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pretraining/mathematica/geometry/solids/10368.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.03 & 0.78 & 1. \\
+ 0.565 & 0.054 & 0.636 \\
+ 0.522 & 0.74 & 0.745 \\
+ 0.485 & 0.164 & 0.891 \\
+ 0.792 & 0.149 & 0.454 \\
+ 0.253 & 0.798 & 0.57 \\
+ 0.867 & 0.46 & 0.248 \\
+ 0.219 & 0.05 & 0.089 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.78$ 
+Solid Angle: $0.63$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/10686.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.275 & 0.971 & 0.773 \\
+ 0.199 & 0.556 & 0.868 \\
+ 0.301 & 0.985 & 0.267 \\
+ 0.893 & 0.244 & 0.898 \\
+ 0.043 & 0.948 & 0.532 \\
+ 0.406 & 0.034 & 0.962 \\
+ 0.061 & 0.078 & 0.631 \\
+ 0.758 & 0.152 & 0.063 \\
+ 0.918 & 0.827 & 0.467 \\
+\end{array}
+\right)$. 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: $0.31$ 
+Solid Angle: $2.43$ 
+Surface Area: $2.67$

pretraining/mathematica/geometry/solids/13767.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.557 & 0.321 & 0.973 \\
+ 0.979 & 0.204 & 0.257 \\
+ 0.333 & 0.768 & 0.521 \\
+ 0.633 & 0.353 & 0.492 \\
+ 0.557 & 0.558 & 0.418 \\
+\end{array}
+\right)$. 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: $0.56$ 
+Volume: $0.01$ 
+Solid Angle: $0.1$

pretraining/mathematica/geometry/solids/17650.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.825 & 0.855 & 0.906 \\
+ 0.204 & 0.49 & 0.584 \\
+ 0.587 & 0.151 & 0.064 \\
+ 0.021 & 0.795 & 0.105 \\
+ 0.156 & 0.682 & 0.414 \\
+ 0.623 & 0.883 & 0.239 \\
+ 0.903 & 0.104 & 0.265 \\
+ 0.948 & 0.913 & 0.529 \\
+ 0.672 & 0.475 & 0.111 \\
+ 0.511 & 0.545 & 0.74 \\
+ 0.239 & 0.146 & 0.082 \\
+\end{array}
+\right)$. 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: $2.11$ 
+Solid Angle: $0.98$ 
+Volume: $0.21$

pretraining/mathematica/geometry/solids/1858.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.794 & 0.135 & 0.852 \\
+ 0.421 & 0.52 & 0.082 \\
+ 0.985 & 0.253 & 0.396 \\
+ 0.92 & 0.488 & 0.521 \\
+ 0.265 & 0.102 & 0.378 \\
+ 0.127 & 0.566 & 0.589 \\
+ 0.602 & 0.909 & 0.715 \\
+ 0.328 & 0.416 & 0.75 \\
+ 0.001 & 0.959 & 0.295 \\
+\end{array}
+\right)$. 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: $1.93$ 
+Solid Angle: $1.29$ 
+Volume: $0.18$

pretraining/mathematica/geometry/solids/20792.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.206 & 0.631 & 0.031 \\
+ 0.971 & 0.372 & 0.637 \\
+ 0.95 & 0.172 & 0.581 \\
+ 0.662 & 0.98 & 0.165 \\
+ 0.31 & 0.339 & 0.145 \\
+ 0.776 & 0.582 & 0.918 \\
+ 0.467 & 0.734 & 0.47 \\
+ 0.61 & 0.977 & 0.371 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $0.67$ 
+Surface Area: $1.36$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/22423.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.627 & 0.972 & 0.36 \\
+ 0.671 & 0.223 & 0.785 \\
+ 0.473 & 0.264 & 0.467 \\
+ 0.869 & 0.583 & 0. \\
+ 0.31 & 0.347 & 0.469 \\
+ 0.734 & 0.502 & 0.99 \\
+ 0.8 & 0.655 & 0.919 \\
+ 0.463 & 0.947 & 0.834 \\
+ 0.95 & 0.813 & 0.336 \\
+ 0.304 & 0.67 & 0.316 \\
+\end{array}
+\right)$. 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: $1.61$ 
+Volume: $0.15$ 
+Solid Angle: $2.7$

