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pretraining/mathematica/geometry/solids/10668.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.736 & 0.925 & 0.081 \\
+ 0.106 & 0.104 & 0.262 \\
+ 0.356 & 0.414 & 0.561 \\
+ 0.569 & 0.417 & 0.846 \\
+ 0.894 & 0.081 & 0.297 \\
+ 0.825 & 0.454 & 0.873 \\
+ 0.53 & 0.131 & 0.856 \\
+ 0.191 & 0.586 & 0.078 \\
+ 0.289 & 0.926 & 0.16 \\
+ 0.242 & 0.018 & 0.568 \\
+\end{array}
+\right)$. 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: $2.11$ 
+Volume: $0.2$ 
+Solid Angle: $1.07$

pretraining/mathematica/geometry/solids/11037.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.17 & 0.902 & 0.73 \\
+ 0.754 & 0.47 & 0.504 \\
+ 0.843 & 0.487 & 0.276 \\
+ 0.102 & 0.91 & 0.587 \\
+ 0.192 & 0.622 & 0.978 \\
+ 0.32 & 0.282 & 0.046 \\
+ 0.524 & 0.533 & 0.884 \\
+ 0.244 & 0.76 & 0.167 \\
+\end{array}
+\right)$. 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: $2.13$ 
+Surface Area: $1.34$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/11879.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.739 & 0.564 & 0.677 \\
+ 0.126 & 0.832 & 0.213 \\
+ 0.777 & 0.836 & 0.194 \\
+ 0.001 & 0.88 & 0.779 \\
+ 0.905 & 0.024 & 0.921 \\
+ 0.215 & 0.223 & 0.033 \\
+ 0.846 & 0.589 & 0. \\
+ 0.594 & 0.78 & 0.741 \\
+ 0.728 & 0.023 & 0.947 \\
+ 0.136 & 0.204 & 0.156 \\
+\end{array}
+\right)$. 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: $0.29$ 
+Solid Angle: $6.01$ 
+Surface Area: $2.59$

pretraining/mathematica/geometry/solids/12094.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.62 & 0.689 & 0.79 \\
+ 0.242 & 0.833 & 0.378 \\
+ 0.921 & 0.69 & 0.418 \\
+ 0.732 & 0.436 & 0.388 \\
+ 0.168 & 0.004 & 0.185 \\
+\end{array}
+\right)$. 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: $0.96$ 
+Solid Angle: $0.94$ 
+Volume: $0.04$

pretraining/mathematica/geometry/solids/13278.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.156 & 0.249 & 0.445 \\
+ 0.755 & 0.429 & 0.09 \\
+ 0.409 & 0.339 & 0.574 \\
+ 0.506 & 0.464 & 0.22 \\
+ 0.399 & 0.577 & 0.543 \\
+ 0.118 & 0.003 & 0.165 \\
+ 0.745 & 0.086 & 0.212 \\
+ 0.99 & 0.964 & 0.523 \\
+ 0.949 & 0.681 & 0.32 \\
+\end{array}
+\right)$. 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: $1.27$ 
+Solid Angle: $1.36$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/14645.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.228 & 0.969 & 0.355 \\
+ 0.522 & 0.006 & 0.568 \\
+ 0.444 & 0.643 & 0.662 \\
+ 0.831 & 0.89 & 0.258 \\
+\end{array}
+\right)$. 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: $0.34$ 
+Surface Area: $0.75$

