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pretraining/mathematica/geometry/solids/10585.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.979 & 0.858 & 0.847 \\
+ 0.441 & 0.675 & 0.254 \\
+ 0.997 & 0.789 & 0.142 \\
+ 0.347 & 0.404 & 0.2 \\
+ 0.39 & 0.381 & 0.399 \\
+ 0.864 & 0.885 & 0.072 \\
+ 0.482 & 0.408 & 0.321 \\
+ 0.302 & 0.491 & 0.803 \\
+ 0.347 & 0.889 & 0.758 \\
+\end{array}
+\right)$. 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: $0.7$ 
+Surface Area: $1.42$

pretraining/mathematica/geometry/solids/10878.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.479 & 0.267 & 0.695 \\
+ 0.629 & 0.895 & 0.156 \\
+ 0.65 & 0.792 & 0.888 \\
+ 0.854 & 0.814 & 0.934 \\
+ 0.134 & 0.116 & 0.654 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $0.95$ 
+Volume: $0.03$ 
+Solid Angle: $2.24$

pretraining/mathematica/geometry/solids/1113.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.893 & 0.95 & 0.642 \\
+ 0.547 & 0.608 & 0.379 \\
+ 0.414 & 0.203 & 0.241 \\
+ 0.974 & 0.22 & 0.582 \\
+ 0.848 & 0.575 & 0.607 \\
+ 0.195 & 0.722 & 0.226 \\
+\end{array}
+\right)$. 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: $0.01$ 
+Surface Area: $0.96$ 
+Solid Angle: $0.1$

pretraining/mathematica/geometry/solids/11422.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.652 & 0.683 & 0.227 \\
+ 0.912 & 0.241 & 0.273 \\
+ 0.005 & 0.059 & 0.317 \\
+ 0.309 & 0.184 & 0.093 \\
+ 0.089 & 0.516 & 0.43 \\
+ 0.442 & 0.472 & 0.905 \\
+ 0.324 & 0.914 & 0.403 \\
+ 0.834 & 0.695 & 0.85 \\
+\end{array}
+\right)$. 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: $1.8$ 
+Solid Angle: $2.94$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/15145.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.733 & 0.004 & 0.275 \\
+ 0.145 & 0.247 & 0.028 \\
+ 0.623 & 0.646 & 0.973 \\
+ 0.998 & 0.247 & 0.683 \\
+ 0.485 & 0.271 & 0.849 \\
+ 0.009 & 0.957 & 0.619 \\
+\end{array}
+\right)$. 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: $0.68$ 
+Volume: $0.1$ 
+Surface Area: $1.84$

pretraining/mathematica/geometry/solids/15492.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.747 & 0.07 & 0.497 \\
+ 0.148 & 0.149 & 0.185 \\
+ 0.062 & 0.191 & 0.419 \\
+ 0.424 & 0.33 & 0.904 \\
+\end{array}
+\right)$. 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: $0.51$ 
+Solid Angle: $0.07$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/16209.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $5.741$ has its base centered at$\{8.849,5.38,7.007\}$ and its tip is at $\{3.604,6.857,1.306\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Volume: $272.17$ 
+Surface Area: $279.47$ 
+Centroid: $\{7.54,5.75,5.58\}$

pretraining/mathematica/geometry/solids/18313.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.579 & 0.758 & 0.024 \\
+ 0.977 & 0.758 & 0.694 \\
+ 0.223 & 0.152 & 0.817 \\
+ 0.351 & 0.32 & 0.899 \\
+\end{array}
+\right)$. 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: $0.84$ 
+Solid Angle: $0.05$

pretraining/mathematica/geometry/solids/19070.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.583 & 0.068 & 0.313 \\
+ 0.141 & 0.724 & 0.899 \\
+ 0.516 & 0.899 & 0.051 \\
+ 0.811 & 0.794 & 0.71 \\
+ 0.049 & 0.157 & 0.92 \\
+ 0.872 & 0.287 & 0.87 \\
+\end{array}
+\right)$. 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: $1.21$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/19442.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.605 & 0.718 & 0.296 \\
+ 0.397 & 0.088 & 0.256 \\
+ 0.911 & 0.29 & 0.429 \\
+ 0.418 & 0.883 & 0.22 \\
+ 0.373 & 0.204 & 0.468 \\
+ 0.841 & 0.153 & 0.784 \\
+ 0.612 & 0.788 & 0.811 \\
+ 0.325 & 0.696 & 0.707 \\
+ 0.699 & 0.548 & 0.145 \\
+ 0.301 & 0.677 & 0.905 \\
+\end{array}
+\right)$. 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: $0.13$ 
+Surface Area: $1.53$ 
+Solid Angle: $5.58$