pretraining/mathematica/geometry/solids/22491.txt

@@ -0,0 +1,55 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.5 & 1.207 & -1.914 \\
+ -0.5 & 1.207 & 1.914 \\
+ -0.5 & -1.207 & -1.914 \\
+ -0.5 & -1.207 & 1.914 \\
+ -0.5 & -1.914 & 1.207 \\
+ -0.5 & -1.914 & -1.207 \\
+ -0.5 & 1.914 & 1.207 \\
+ -0.5 & 1.914 & -1.207 \\
+ 0.5 & 1.207 & -1.914 \\
+ 0.5 & 1.207 & 1.914 \\
+ 0.5 & -1.207 & -1.914 \\
+ 0.5 & -1.207 & 1.914 \\
+ 0.5 & -1.914 & 1.207 \\
+ 0.5 & -1.914 & -1.207 \\
+ 0.5 & 1.914 & 1.207 \\
+ 0.5 & 1.914 & -1.207 \\
+ 1.207 & -0.5 & -1.914 \\
+ 1.207 & -0.5 & 1.914 \\
+ 1.207 & 0.5 & -1.914 \\
+ 1.207 & 0.5 & 1.914 \\
+ 1.207 & -1.914 & -0.5 \\
+ 1.207 & -1.914 & 0.5 \\
+ 1.207 & 1.914 & -0.5 \\
+ 1.207 & 1.914 & 0.5 \\
+ -1.207 & -0.5 & -1.914 \\
+ -1.207 & -0.5 & 1.914 \\
+ -1.207 & 0.5 & -1.914 \\
+ -1.207 & 0.5 & 1.914 \\
+ -1.207 & -1.914 & -0.5 \\
+ -1.207 & -1.914 & 0.5 \\
+ -1.207 & 1.914 & -0.5 \\
+ -1.207 & 1.914 & 0.5 \\
+ -1.914 & -0.5 & 1.207 \\
+ -1.914 & -0.5 & -1.207 \\
+ -1.914 & 0.5 & 1.207 \\
+ -1.914 & 0.5 & -1.207 \\
+ -1.914 & 1.207 & -0.5 \\
+ -1.914 & 1.207 & 0.5 \\
+ -1.914 & -1.207 & -0.5 \\
+ -1.914 & -1.207 & 0.5 \\
+ 1.914 & -0.5 & 1.207 \\
+ 1.914 & -0.5 & -1.207 \\
+ 1.914 & 0.5 & 1.207 \\
+ 1.914 & 0.5 & -1.207 \\
+ 1.914 & 1.207 & -0.5 \\
+ 1.914 & 1.207 & 0.5 \\
+ 1.914 & -1.207 & -0.5 \\
+ 1.914 & -1.207 & 0.5 \\
+\end{array}
+\right)$. Determine the Circumdiameter.
+Answer:
+$4.64$

pretraining/mathematica/geometry/solids/23336.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.461 & 0.735 & 0.609 \\
+ 0.681 & 0.641 & 0.331 \\
+ 0.781 & 0.671 & 0.742 \\
+ 0.24 & 0.758 & 0.964 \\
+ 0.869 & 0.334 & 0.148 \\
+\end{array}
+\right)$. 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: $0.51$ 
+Solid Angle: $2.09$

pretraining/mathematica/geometry/solids/24741.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.335 & 0.95 & 0.774 \\
+ 0.091 & 0.322 & 0.161 \\
+ 0.464 & 0.354 & 0.663 \\
+ 0.342 & 0.031 & 0.339 \\
+ 0.016 & 0.133 & 0.858 \\
+ 0.893 & 0.922 & 0.528 \\
+\end{array}
+\right)$. 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: $1.54$ 
+Solid Angle: $0.86$