pretraining/mathematica/geometry/solids/16212.txt

@@ -0,0 +1,20 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.48 & 0.592 & 0.902 \\
+ 0.329 & 0.339 & 0.848 \\
+ 0.7 & 0.245 & 0.945 \\
+ 0.787 & 0.141 & 0.448 \\
+ 0.483 & 0.538 & 0.202 \\
+ 0.177 & 0.313 & 0.029 \\
+ 0.718 & 0.471 & 0.744 \\
+ 0.553 & 0.513 & 0.311 \\
+ 0.687 & 0.232 & 0.351 \\
+ 0.555 & 0.248 & 0.9 \\
+ 0.171 & 0.61 & 0.64 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.24$ 
+Solid Angle: $1.96$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/19529.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.819 & 0.976 & 0.378 \\
+ 0.895 & 0.63 & 0.878 \\
+ 0.337 & 0.054 & 0.48 \\
+ 0.732 & 0.434 & 0.58 \\
+ 0.747 & 0.577 & 0.894 \\
+\end{array}
+\right)$. 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: $0.65$ 
+Solid Angle: $0.16$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/21064.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.116 & 0.739 & 0.243 \\
+ 0.199 & 0.641 & 0.226 \\
+ 0.628 & 0.911 & 0.113 \\
+ 0.718 & 0.222 & 0.919 \\
+ 0.328 & 0.782 & 0.951 \\
+ 0.86 & 0.465 & 0.625 \\
+ 0.348 & 0.555 & 0.33 \\
+ 0.719 & 0.971 & 0.607 \\
+ 0.262 & 0.798 & 0.569 \\
+\end{array}
+\right)$. 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: $1.43$ 
+Solid Angle: $1.26$ 
+Volume: $0.1$

pretraining/mathematica/geometry/solids/21085.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.814 & 0.614 & 0.448 \\
+ 0.237 & 0.776 & 0.994 \\
+ 0.171 & 0.199 & 0.352 \\
+ 0.169 & 0.494 & 0.465 \\
+ 0.398 & 0.882 & 0.052 \\
+ 0.258 & 0.425 & 0.065 \\
+\end{array}
+\right)$. 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: $0.95$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/22351.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.858 & 0.815 & 0.806 \\
+ 0.933 & 0.154 & 0.64 \\
+ 0.833 & 0.906 & 0.606 \\
+ 0.019 & 0.457 & 0.189 \\
+ 0.228 & 0.62 & 0.107 \\
+ 0.156 & 0.949 & 0.238 \\
+ 0.302 & 0.21 & 0.504 \\
+\end{array}
+\right)$. 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: $1.18$ 
+Volume: $0.09$ 
+Surface Area: $1.55$

pretraining/mathematica/geometry/solids/23039.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.224 & 0.718 & 0.856 \\
+ 0.729 & 0.518 & 0.73 \\
+ 0.713 & 0.962 & 0.412 \\
+ 0.311 & 0.474 & 0.633 \\
+ 0.287 & 0.591 & 0.381 \\
+ 0.271 & 0.846 & 0.368 \\
+ 0.332 & 0.793 & 0.311 \\
+ 0.55 & 0.845 & 0.957 \\
+ 0.812 & 0.543 & 0.898 \\
+ 0.36 & 0.352 & 0.495 \\
+\end{array}
+\right)$. 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: $1.03$ 
+Volume: $0.07$ 
+Solid Angle: $1.58$

pretraining/mathematica/geometry/solids/24086.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-4.764,-5.611,8.819\}$ has radius $9.292$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $3360.74$ 
+Surface Area: $1085.03$

pretraining/mathematica/geometry/solids/25998.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.733 & 0.972 & 0.788 \\
+ 0.248 & 0.661 & 0.567 \\
+ 0.848 & 0.125 & 0.461 \\
+ 0.206 & 0.789 & 0.009 \\
+ 0.861 & 0.151 & 0.418 \\
+ 0.212 & 0.018 & 0.58 \\
+ 0.948 & 0.107 & 0.77 \\
+ 0.25 & 0.198 & 0.061 \\
+ 0.267 & 0.215 & 0.667 \\
+ 0.756 & 0.183 & 0.278 \\
+ 0.863 & 0.484 & 0.299 \\
+ 0.559 & 0.762 & 0.327 \\
+\end{array}
+\right)$. 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: $2.21$ 
+Solid Angle: $0.9$ 
+Volume: $0.23$