pretraining/mathematica/geometry/solids/20954.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.724 & 0.486 & 0.333 \\
+ 0.09 & 0.108 & 0.532 \\
+ 0.024 & 0.713 & 0.621 \\
+ 0.046 & 0.812 & 0.239 \\
+ 0.985 & 0.874 & 0.568 \\
+ 0.964 & 0.401 & 0.906 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $2.75$ 
+Surface Area: $1.71$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/21971.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.805 & 0.326 & 0.098 \\
+ 0.729 & 0.865 & 0.153 \\
+ 0.331 & 0.702 & 0.862 \\
+ 0.01 & 0.534 & 0.322 \\
+\end{array}
+\right)$. 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: $0.41$ 
+Surface Area: $0.96$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/22405.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.231 & 0.507 & 0.485 \\
+ 0.778 & 0.341 & 0.229 \\
+ 0.82 & 0.059 & 0.37 \\
+ 0.602 & 0.768 & 0.15 \\
+ 0.738 & 0.209 & 0.781 \\
+ 0.286 & 0.719 & 0.602 \\
+ 0.256 & 0.506 & 0.634 \\
+ 0.481 & 0.672 & 0.658 \\
+\end{array}
+\right)$. 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: $2.41$ 
+Surface Area: $0.98$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/22661.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.866 & 0.227 & 0.639 \\
+ 0.603 & 0.94 & 0.315 \\
+ 0.8 & 0.672 & 0.124 \\
+ 0.883 & 0.925 & 0.372 \\
+ 0.37 & 0.077 & 0.036 \\
+ 0.034 & 0.695 & 0.958 \\
+ 0.116 & 0.192 & 0.765 \\
+\end{array}
+\right)$. 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: $1.55$ 
+Surface Area: $2.07$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/2476.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.346 & 0.909 & 0.556 \\
+ 0.92 & 0.913 & 0.499 \\
+ 0.895 & 0.497 & 0.502 \\
+ 0.385 & 0.693 & 0.268 \\
+ 0.406 & 0.12 & 0.208 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.03$ 
+Solid Angle: $0.62$ 
+Surface Area: $0.81$

pretraining/mathematica/geometry/solids/25231.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.722 & 0.631 & 0.927 \\
+ 0.516 & 0.961 & 0.949 \\
+ 0.981 & 0.411 & 0.005 \\
+ 0.571 & 0.033 & 0.614 \\
+ 0.304 & 0.711 & 0.092 \\
+ 0.717 & 0.1 & 0.208 \\
+ 0.982 & 0.782 & 0.283 \\
+ 0.942 & 0.893 & 0.917 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.19$ 
+Surface Area: $2.07$ 
+Solid Angle: $3.72$

pretraining/mathematica/geometry/solids/25820.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.071 & 0.555 & 0.358 \\
+ 0.072 & 0.814 & 0.757 \\
+ 0.805 & 0.442 & 0.479 \\
+ 0.437 & 0.607 & 0.958 \\
+ 0.18 & 0.244 & 0.047 \\
+ 0.458 & 0.171 & 0.038 \\
+ 0.546 & 0.515 & 0.044 \\
+ 0.073 & 0.453 & 0.47 \\
+ 0.478 & 0.728 & 0.791 \\
+\end{array}
+\right)$. 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: $1.38$ 
+Volume: $0.09$ 
+Solid Angle: $3.26$

pretraining/mathematica/geometry/solids/26183.txt

@@ -0,0 +1,6 @@
+Problem:
+A cylinder with radius $5.563$ is around the line from $\{2.97,7.771,8.765\}$ to $\{6.292,3.581,1.204\}$. Estimate the cylinder's surface area, volume, and centroid.
+Answer:
+Surface Area: $518.17$ 
+Volume: $900.43$ 
+Centroid: $\{4.63,5.68,4.98\}$

