: $3.05$

pretraining/mathematica/geometry/solids/58043.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.661 & 0.473 & 0.143 \\
+ 0.899 & 0.858 & 0.827 \\
+ 0.84 & 0.971 & 0.639 \\
+ 0.305 & 0.874 & 0.585 \\
+\end{array}
+\right)$. 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: $0.53$ 
+Solid Angle: $0.11$

pretraining/mathematica/geometry/solids/59123.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.018 & 0.688 & 0.488 \\
+ 0.278 & 0.448 & 0.076 \\
+ 0.258 & 0.906 & 0.441 \\
+ 0.998 & 0.19 & 0.941 \\
+ 0.93 & 0.433 & 0.555 \\
+ 0.749 & 0.286 & 0.875 \\
+\end{array}
+\right)$. 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: $1.16$ 
+Solid Angle: $1.03$ 
+Volume: $0.06$

pretraining/mathematica/geometry/solids/60456.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.13 & 0.745 & 0.654 \\
+ 0.572 & 0.971 & 0.269 \\
+ 0.936 & 0.814 & 0.774 \\
+ 0.374 & 0.957 & 0.01 \\
+ 0.99 & 0.943 & 0.571 \\
+ 0.191 & 0.31 & 0.957 \\
+ 0.038 & 0.137 & 0.612 \\
+ 0.348 & 0.726 & 0.01 \\
+ 0.95 & 0.794 & 0.057 \\
+ 0.495 & 0.686 & 0.793 \\
+ 0.58 & 0.223 & 0.746 \\
+\end{array}
+\right)$. 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:
+Volume: $0.23$ 
+Solid Angle: $2.62$ 
+Surface Area: $2.29$

pretraining/mathematica/geometry/solids/61174.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.542 & 0.135 & 0.559 \\
+ 0.959 & 0.741 & 0.082 \\
+ 0.851 & 0.273 & 0.185 \\
+ 0.858 & 0.406 & 0.381 \\
+ 0.585 & 0.299 & 0.184 \\
+ 0.624 & 0.639 & 1. \\
+ 0.175 & 0.4 & 0.168 \\
+ 0.347 & 0.368 & 0.537 \\
+ 0.511 & 0.016 & 0.284 \\
+ 0.308 & 0.847 & 0.795 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.72$ 
+Solid Angle: $2.62$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/62143.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.517 & 0.972 & 0.485 \\
+ 0.32 & 0.386 & 0.296 \\
+ 0.344 & 0.916 & 0.176 \\
+ 0.996 & 0.051 & 0.528 \\
+ 0.37 & 0.956 & 0.831 \\
+ 0.032 & 0.256 & 0.265 \\
+\end{array}
+\right)$. 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: $2.69$ 
+Volume: $0.1$ 
+Surface Area: $1.5$

pretraining/mathematica/geometry/solids/63012.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.704 & 0.813 & 0.734 \\
+ 0.854 & 0.313 & 0.792 \\
+ 0.107 & 0.006 & 0.561 \\
+ 0.738 & 0.208 & 0.434 \\
+ 0.13 & 0.899 & 0.216 \\
+\end{array}
+\right)$. 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: $0.07$ 
+Surface Area: $1.36$ 
+Solid Angle: $0.69$

pretraining/mathematica/geometry/solids/64021.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.853 & 0.276 & 0.02 \\
+ 0.436 & 0.287 & 0.734 \\
+ 0.594 & 0.672 & 0.733 \\
+ 0.244 & 0.591 & 0.079 \\
+\end{array}
+\right)$. 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: $0.27$ 
+Volume: $0.03$ 
+Surface Area: $0.81$

pretraining/mathematica/geometry/solids/64564.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.738 & 0.79 & 0.138 \\
+ 0.144 & 0.498 & 0.728 \\
+ 0.94 & 0.98 & 0.737 \\
+ 0.961 & 0.496 & 0.862 \\
+ 0.841 & 0.637 & 0.418 \\
+\end{array}
+\right)$. 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: $0.98$ 
+Volume: $0.05$ 
+Solid Angle: $0.46$

pretraining/mathematica/geometry/solids/65494.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.066 & 0.473 & 0.176 \\
+ 0.289 & 0.569 & 0.815 \\
+ 0.579 & 0.281 & 0.855 \\
+ 0.944 & 0.916 & 0.812 \\
+ 0.655 & 0.259 & 0.826 \\
+ 0.444 & 0.949 & 0.98 \\
+ 0.184 & 0.348 & 0.067 \\
+ 0.899 & 0.466 & 0.073 \\
+\end{array}
+\right)$. 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: $1.92$ 
+Solid Angle: $1.09$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/68070.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.94 & 0.496 & 0.403 \\
+ 0.31 & 0.462 & 0.03 \\
+ 0.699 & 0.77 & 0.269 \\
+ 0.114 & 0.473 & 0.869 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.49$ 
+Volume: $0.03$ 
+Surface Area: $0.81$