pretraining/mathematica/geometry/solids/25632.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.281 & 0.46 & 0.751 \\
+ 0.734 & 0.3 & 0.839 \\
+ 0.279 & 0.167 & 0.543 \\
+ 0.637 & 0.751 & 0.362 \\
+ 0.08 & 0.703 & 0.606 \\
+ 0.711 & 0.204 & 0.954 \\
+ 0.11 & 0.188 & 0.614 \\
+ 0.807 & 0.405 & 0.158 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.22$ 
+Solid Angle: $4.93$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/29166.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.803 & 0.414 & 0.937 \\
+ 0.14 & 0.197 & 0.725 \\
+ 0.022 & 0.324 & 0.031 \\
+ 0.206 & 0.984 & 0.635 \\
+ 0.933 & 0.376 & 0.365 \\
+ 0.859 & 0.069 & 0.759 \\
+ 0.104 & 0.651 & 0.728 \\
+ 0.476 & 0.877 & 0.943 \\
+ 0.262 & 0.413 & 0.154 \\
+\end{array}
+\right)$. 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: $0.21$ 
+Surface Area: $2.15$ 
+Solid Angle: $2.14$

pretraining/mathematica/geometry/solids/32323.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.776 & 0.805 & 0.586 \\
+ 0.574 & 0.928 & 0.518 \\
+ 0.691 & 0.698 & 0.461 \\
+ 0.651 & 0.073 & 0.297 \\
+ 0.921 & 0.018 & 0.81 \\
+ 0.706 & 0.154 & 0.334 \\
+ 0.159 & 0.859 & 0.435 \\
+ 0.267 & 0.324 & 0.274 \\
+ 0.063 & 0.349 & 0.408 \\
+\end{array}
+\right)$. 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: $1.14$ 
+Surface Area: $1.45$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/33307.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.88 & 0.179 & 0.009 \\
+ 0.663 & 0.098 & 0.79 \\
+ 0.725 & 0.16 & 0.707 \\
+ 0.483 & 0.895 & 0.665 \\
+ 0.066 & 0.624 & 0.086 \\
+ 0.678 & 0.907 & 0.613 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.53$ 
+Volume: $0.1$ 
+Solid Angle: $0.55$

pretraining/mathematica/geometry/solids/33610.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.741 & 0.307 & 0.615 \\
+ 0.423 & 0.186 & 0.856 \\
+ 0.495 & 0.507 & 0.013 \\
+ 0.867 & 0.154 & 0.228 \\
+ 0.107 & 0.545 & 0.769 \\
+ 0.396 & 0.254 & 0.679 \\
+ 0.438 & 0.164 & 0.985 \\
+ 0.928 & 0.626 & 0.241 \\
+\end{array}
+\right)$. 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: $1.3$ 
+Volume: $0.08$ 
+Solid Angle: $4.19$

pretraining/mathematica/geometry/solids/34050.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.673 & 1. & 0.956 \\
+ 0.767 & 0.107 & 0.009 \\
+ 0.036 & 0.809 & 0.023 \\
+ 0.038 & 0.814 & 0.837 \\
+\end{array}
+\right)$. 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: $1.66$ 
+Solid Angle: $0.35$

pretraining/mathematica/geometry/solids/34947.txt

@@ -0,0 +1,29 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & 1.401 \\
+ 0. & 0. & -1.401 \\
+ 0.178 & -1.309 & 0.467 \\
+ 0.178 & 1.309 & 0.467 \\
+ 0.467 & -0.809 & -1.044 \\
+ 0.467 & 0.809 & -1.044 \\
+ 1.044 & -0.809 & 0.467 \\
+ 1.044 & 0.809 & 0.467 \\
+ -1.223 & -0.5 & 0.467 \\
+ -1.223 & 0.5 & 0.467 \\
+ 1.223 & -0.5 & -0.467 \\
+ 1.223 & 0.5 & -0.467 \\
+ -0.934 & 0. & -1.044 \\
+ -0.467 & -0.809 & 1.044 \\
+ -0.467 & 0.809 & 1.044 \\
+ 0.934 & 0. & 1.044 \\
+ -1.044 & -0.809 & -0.467 \\
+ -1.044 & 0.809 & -0.467 \\
+ 0.995 & 0. & -1.303 \\
+ -0.995 & 0. & 1.303 \\
+ -0.178 & -1.309 & -0.467 \\
+ -0.178 & 1.309 & -0.467 \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$3.28$