pretraining/mathematica/geometry/solids/26965.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.262 & 0.678 & 0.731 \\
+ 0.856 & 0.681 & 0.98 \\
+ 0.154 & 0.298 & 0.114 \\
+ 0.882 & 0.687 & 0.608 \\
+ 0.426 & 0.02 & 0.616 \\
+ 0.034 & 0.687 & 0.607 \\
+ 0.996 & 0.217 & 0.217 \\
+ 0.185 & 0.064 & 0.552 \\
+ 0.468 & 0.309 & 0.084 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.17$ 
+Solid Angle: $3.39$ 
+Surface Area: $1.94$

pretraining/mathematica/geometry/solids/27049.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.712 & 0.49 & 0.206 \\
+ 0.295 & 0.258 & 0.514 \\
+ 0.486 & 0.067 & 0.882 \\
+ 0.61 & 0.638 & 0.174 \\
+ 0.427 & 0.743 & 0.72 \\
+\end{array}
+\right)$. 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: $0.91$ 
+Surface Area: $0.71$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/28034.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.919 & 0.408 & 0.776 \\
+ 0.221 & 0.004 & 0.21 \\
+ 0.666 & 0.425 & 0.966 \\
+ 0.24 & 0.741 & 0.954 \\
+ 0.058 & 0.022 & 0.131 \\
+ 0.686 & 0.909 & 0.623 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.42$ 
+Volume: $0.09$ 
+Solid Angle: $1.14$

pretraining/mathematica/geometry/solids/32001.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.112 & 0.975 & 0.954 \\
+ 0.951 & 0.686 & 0.613 \\
+ 0.022 & 0.918 & 0.298 \\
+ 0.057 & 0.415 & 0.325 \\
+ 0.188 & 0.649 & 0.818 \\
+ 0.746 & 0.222 & 0.894 \\
+\end{array}
+\right)$. 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: $0.12$ 
+Solid Angle: $0.88$

pretraining/mathematica/geometry/solids/3209.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.693 & 0.628 & 0.014 \\
+ 0.244 & 0.621 & 0.844 \\
+ 0.319 & 0.288 & 0.124 \\
+ 0.145 & 0.184 & 0.814 \\
+ 0.384 & 0.79 & 0.239 \\
+\end{array}
+\right)$. 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: $0.91$ 
+Solid Angle: $0.43$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/3451.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.301 & 0.117 & 0.706 \\
+ 0.653 & 0.151 & 0.02 \\
+ 0.96 & 0.886 & 0.882 \\
+ 0.263 & 0.423 & 0.458 \\
+ 0.185 & 0.962 & 0.951 \\
+\end{array}
+\right)$. 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:
+Volume: $0.09$ 
+Solid Angle: $1.13$ 
+Surface Area: $1.59$

pretraining/mathematica/geometry/solids/38014.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.132 & 0.654 & 0.894 \\
+ 0.138 & 0.826 & 0.664 \\
+ 0.218 & 0.733 & 0.257 \\
+ 0.186 & 0.811 & 0.42 \\
+\end{array}
+\right)$. 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: $0.11$ 
+Solid Angle: $0.$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/38978.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.298 & 0.735 & 0.645 \\
+ 0.259 & 0.756 & 0.172 \\
+ 0.795 & 0.276 & 0.517 \\
+ 0.301 & 0.36 & 0.082 \\
+ 0.484 & 0.703 & 0.572 \\
+\end{array}
+\right)$. 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: $0.02$ 
+Solid Angle: $0.77$ 
+Surface Area: $0.63$

pretraining/mathematica/geometry/solids/39511.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.385 & 0.643 & 0.947 \\
+ 0.998 & 0.936 & 0.037 \\
+ 0.078 & 0.664 & 0.594 \\
+ 0.783 & 0.342 & 0.323 \\
+ 0.823 & 0.129 & 0.404 \\
+ 0.051 & 0.56 & 0.953 \\
+ 0.065 & 0.514 & 0.875 \\
+ 0.93 & 0.701 & 0.742 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.91$ 
+Surface Area: $1.7$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/41226.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.035 & 0.763 & 0.18 \\
+ 0.179 & 0.921 & 0.34 \\
+ 0.389 & 0.783 & 0.542 \\
+ 0.352 & 0.742 & 0.222 \\
+\end{array}
+\right)$. 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: $0.16$ 
+Solid Angle: $0.21$