pretraining/mathematica/geometry/solids/26772.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.95 & 0.889 & 0.556 \\
+ 0.825 & 0.27 & 0.063 \\
+ 0.591 & 0.831 & 0.399 \\
+ 0.023 & 0.022 & 0.853 \\
+ 0.101 & 0.482 & 0.285 \\
+ 0.063 & 0.014 & 0.989 \\
+ 0.979 & 0.577 & 0.815 \\
+ 0.558 & 0.987 & 0.202 \\
+\end{array}
+\right)$. 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: $1.39$ 
+Surface Area: $2.23$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/27895.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.408 & 0.238 & 0.887 \\
+ 0.165 & 0.409 & 0.634 \\
+ 0.658 & 0.623 & 0.067 \\
+ 0.848 & 0.696 & 0.75 \\
+ 0.693 & 0.965 & 0.81 \\
+ 0.47 & 0.169 & 0.164 \\
+ 0.193 & 0.92 & 0.038 \\
+ 0.672 & 0.212 & 0.776 \\
+\end{array}
+\right)$. 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: $1.87$ 
+Volume: $0.18$ 
+Solid Angle: $1.72$

pretraining/mathematica/geometry/solids/28130.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.751 & 0.257 & 0.101 \\
+ 0.325 & 0.121 & 0.358 \\
+ 0.686 & 0.631 & 0.139 \\
+ 0.35 & 0.392 & 0.253 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.15$ 
+Surface Area: $0.28$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/2913.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.099 & 0.493 & 0.106 \\
+ 0.927 & 0.376 & 0.684 \\
+ 0.79 & 0.946 & 0.22 \\
+ 0.895 & 0.658 & 0.75 \\
+ 0.201 & 0.911 & 0.714 \\
+\end{array}
+\right)$. 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: $0.43$ 
+Volume: $0.08$ 
+Surface Area: $1.29$

pretraining/mathematica/geometry/solids/3021.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.221 & 0.616 & 0.279 \\
+ 0.312 & 0.956 & 0.001 \\
+ 0.971 & 0.43 & 0.448 \\
+ 0.888 & 0.725 & 0.215 \\
+ 0.225 & 0.279 & 0.084 \\
+ 0.982 & 0.072 & 0.82 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $2.41$ 
+Surface Area: $1.34$ 
+Volume: $0.06$

pretraining/mathematica/geometry/solids/3200.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.795 & 0.305 & 0.258 \\
+ 0.609 & 0.393 & 0.313 \\
+ 0.572 & 0.501 & 0.322 \\
+ 0.916 & 0.449 & 0.342 \\
+\end{array}
+\right)$. 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: $0.08$ 
+Volume: $0.$ 
+Solid Angle: $0.16$

pretraining/mathematica/geometry/solids/32693.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.716 & 0.974 & 0.969 \\
+ 0.539 & 0.127 & 0.175 \\
+ 0.237 & 0.913 & 0.894 \\
+ 0.584 & 0.742 & 0.322 \\
+ 0.54 & 0.492 & 0.776 \\
+ 0.747 & 0.151 & 0.635 \\
+ 0.02 & 0.13 & 0.306 \\
+ 0.961 & 0.188 & 0.067 \\
+ 0.336 & 0.751 & 0.065 \\
+ 0.488 & 0.379 & 0.061 \\
+\end{array}
+\right)$. 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: $2.34$ 
+Volume: $0.24$ 
+Solid Angle: $0.86$

pretraining/mathematica/geometry/solids/32973.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.571 & 0.715 & 0.063 \\
+ 0.182 & 0.396 & 0.168 \\
+ 0.545 & 0.104 & 0.507 \\
+ 0.009 & 0.117 & 0.614 \\
+ 0.33 & 0.026 & 0.712 \\
+ 0.034 & 0.051 & 0.047 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.27$ 
+Surface Area: $0.98$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/37432.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.304 & 0.828 & 0.362 \\
+ 0.576 & 0.787 & 0.994 \\
+ 0.495 & 0.063 & 0.481 \\
+ 0.176 & 0.243 & 0.929 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.04$ 
+Solid Angle: $0.37$ 
+Surface Area: $0.9$