pretraining/mathematica/geometry/solids/35237.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.99 & 0.764 & 0.18 \\
+ 0.903 & 0.807 & 0.544 \\
+ 0.995 & 0.339 & 0.912 \\
+ 0.854 & 0.705 & 0.318 \\
+ 0.998 & 0.118 & 0.579 \\
+ 0.398 & 0.327 & 0.404 \\
+ 0.637 & 0.242 & 0.244 \\
+\end{array}
+\right)$. 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: $1.05$ 
+Solid Angle: $0.8$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/36062.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.791 & 0.949 & 0.31 \\
+ 0.151 & 0.708 & 0.665 \\
+ 0.591 & 0.197 & 0.83 \\
+ 0.113 & 0.744 & 0.732 \\
+ 0.453 & 0.391 & 0.265 \\
+ 0.691 & 0.136 & 0.756 \\
+\end{array}
+\right)$. 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: $1.$ 
+Volume: $0.05$ 
+Solid Angle: $0.38$

pretraining/mathematica/geometry/solids/36643.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.116 & 0.701 & 0.979 \\
+ 0.206 & 0.602 & 0.576 \\
+ 0.949 & 0.588 & 0.384 \\
+ 0.711 & 0.359 & 0.441 \\
+ 0.456 & 0.911 & 0.33 \\
+ 0.545 & 0.07 & 0.099 \\
+ 0.917 & 0.454 & 0.211 \\
+ 0.32 & 0.837 & 0.038 \\
+ 0.667 & 0.84 & 0.993 \\
+\end{array}
+\right)$. 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: $0.72$ 
+Surface Area: $1.87$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/38480.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.139 & 0.51 & 0.054 \\
+ 0.886 & 0.648 & 0.529 \\
+ 0.976 & 0.012 & 0.696 \\
+ 0.908 & 0.839 & 0.953 \\
+ 0.9 & 0.347 & 0.445 \\
+ 0.636 & 0.112 & 0.674 \\
+ 0.026 & 0.396 & 0.701 \\
+ 0.237 & 0.988 & 0.279 \\
+\end{array}
+\right)$. 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: $1.17$ 
+Surface Area: $2.16$

pretraining/mathematica/geometry/solids/39780.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1. & 0. & 0. \\
+ -0.5 & -0.5 & -0.707 \\
+ -0.5 & -0.5 & 0.707 \\
+ -0.5 & 0.5 & -0.707 \\
+ -0.5 & 0.5 & 0.707 \\
+ 0. & -1. & 0. \\
+ 0. & 1. & 0. \\
+ 0.5 & -0.5 & -0.707 \\
+ 0.5 & -0.5 & 0.707 \\
+ 0.5 & 0.5 & -0.707 \\
+ 0.5 & 0.5 & 0.707 \\
+ 1. & 0. & 0. \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$24.$

pretraining/mathematica/geometry/solids/41329.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.132 & 0.488 & 0.539 \\
+ 0.104 & 0.136 & 0.153 \\
+ 0.522 & 0.854 & 0.002 \\
+ 0.085 & 0.198 & 0.56 \\
+ 0.095 & 0.071 & 0.73 \\
+\end{array}
+\right)$. 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: $0.66$ 
+Solid Angle: $1.17$

pretraining/mathematica/geometry/solids/43592.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.204 & 0.263 & 0.967 \\
+ 0.323 & 0.761 & 0.88 \\
+ 0.819 & 0.943 & 0.069 \\
+ 0.201 & 0.877 & 0.835 \\
+ 0.269 & 0.892 & 0.357 \\
+ 0.482 & 0.536 & 0.263 \\
+\end{array}
+\right)$. 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: $1.05$ 
+Solid Angle: $0.28$