pretraining/mathematica/geometry/solids/44625.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.222 & 0.89 & 0.601 \\
+ 0.811 & 0.087 & 0.752 \\
+ 0.095 & 0.337 & 0.992 \\
+ 0.969 & 0.43 & 0.054 \\
+ 0.929 & 0.734 & 0.565 \\
+ 0.388 & 0.134 & 0.059 \\
+\end{array}
+\right)$. 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: $1.13$ 
+Surface Area: $2.1$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/45059.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.525 & 0.319 & 0.989 \\
+ 0.962 & 0.75 & 0.245 \\
+ 0.894 & 0.158 & 0.064 \\
+ 0.124 & 0.897 & 0.015 \\
+ 0.118 & 0.611 & 0.791 \\
+\end{array}
+\right)$. 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: $1.86$ 
+Volume: $0.12$ 
+Solid Angle: $0.56$

pretraining/mathematica/geometry/solids/45958.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.618 & 0.667 & 0.966 \\
+ 0.501 & 0.422 & 0.69 \\
+ 0.276 & 0.479 & 0.082 \\
+ 0.9 & 0.37 & 0.096 \\
+ 0.535 & 0.559 & 0.284 \\
+ 0.251 & 0.858 & 0.887 \\
+ 0.792 & 0.362 & 0.623 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.05$ 
+Surface Area: $1.15$ 
+Solid Angle: $0.81$

pretraining/mathematica/geometry/solids/47217.txt

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

pretraining/mathematica/geometry/solids/49393.txt

@@ -0,0 +1,37 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -0.707 & 0. \\
+ 0. & 0.707 & 0. \\
+ 0.115 & -0.354 & -0.602 \\
+ 0.115 & 0.354 & -0.602 \\
+ 0.186 & -0.572 & 0.372 \\
+ 0.186 & 0.572 & 0.372 \\
+ 0.301 & -0.219 & 0.602 \\
+ 0.301 & 0.219 & 0.602 \\
+ 0.487 & -0.354 & -0.372 \\
+ 0.487 & 0.354 & -0.372 \\
+ -0.602 & 0. & -0.372 \\
+ -0.301 & -0.219 & -0.602 \\
+ -0.301 & 0.219 & -0.602 \\
+ 0.602 & 0. & 0.372 \\
+ 0.416 & -0.572 & 0. \\
+ 0.416 & 0.572 & 0. \\
+ 0.372 & 0. & -0.602 \\
+ -0.416 & -0.572 & 0. \\
+ -0.416 & 0.572 & 0. \\
+ -0.672 & -0.219 & 0. \\
+ -0.672 & 0.219 & 0. \\
+ 0.672 & -0.219 & 0. \\
+ 0.672 & 0.219 & 0. \\
+ -0.372 & 0. & 0.602 \\
+ -0.487 & -0.354 & 0.372 \\
+ -0.487 & 0.354 & 0.372 \\
+ -0.186 & -0.572 & -0.372 \\
+ -0.186 & 0.572 & -0.372 \\
+ -0.115 & -0.354 & 0.602 \\
+ -0.115 & 0.354 & 0.602 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$60.$

pretraining/mathematica/geometry/solids/51115.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.892 & 0.531 & 0.685 \\
+ 0.813 & 0.774 & 0.905 \\
+ 0.089 & 0.728 & 0.558 \\
+ 0.393 & 0.018 & 0.959 \\
+ 0.508 & 0.327 & 0.48 \\
+ 0.933 & 0.243 & 0.138 \\
+ 0.579 & 0.719 & 0.178 \\
+ 0.388 & 0.252 & 0.954 \\
+ 0.874 & 0.298 & 0.105 \\
+\end{array}
+\right)$. 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: $3.54$ 
+Surface Area: $1.77$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/51120.txt