pretraining/mathematica/geometry/solids/37615.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.086 & 0.631 & 0.237 \\
+ 0.801 & 0.464 & 0.757 \\
+ 0.844 & 0.981 & 0.078 \\
+ 0.789 & 0.22 & 0.761 \\
+ 0.089 & 0.114 & 0.781 \\
+ 0.587 & 0.679 & 0.837 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.61$ 
+Solid Angle: $0.59$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/41968.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.038 & 0.673 & 0.586 \\
+ 0.888 & 0.969 & 0.577 \\
+ 0.767 & 0.244 & 0.576 \\
+ 0.872 & 0.556 & 0.135 \\
+ 0.712 & 0.387 & 0.226 \\
+ 0.893 & 0.995 & 0.719 \\
+ 0.43 & 0.159 & 0.422 \\
+ 0.159 & 0.537 & 0.777 \\
+\end{array}
+\right)$. 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: $1.09$ 
+Surface Area: $1.44$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/43653.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.04 & 0.012 & 0.01 \\
+ 0.964 & 0.416 & 0.505 \\
+ 0.459 & 0.842 & 0.833 \\
+ 0.988 & 0.81 & 0.621 \\
+ 0.993 & 0.355 & 0.091 \\
+ 0.889 & 0.354 & 0.292 \\
+ 0.089 & 0.424 & 0.567 \\
+\end{array}
+\right)$. 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: $0.15$ 
+Volume: $0.07$ 
+Surface Area: $1.76$

pretraining/mathematica/geometry/solids/44053.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.941 & 0.117 & 0.787 \\
+ 0.429 & 0.744 & 0.504 \\
+ 0.948 & 0.235 & 0.461 \\
+ 0.023 & 0.378 & 0.456 \\
+ 0.317 & 0.514 & 0.117 \\
+ 0.799 & 0.351 & 0.832 \\
+ 0.16 & 0.342 & 0.801 \\
+ 0.23 & 0.004 & 0.624 \\
+\end{array}
+\right)$. 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: $1.47$ 
+Volume: $0.12$ 
+Solid Angle: $1.13$

pretraining/mathematica/geometry/solids/48581.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & 1.192 \\
+ 0.607 & 0. & 0.397 \\
+ -0.304 & 0.526 & 0.397 \\
+ 0.304 & 0.526 & -0.397 \\
+ -0.304 & -0.526 & 0.397 \\
+ 0.304 & -0.526 & -0.397 \\
+ -0.607 & 0. & -0.397 \\
+ 0. & 0. & -1.192 \\
+\end{array}
+\right)$. Determine the Inradius.
+Answer:
+$0.43$

pretraining/mathematica/geometry/solids/50997.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.912 & 0.994 & 0.491 \\
+ 0.069 & 0.146 & 0.957 \\
+ 0.344 & 0.895 & 0.339 \\
+ 0.984 & 0.514 & 0.592 \\
+ 0.256 & 0.785 & 0.692 \\
+ 0.742 & 0.045 & 0.102 \\
+ 0.809 & 0.12 & 0.521 \\
+ 0.137 & 0.616 & 0.874 \\
+ 0.508 & 0.95 & 0.863 \\
+ 0.646 & 0.03 & 0.908 \\
+\end{array}
+\right)$. 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: $2.47$ 
+Volume: $0.27$

pretraining/mathematica/geometry/solids/53434.txt

@@ -0,0 +1,25 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{i \left(1+(-1)^{2/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{(-1)^{11/18}}{(-1)^{2/9}-1} & 0 & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{(-1)^{11/18}}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{11/18} \sqrt{3}}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{(-1)^{11/18}}{(-1)^{2/9}-1} & 0 & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{(-1)^{7/18} \left(1+(-1)^{4/9}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{8/9} \left((-1)^{4/9}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ -\frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & -\frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+ \frac{\sqrt[18]{-1} \left(\sqrt[9]{-1}-1\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{(-1)^{5/9} \left(1+\sqrt[9]{-1}\right)}{2 \left((-1)^{2/9}-1\right)} & \frac{1}{2} \sqrt{1-\frac{\sqrt[9]{-1}}{\left(1+\sqrt[9]{-1}\right)^2}} \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$\frac{1}{2} \sqrt{\frac{3}{2} \left(26+\frac{70}{\sqrt[3]{\frac{1}{2} \left(37+i \sqrt{3}\right)}}+5\ 2^{2/3} \sqrt[3]{37+i \sqrt{3}}\right)}$