pretraining/mathematica/geometry/solids/45921.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.779 & 0.001 & 0.318 \\
+ 0.325 & 0.668 & 0.022 \\
+ 0.075 & 0.585 & 0.239 \\
+ 0.532 & 0.979 & 0.419 \\
+ 0.343 & 0.898 & 0.904 \\
+ 0.967 & 0.727 & 0.591 \\
+ 0.781 & 0.252 & 0.353 \\
+ 0.353 & 0.531 & 0.956 \\
+ 0.255 & 0.973 & 0.374 \\
+\end{array}
+\right)$. 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: $1.94$ 
+Volume: $0.19$ 
+Solid Angle: $0.67$

pretraining/mathematica/geometry/solids/46955.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.416 & 0.811 & 0.809 \\
+ 0.114 & 0.924 & 0.529 \\
+ 0.634 & 0.13 & 0.989 \\
+ 0.82 & 0.461 & 0.39 \\
+ 0.403 & 0.012 & 0.008 \\
+ 0.702 & 0.727 & 0.613 \\
+ 0.923 & 0.26 & 0.332 \\
+ 0.736 & 0.666 & 0.678 \\
+ 0.389 & 0.71 & 0.434 \\
+\end{array}
+\right)$. 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:
+Volume: $0.14$ 
+Surface Area: $1.75$ 
+Solid Angle: $2.16$

pretraining/mathematica/geometry/solids/48255.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.202 & 0.218 & 0.782 \\
+ 0.137 & 0.772 & 0.577 \\
+ 0.49 & 0.566 & 0.651 \\
+ 0.338 & 0.769 & 0.927 \\
+ 0.954 & 0.949 & 0.66 \\
+ 0.868 & 0.367 & 0.918 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.1$ 
+Volume: $0.06$ 
+Solid Angle: $0.55$

pretraining/mathematica/geometry/solids/49284.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.522 & 0.338 & 0.211 \\
+ 0.407 & 0.526 & 0.891 \\
+ 0.653 & 0.259 & 0.451 \\
+ 0.804 & 0.278 & 0.233 \\
+\end{array}
+\right)$. 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: $0.25$ 
+Solid Angle: $0.33$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/53036.txt

@@ -0,0 +1,39 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & -\frac{1}{2} \sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)} \\
+ 0 & 0 & \sqrt{\frac{9}{8}+\frac{3 \sqrt{5}}{8}} \\
+ \sqrt{\frac{1}{8}-\frac{\sqrt{5}}{24}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} \\
+ \sqrt{\frac{1}{8}-\frac{\sqrt{5}}{24}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{5 \sqrt{5}}{24}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{5 \sqrt{5}}{24}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} \\
+ \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} \\
+ \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} \\
+ -\sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2 \sqrt{3}} \\
+ -\sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2 \sqrt{3}} \\
+ \sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2 \sqrt{3}} \\
+ \sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2 \sqrt{3}} \\
+ -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & 0 & -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & 0 & \sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} \\
+ \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & 0 & \sqrt{\frac{5}{8}+\frac{5 \sqrt{5}}{24}} \\
+ \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & 0 & -\sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} \\
+ -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{5}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{5 \sqrt{5}}{24}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} \\
+ \sqrt{\frac{5}{8}+\frac{5 \sqrt{5}}{24}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} \\
+ -\sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} & -\frac{1}{2} & \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} \\
+ -\sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} & \frac{1}{2} & \sqrt{\frac{1}{8}+\frac{\sqrt{5}}{24}} \\
+ \sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ \sqrt{\frac{3}{4}+\frac{\sqrt{5}}{3}} & \frac{1}{2} & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\sqrt{\frac{7}{6}+\frac{\sqrt{5}}{2}} & 0 & -\frac{1}{2 \sqrt{3}} \\
+ \sqrt{\frac{1}{6} \left(7+3 \sqrt{5}\right)} & 0 & \frac{1}{2 \sqrt{3}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$60$