@@ -0,0 +1,23 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & 0 & \sqrt{\text{Root}\left[268435456 \text{$\#$1}^{12}-268435456 \text{$\#$1}^{11}-352321536 \text{$\#$1}^{10}-79691776 \text{$\#$1}^9+70647808 \text{$\#$1}^8+799604736 \text{$\#$1}^7+166375424 \text{$\#$1}^6-526660608 \text{$\#$1}^5-111801072 \text{$\#$1}^4+48951424 \text{$\#$1}^3-4124152 \text{$\#$1}^2+631136 \text{$\#$1}+14641\&,7\right]} \\
+ -\frac{1}{2} & -\text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & \sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ -\frac{1}{2} & \text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & \sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ 0 & -\frac{1}{2} & -\sqrt{\text{Root}\left[268435456 \text{$\#$1}^{12}-268435456 \text{$\#$1}^{11}-352321536 \text{$\#$1}^{10}-79691776 \text{$\#$1}^9+70647808 \text{$\#$1}^8+799604736 \text{$\#$1}^7+166375424 \text{$\#$1}^6-526660608 \text{$\#$1}^5-111801072 \text{$\#$1}^4+48951424 \text{$\#$1}^3-4124152 \text{$\#$1}^2+631136 \text{$\#$1}+14641\&,7\right]} \\
+ 0 & \frac{1}{2} & -\sqrt{\text{Root}\left[268435456 \text{$\#$1}^{12}-268435456 \text{$\#$1}^{11}-352321536 \text{$\#$1}^{10}-79691776 \text{$\#$1}^9+70647808 \text{$\#$1}^8+799604736 \text{$\#$1}^7+166375424 \text{$\#$1}^6-526660608 \text{$\#$1}^5-111801072 \text{$\#$1}^4+48951424 \text{$\#$1}^3-4124152 \text{$\#$1}^2+631136 \text{$\#$1}+14641\&,7\right]} \\
+ 0 & -\text{Root}\left[61440 \text{$\#$1}^{12}-221184 \text{$\#$1}^{11}-83968 \text{$\#$1}^{10}+1230848 \text{$\#$1}^9-1328896 \text{$\#$1}^8-711680 \text{$\#$1}^7+1587456 \text{$\#$1}^6-14080 \text{$\#$1}^5-688112 \text{$\#$1}^4+31520 \text{$\#$1}^3+129208 \text{$\#$1}^2+11048 \text{$\#$1}-3337\&,7\right] & -\sqrt{\text{Root}\left[60397977600 \text{$\#$1}^{12}+512980156416 \text{$\#$1}^{11}+1595228028928 \text{$\#$1}^{10}+2202231898112 \text{$\#$1}^9+1208931975168 \text{$\#$1}^8-50417041408 \text{$\#$1}^7-258660978688 \text{$\#$1}^6-65149629440 \text{$\#$1}^5-12078370032 \text{$\#$1}^4+376397056 \text{$\#$1}^3+216014664 \text{$\#$1}^2-1436544 \text{$\#$1}+2209\&,7\right]} \\
+ 0 & \text{Root}\left[61440 \text{$\#$1}^{12}-221184 \text{$\#$1}^{11}-83968 \text{$\#$1}^{10}+1230848 \text{$\#$1}^9-1328896 \text{$\#$1}^8-711680 \text{$\#$1}^7+1587456 \text{$\#$1}^6-14080 \text{$\#$1}^5-688112 \text{$\#$1}^4+31520 \text{$\#$1}^3+129208 \text{$\#$1}^2+11048 \text{$\#$1}-3337\&,7\right] & -\sqrt{\text{Root}\left[60397977600 \text{$\#$1}^{12}+512980156416 \text{$\#$1}^{11}+1595228028928 \text{$\#$1}^{10}+2202231898112 \text{$\#$1}^9+1208931975168 \text{$\#$1}^8-50417041408 \text{$\#$1}^7-258660978688 \text{$\#$1}^6-65149629440 \text{$\#$1}^5-12078370032 \text{$\#$1}^4+376397056 \text{$\#$1}^3+216014664 \text{$\#$1}^2-1436544 \text{$\#$1}+2209\&,7\right]} \\
+ \frac{1}{2} & 0 & \sqrt{\text{Root}\left[268435456 \text{$\#$1}^{12}-268435456 \text{$\#$1}^{11}-352321536 \text{$\#$1}^{10}-79691776 \text{$\#$1}^9+70647808 \text{$\#$1}^8+799604736 \text{$\#$1}^7+166375424 \text{$\#$1}^6-526660608 \text{$\#$1}^5-111801072 \text{$\#$1}^4+48951424 \text{$\#$1}^3-4124152 \text{$\#$1}^2+631136 \text{$\#$1}+14641\&,7\right]} \\
+ \frac{1}{2} & -\text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & \sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ \frac{1}{2} & \text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & \sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ -\text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & -\frac{1}{2} & -\sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ -\text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & \frac{1}{2} & -\sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ \text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & -\frac{1}{2} & -\sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ \text{Root}\left[256 \text{$\#$1}^{12}-512 \text{$\#$1}^{11}-1664 \text{$\#$1}^{10}+3712 \text{$\#$1}^9+1552 \text{$\#$1}^8-6592 \text{$\#$1}^7+1248 \text{$\#$1}^6+4352 \text{$\#$1}^5-2024 \text{$\#$1}^4-944 \text{$\#$1}^3+672 \text{$\#$1}^2-24 \text{$\#$1}-23\&,5\right] & \frac{1}{2} & -\sqrt{\text{Root}\left[4294967296 \text{$\#$1}^{12}+25769803776 \text{$\#$1}^{11}+41070624768 \text{$\#$1}^{10}+8925478912 \text{$\#$1}^9-17811111936 \text{$\#$1}^8-3897556992 \text{$\#$1}^7+4095672320 \text{$\#$1}^6+36028416 \text{$\#$1}^5-421654272 \text{$\#$1}^4+89818880 \text{$\#$1}^3-5653664 \text{$\#$1}^2-51280 \text{$\#$1}+5041\&,7\right]} \\
+ -\text{Root}\left[61440 \text{$\#$1}^{12}-221184 \text{$\#$1}^{11}-83968 \text{$\#$1}^{10}+1230848 \text{$\#$1}^9-1328896 \text{$\#$1}^8-711680 \text{$\#$1}^7+1587456 \text{$\#$1}^6-14080 \text{$\#$1}^5-688112 \text{$\#$1}^4+31520 \text{$\#$1}^3+129208 \text{$\#$1}^2+11048 \text{$\#$1}-3337\&,7\right] & 0 & \sqrt{\text{Root}\left[60397977600 \text{$\#$1}^{12}+512980156416 \text{$\#$1}^{11}+1595228028928 \text{$\#$1}^{10}+2202231898112 \text{$\#$1}^9+1208931975168 \text{$\#$1}^8-50417041408 \text{$\#$1}^7-258660978688 \text{$\#$1}^6-65149629440 \text{$\#$1}^5-12078370032 \text{$\#$1}^4+376397056 \text{$\#$1}^3+216014664 \text{$\#$1}^2-1436544 \text{$\#$1}+2209\&,7\right]} \\
+ \text{Root}\left[61440 \text{$\#$1}^{12}-221184 \text{$\#$1}^{11}-83968 \text{$\#$1}^{10}+1230848 \text{$\#$1}^9-1328896 \text{$\#$1}^8-711680 \text{$\#$1}^7+1587456 \text{$\#$1}^6-14080 \text{$\#$1}^5-688112 \text{$\#$1}^4+31520 \text{$\#$1}^3+129208 \text{$\#$1}^2+11048 \text{$\#$1}-3337\&,7\right] & 0 & \sqrt{\text{Root}\left[60397977600 \text{$\#$1}^{12}+512980156416 \text{$\#$1}^{11}+1595228028928 \text{$\#$1}^{10}+2202231898112 \text{$\#$1}^9+1208931975168 \text{$\#$1}^8-50417041408 \text{$\#$1}^7-258660978688 \text{$\#$1}^6-65149629440 \text{$\#$1}^5-12078370032 \text{$\#$1}^4+376397056 \text{$\#$1}^3+216014664 \text{$\#$1}^2-1436544 \text{$\#$1}+2209\&,7\right]} \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$\sqrt{\text{Root}\left[16 \text{$\#$1}^{12}-160 \text{$\#$1}^{11}+280 \text{$\#$1}^{10}+56 \text{$\#$1}^9+481 \text{$\#$1}^8+45080 \text{$\#$1}^7-122712 \text{$\#$1}^6-387808 \text{$\#$1}^5+792880 \text{$\#$1}^4+355712 \text{$\#$1}^3-1410688 \text{$\#$1}^2+1044480 \text{$\#$1}-235264\&,7\right]}$