pretraining/mathematica/geometry/solids/53318.txt

@@ -0,0 +1,30 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & 1.401 \\
+ 0. & 0. & -1.401 \\
+ 0.178 & -1.309 & 0.467 \\
+ 0.178 & 1.309 & 0.467 \\
+ 0.467 & -0.809 & -1.044 \\
+ 0.467 & 0.809 & -1.044 \\
+ 1.044 & -0.809 & 0.467 \\
+ 1.044 & 0.809 & 0.467 \\
+ -1.223 & -0.5 & 0.467 \\
+ -1.223 & 0.5 & 0.467 \\
+ 1.223 & -0.5 & -0.467 \\
+ 1.223 & 0.5 & -0.467 \\
+ -0.934 & 0. & -1.044 \\
+ -0.467 & -0.809 & 1.044 \\
+ -0.467 & 0.809 & 1.044 \\
+ 0.934 & 0. & 1.044 \\
+ -1.044 & -0.809 & -0.467 \\
+ -1.044 & 0.809 & -0.467 \\
+ -0.995 & 0. & 1.303 \\
+ -0.178 & -1.309 & -0.467 \\
+ -0.178 & 1.309 & -0.467 \\
+ 0.805 & 1.394 & -0.308 \\
+ 0.805 & -1.394 & -0.308 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$45.$

pretraining/mathematica/geometry/solids/55449.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.914 & 0.027 & 0.8 \\
+ 0.874 & 0.128 & 0.436 \\
+ 0.266 & 0.268 & 0.182 \\
+ 0.19 & 0.827 & 0.986 \\
+ 0.244 & 0.832 & 0.918 \\
+ 0.175 & 0.593 & 0.342 \\
+ 0.454 & 0.181 & 0.119 \\
+ 0.091 & 0.051 & 0.572 \\
+ 0.45 & 0.633 & 0.702 \\
+\end{array}
+\right)$. 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: $1.73$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/56107.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.145 & 0.109 & 0.912 \\
+ 0.304 & 0.518 & 0.861 \\
+ 0.456 & 0.457 & 0.488 \\
+ 0.531 & 0.93 & 0.251 \\
+ 0.453 & 0.878 & 0.735 \\
+ 0.205 & 0.223 & 0.419 \\
+ 0.754 & 0.787 & 0.39 \\
+ 0.057 & 0.114 & 0.438 \\
+ 0.002 & 0.463 & 0.066 \\
+ 0.026 & 0.86 & 0.922 \\
+\end{array}
+\right)$. 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: $1.81$ 
+Volume: $0.16$ 
+Solid Angle: $0.87$

pretraining/mathematica/geometry/solids/57520.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.167 & 0.26 & 0.907 \\
+ 0.013 & 0.623 & 0.44 \\
+ 0.811 & 0.703 & 0.792 \\
+ 0.367 & 0.051 & 0.734 \\
+ 0.048 & 0.295 & 0.366 \\
+ 0.59 & 0.572 & 0.11 \\
+ 0.936 & 0.642 & 0.504 \\
+ 0.941 & 0.491 & 0.43 \\
+ 0.851 & 0.295 & 0.874 \\
+ 0.805 & 0.458 & 0.261 \\
+ 0.293 & 0.292 & 0.994 \\
+\end{array}
+\right)$. 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: $2.51$ 
+Surface Area: $1.8$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/57665.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.224 & 0.566 & 0.795 \\
+ 0.074 & 0.502 & 0.742 \\
+ 0.204 & 0.008 & 0.392 \\
+ 0.303 & 0.959 & 0.981 \\
+\end{array}
+\right)$. 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: $4.11$ 
+Volume: $0.$ 
+Surface Area: $0.22$