pretraining/mathematica/geometry/solids/52945.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.5 & -0.5 & -0.5 \\
+ -0.5 & -0.5 & 0.5 \\
+ -0.5 & 0.5 & -0.5 \\
+ -0.5 & 0.5 & 0.5 \\
+ 0.5 & -0.5 & -0.5 \\
+ 0.5 & -0.5 & 0.5 \\
+ 0.5 & 0.5 & -0.5 \\
+ 0.5 & 0.5 & 0.5 \\
+\end{array}
+\right)$. Determine the Circumcenter.
+Answer:
+$\{0.,0.,0.\}$

pretraining/mathematica/geometry/solids/53576.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.866 & 0.177 & 0.043 \\
+ 0.248 & 0.892 & 0.509 \\
+ 0.063 & 0.154 & 0.796 \\
+ 0.709 & 0.274 & 0.468 \\
+ 0.104 & 0.496 & 0.926 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.14$ 
+Solid Angle: $0.19$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/54473.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.558 & 0.538 & 0.569 \\
+ 0.962 & 0.509 & 0.11 \\
+ 0.937 & 0.165 & 0.157 \\
+ 0.43 & 0.7 & 0.328 \\
+ 0.367 & 0.325 & 0.048 \\
+ 0.273 & 0.728 & 0.38 \\
+ 0.71 & 0.998 & 0.982 \\
+ 0.539 & 0.145 & 0.086 \\
+ 0.945 & 0.405 & 0.379 \\
+ 0.654 & 0.663 & 0.683 \\
+\end{array}
+\right)$. 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: $0.08$ 
+Surface Area: $1.4$

pretraining/mathematica/geometry/solids/5649.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.967 & 0.472 & 0.285 \\
+ 0.019 & 0.109 & 0.389 \\
+ 0.756 & 0.871 & 0.407 \\
+ 0.401 & 0.639 & 0.457 \\
+ 0.245 & 0.323 & 0.332 \\
+ 0.539 & 0.985 & 0.453 \\
+\end{array}
+\right)$. 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: $0.1$ 
+Surface Area: $0.75$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/56568.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.93 & 0.197 & 0.643 \\
+ 0.222 & 0.482 & 0.991 \\
+ 0.958 & 0.678 & 0.448 \\
+ 0.31 & 0.284 & 0.329 \\
+ 0.784 & 0.535 & 0.239 \\
+ 0.686 & 0.773 & 0.694 \\
+ 0.907 & 0.687 & 0.31 \\
+\end{array}
+\right)$. 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: $1.18$ 
+Solid Angle: $1.09$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/567.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.35 & 0.761 & 0.091 \\
+ 0.079 & 0.155 & 0.864 \\
+ 0.663 & 0.708 & 0.576 \\
+ 0.96 & 0.1 & 0.054 \\
+ 0.602 & 0.888 & 0.87 \\
+ 0.711 & 0.617 & 0.852 \\
+ 0.193 & 0.567 & 0.126 \\
+ 0.49 & 0.899 & 0.794 \\
+\end{array}
+\right)$. 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: $1.96$ 
+Volume: $0.17$ 
+Solid Angle: $1.68$

pretraining/mathematica/geometry/solids/57199.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.173 & 0.459 & 0.733 \\
+ 0.632 & 0.094 & 0.884 \\
+ 0.048 & 0.223 & 0.165 \\
+ 0.688 & 0.676 & 0.396 \\
+\end{array}
+\right)$. 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: $0.88$ 
+Solid Angle: $1.23$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/58520.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.632 & 0.322 & 0.88 \\
+ 0.469 & 0.817 & 0.822 \\
+ 0.797 & 0.609 & 0.108 \\
+ 0.024 & 0.84 & 0.38 \\
+ 0.019 & 0.19 & 0.961 \\
+\end{array}
+\right)$. 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: $1.47$ 
+Solid Angle: $1.32$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/62355.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/62370.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.759 & 0.885 & 0.57 \\
+ 0.344 & 0.442 & 0.121 \\
+ 0.013 & 0.567 & 0.689 \\
+ 0.724 & 0.974 & 0.241 \\
+ 0.892 & 0.723 & 0.655 \\
+ 0.775 & 0.508 & 0.716 \\
+ 0.637 & 0.227 & 0.006 \\
+ 0.601 & 0.893 & 0.451 \\
+\end{array}
+\right)$. 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: $3.14$ 
+Surface Area: $1.37$

pretraining/mathematica/geometry/solids/63819.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.188 & 0.986 & 0.076 \\
+ 0.383 & 0.75 & 0.682 \\
+ 0.9 & 0.403 & 0.737 \\
+ 0.135 & 0.15 & 0.012 \\
+ 0.607 & 0.692 & 0.673 \\
+ 0.546 & 0.933 & 0.27 \\
+ 0.904 & 0.621 & 0.323 \\
+\end{array}
+\right)$. 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: $0.75$

pretraining/mathematica/geometry/solids/63830.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.876 & 0.478 & 0.788 \\
+ 0.144 & 0.244 & 0.178 \\
+ 0.782 & 0.042 & 0.757 \\
+ 0.171 & 0.329 & 0.057 \\
+ 0.868 & 0.626 & 0.212 \\
+ 0.454 & 0.119 & 0.678 \\
+ 0.867 & 0.951 & 0.208 \\
+ 0.858 & 0.299 & 0.575 \\
+ 0.173 & 0.876 & 0.885 \\
+ 0.144 & 0.911 & 0.532 \\
+\end{array}
+\right)$. 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: $0.24$ 
+Surface Area: $2.22$ 
+Solid Angle: $2.54$

pretraining/mathematica/geometry/solids/64045.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.87 & 0.269 & 0.735 \\
+ 0.238 & 0.984 & 0.864 \\
+ 0.362 & 0.283 & 0.595 \\
+ 0.604 & 0.316 & 0.87 \\
+\end{array}
+\right)$. 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: $0.48$ 
+Solid Angle: $0.28$

pretraining/mathematica/geometry/solids/64067.txt

@@ -0,0 +1,42 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & \frac{1}{2} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \\
+ 0 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \\
+ 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & -\frac{1}{2} \\
+ 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{10} \left(5+\sqrt{10 \left(5+\sqrt{5}\right)}\right) \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{10} \left(5+\sqrt{10 \left(5+\sqrt{5}\right)}\right) \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \frac{1}{10} \left(-5-\sqrt{10 \left(5-\sqrt{5}\right)}\right) \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \frac{1}{10} \left(5+2 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} & \frac{1}{10} \left(-5-\sqrt{10 \left(5-\sqrt{5}\right)}\right) \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} & \frac{1}{10} \left(5+2 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\
+ -\sqrt{\frac{1}{2}+\frac{1}{2 \sqrt{5}}} & 0 & \frac{1}{10} \left(-5-\sqrt{10 \left(5-\sqrt{5}\right)}\right) \\
+ -\sqrt{\frac{1}{2}+\frac{1}{2 \sqrt{5}}} & 0 & \frac{1}{10} \left(5+2 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{10} \left(5+\sqrt{10 \left(5+\sqrt{5}\right)}\right) \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{10} \left(5+\sqrt{10 \left(5+\sqrt{5}\right)}\right) \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{10} \left(-5-\sqrt{10 \left(5-\sqrt{5}\right)}\right) \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{10} \left(5+2 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{10} \left(-5-\sqrt{10 \left(5-\sqrt{5}\right)}\right) \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{10} \left(5+2 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\
+ -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \\
+ -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \\
+ -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \\
+ -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \\
+ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \\
+ -\sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & -\frac{1}{2} & -\frac{1}{2} \\
+ -\sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & -\frac{1}{2} & \frac{1}{2} \\
+ -\sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & \frac{1}{2} & -\frac{1}{2} \\
+ -\sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & \frac{1}{2} & \frac{1}{2} \\
+ \sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & -\frac{1}{2} & -\frac{1}{2} \\
+ \sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & -\frac{1}{2} & \frac{1}{2} \\
+ \sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & \frac{1}{2} & -\frac{1}{2} \\
+ \sqrt{\frac{5}{4}+\frac{\sqrt{5}}{2}} & \frac{1}{2} & \frac{1}{2} \\
+ \sqrt{\frac{1}{5} \left(5+2 \sqrt{5}\right)} & 0 & \frac{1}{10} \left(5+\sqrt{10 \left(5+\sqrt{5}\right)}\right) \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$\frac{5}{12} \left(11+5 \sqrt{5}+6 \sqrt{5+2 \sqrt{5}}\right)$

pretraining/mathematica/geometry/solids/64621.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.982 & 0.313 & 0.562 \\
+ 0.172 & 0.53 & 0.512 \\
+ 0.46 & 0.008 & 0.85 \\
+ 0.183 & 0.92 & 0.293 \\
+ 0.205 & 0.308 & 0.973 \\
+ 0.868 & 0.591 & 0.8 \\
+ 0.997 & 0.617 & 0.324 \\
+\end{array}
+\right)$. 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: $1.6$ 
+Volume: $0.11$ 
+Solid Angle: $1.57$