pretraining/mathematica/geometry/solids/61433.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.413 & 0.043 & 0.387 \\
+ 0.058 & 0.272 & 0.165 \\
+ 0.09 & 0.414 & 0.478 \\
+ 0.105 & 0.947 & 0.869 \\
+ 0.948 & 0.552 & 0.693 \\
+ 0.844 & 0.713 & 0.101 \\
+\end{array}
+\right)$. 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: $1.36$ 
+Surface Area: $1.77$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/61647.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.752 & 0.405 & 0.676 \\
+ 0.541 & 0.961 & 0.627 \\
+ 0.899 & 0.621 & 0.79 \\
+ 0.995 & 0.902 & 0.274 \\
+ 0.396 & 0.844 & 0.757 \\
+ 0.715 & 0.803 & 0.191 \\
+ 0.254 & 0.036 & 0.781 \\
+ 0.217 & 0.695 & 0.896 \\
+ 0.936 & 0.768 & 0.953 \\
+ 0.389 & 0.497 & 0.939 \\
+ 0.085 & 0.005 & 0.521 \\
+\end{array}
+\right)$. 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: $1.89$ 
+Solid Angle: $4.01$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/61816.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-0.974,-4.014,-8.585\}$ has radius $0.299$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $1.13$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/6244.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.101 & 0.316 & 0.156 \\
+ 0.992 & 0.163 & 0.313 \\
+ 0.271 & 0.179 & 0.142 \\
+ 0.455 & 0.416 & 0.902 \\
+ 0.852 & 0.161 & 0.815 \\
+ 0.118 & 0.445 & 0.359 \\
+ 0.878 & 0.222 & 0.629 \\
+ 0.354 & 0.251 & 0.81 \\
+\end{array}
+\right)$. 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: $0.71$

pretraining/mathematica/geometry/solids/63927.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.685 & 0.87 & 0.688 \\
+ 0.227 & 0.325 & 0.821 \\
+ 0.322 & 0.368 & 0.772 \\
+ 0.072 & 0.982 & 0.913 \\
+ 0.937 & 0.541 & 0.332 \\
+\end{array}
+\right)$. 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: $0.83$ 
+Volume: $0.02$ 
+Solid Angle: $0.86$

pretraining/mathematica/geometry/solids/64674.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.862 & 0.798 & 0.113 \\
+ 0.371 & 0.334 & 0.499 \\
+ 0.378 & 0.252 & 0.404 \\
+ 0.454 & 0.732 & 0.105 \\
+ 0.409 & 0.252 & 0.473 \\
+ 0.191 & 0.733 & 0.082 \\
+ 0.73 & 0.8 & 0.437 \\
+\end{array}
+\right)$. 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:
+Surface Area: $0.68$ 
+Volume: $0.03$ 
+Solid Angle: $0.5$

pretraining/mathematica/geometry/solids/65920.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.288 & 0.85 & 0.627 \\
+ 0.641 & 0.594 & 0.788 \\
+ 0.876 & 0.335 & 0.22 \\
+ 0.752 & 0.05 & 0.053 \\
+ 0.176 & 0.805 & 0.209 \\
+ 0.845 & 0.026 & 0.216 \\
+ 0.127 & 0.421 & 0.493 \\
+ 0.073 & 0.695 & 0.087 \\
+\end{array}
+\right)$. 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: $1.48$ 
+Solid Angle: $1.3$

pretraining/mathematica/geometry/solids/65987.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.67 & 0.62 & 0.237 \\
+ 0.788 & 0.045 & 0.207 \\
+ 0.969 & 0.306 & 0.789 \\
+ 0.622 & 0.088 & 0.804 \\
+ 0.267 & 0.4 & 0.748 \\
+ 0.33 & 0.703 & 0.397 \\
+ 0.94 & 0.163 & 0.377 \\
+\end{array}
+\right)$. 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: $1.18$ 
+Volume: $0.08$ 
+Solid Angle: $1.52$

pretraining/mathematica/geometry/solids/66120.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.029 & 0.139 & 0.631 \\
+ 0.091 & 0.617 & 0.676 \\
+ 0.174 & 0.181 & 0.879 \\
+ 0.608 & 0.813 & 0.008 \\
+ 0.982 & 0.999 & 0.659 \\
+ 0.59 & 0.44 & 0.791 \\
+ 0.938 & 0.84 & 0.87 \\
+ 0.106 & 0.137 & 0.2 \\
+ 0.897 & 0.92 & 0.107 \\
+\end{array}
+\right)$. 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: $1.86$ 
+Volume: $0.16$ 
+Surface Area: $2.17$

pretraining/mathematica/geometry/solids/66128.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.539 & 0.679 & 0.897 \\
+ 0.319 & 0.713 & 0.611 \\
+ 0.976 & 0.842 & 0.512 \\
+ 0.859 & 0.036 & 0.688 \\
+ 0.256 & 0.045 & 0.669 \\
+ 0.381 & 0.17 & 0.878 \\
+ 0.73 & 0.066 & 0.081 \\
+ 0.894 & 0.452 & 0.791 \\
+\end{array}
+\right)$. 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: $1.74$ 
+Solid Angle: $2.35$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/67111.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.422 & 0.6 & 0.663 \\
+ 0.204 & 0.927 & 0.751 \\
+ 0.06 & 0.472 & 0.291 \\
+ 0.935 & 0.887 & 0.137 \\
+ 0.424 & 0.105 & 0.155 \\
+ 0.987 & 0.189 & 0.151 \\
+ 0.975 & 0.275 & 0.304 \\
+ 0.231 & 0.404 & 0.61 \\
+\end{array}
+\right)$. 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: $4.3$ 
+Volume: $0.13$ 
+Surface Area: $1.75$

pretraining/mathematica/geometry/solids/6762.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.405 & 0.439 & 0.042 \\
+ 0.211 & 0.997 & 0.285 \\
+ 0.109 & 0.598 & 0.802 \\
+ 0.011 & 0.038 & 0.456 \\
+ 0.52 & 0.061 & 0.699 \\
+ 0.983 & 0.62 & 0.701 \\
+ 0.534 & 0.19 & 0.053 \\
+ 0.697 & 0.084 & 0.728 \\
+\end{array}
+\right)$. 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: $2.12$ 
+Solid Angle: $2.64$ 
+Volume: $0.22$

pretraining/mathematica/geometry/solids/6894.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & -\frac{1}{2} \sqrt{\frac{1}{6} \left(23+12 \sqrt{2}+\sqrt{5 \left(41+24 \sqrt{2}\right)}\right)} \\
+ -\frac{1}{\sqrt{3}} & 0 & -\sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & \sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & \sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} \\
+ \frac{1}{2 \sqrt{3}} & -\frac{1}{2} & -\sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} \\
+ \frac{1}{2 \sqrt{3}} & \frac{1}{2} & -\sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} \\
+ \frac{1}{\sqrt{3}} & 0 & \sqrt{\frac{7}{24}+\frac{\sqrt{5}}{8}} \\
+ -\sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & 0 & \frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{6} \left(3+\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{6} \left(3-\sqrt{5}\right)} \\
+\end{array}
+\right)$. Determine the FaceCount.
+Answer:
+$10$

pretraining/mathematica/geometry/solids/71921.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.582 & 0.72 & 0.008 \\
+ 0.441 & 0.709 & 0.166 \\
+ 0.773 & 0.153 & 0.906 \\
+ 0.758 & 0.341 & 0.535 \\
+\end{array}
+\right)$. 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: $0.27$ 
+Solid Angle: $0.01$

pretraining/mathematica/geometry/solids/74787.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.121 & 0.965 & 0.524 \\
+ 0.798 & 0.36 & 0.728 \\
+ 0.318 & 0.699 & 0.917 \\
+ 0.84 & 0.113 & 0.318 \\
+ 0.903 & 0.929 & 0.403 \\
+ 0.534 & 0.932 & 0.012 \\
+ 0.191 & 0.005 & 0.113 \\
+ 0.76 & 0.37 & 0.066 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.27$ 
+Surface Area: $2.36$ 
+Solid Angle: $1.54$

pretraining/mathematica/geometry/solids/76729.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.94 & 0.428 & 0.85 \\
+ 0.541 & 0.746 & 0.673 \\
+ 0.261 & 0.032 & 0.642 \\
+ 0.246 & 0.716 & 0.533 \\
+ 0.139 & 0.205 & 0.432 \\
+ 0.73 & 0.163 & 0.709 \\
+ 0.897 & 0.34 & 0.466 \\
+ 0.229 & 0.069 & 0.479 \\
+\end{array}
+\right)$. 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: $0.91$ 
+Surface Area: $1.09$