pretraining/mathematica/geometry/solids/57106.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.061 & 0.172 & 0.495 \\
+ 0.004 & 0.627 & 0.704 \\
+ 0.109 & 0.478 & 0.287 \\
+ 0.746 & 0.456 & 0.925 \\
+ 0.931 & 0.647 & 0.074 \\
+ 0.969 & 0.361 & 0.091 \\
+ 0.717 & 0.885 & 0.525 \\
+ 0.073 & 0.743 & 0.673 \\
+ 0.158 & 0.743 & 0.614 \\
+ 0.051 & 0.368 & 0.786 \\
+\end{array}
+\right)$. 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: $1.92$ 
+Volume: $0.18$ 
+Solid Angle: $1.65$

pretraining/mathematica/geometry/solids/57134.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-3.149,-0.09,-3.843\}$ has radius $7.936$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $791.49$ 
+Volume: $2093.83$

pretraining/mathematica/geometry/solids/57137.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.613 & 0.108 & 0.046 \\
+ 0.161 & 0.818 & 0.649 \\
+ 0.811 & 0.845 & 0.927 \\
+ 0.223 & 0.087 & 0.667 \\
+ 0.564 & 0.737 & 0.313 \\
+ 0.577 & 0.415 & 0.035 \\
+ 0.137 & 0.222 & 0.455 \\
+\end{array}
+\right)$. 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: $1.59$ 
+Solid Angle: $0.81$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/57142.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.832 & 0.955 & 0.35 \\
+ 0.133 & 0.246 & 0.075 \\
+ 0.188 & 0.011 & 0.334 \\
+ 0.035 & 0.98 & 0.352 \\
+ 0.487 & 0.311 & 0.971 \\
+ 0.662 & 0.789 & 0.575 \\
+\end{array}
+\right)$. 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: $0.65$ 
+Surface Area: $1.71$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/58786.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.992 & 0.549 & 0.853 \\
+ 0.303 & 0.435 & 0.005 \\
+ 0.395 & 0.06 & 0.869 \\
+ 0.94 & 0.228 & 0.535 \\
+ 0.175 & 0.414 & 0.957 \\
+ 0.852 & 0.686 & 0.31 \\
+ 0.556 & 0.922 & 0.886 \\
+ 0.775 & 0.021 & 0.498 \\
+ 0.011 & 0.695 & 0.654 \\
+\end{array}
+\right)$. 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: $2.29$ 
+Solid Angle: $1.86$ 
+Volume: $0.25$

pretraining/mathematica/geometry/solids/59056.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.277 & 0.754 & 0.27 \\
+ 0.774 & 0.296 & 0.878 \\
+ 0.868 & 0.891 & 0.796 \\
+ 0.748 & 0.646 & 0.954 \\
+ 0.919 & 0.625 & 0.743 \\
+ 0.568 & 0.285 & 0.578 \\
+ 0.028 & 0.476 & 0.997 \\
+\end{array}
+\right)$. 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: $1.31$ 
+Solid Angle: $0.64$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/65769.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.494 & 0.566 & 0.021 \\
+ 0.8 & 0.263 & 0.772 \\
+ 0.27 & 0.27 & 0.897 \\
+ 0.427 & 0.992 & 0.901 \\
+\end{array}
+\right)$. 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: $1.1$ 
+Solid Angle: $0.25$

pretraining/mathematica/geometry/solids/68795.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1.038 & -0.5 & 0.429 \\
+ -1.038 & 0.5 & 0.429 \\
+ 1.038 & -0.5 & -0.429 \\
+ 1.038 & 0.5 & -0.429 \\
+ -1.152 & 0. & -0.429 \\
+ 1.152 & 0. & 0.429 \\
+ -0.256 & -1.123 & 0.429 \\
+ -0.256 & 1.123 & 0.429 \\
+ 0.256 & -1.123 & -0.429 \\
+ 0.256 & 1.123 & -0.429 \\
+ -0.718 & -0.901 & -0.429 \\
+ -0.718 & 0.901 & -0.429 \\
+ 0.718 & -0.901 & 0.429 \\
+ 0.718 & 0.901 & 0.429 \\
+\end{array}
+\right)$. Determine the Circumradius.
+Answer:
+$1.23$

pretraining/mathematica/geometry/solids/69430.txt

@@ -0,0 +1,57 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1.633 & -0.816 & -0.577 \\
+ -1.633 & -0.816 & 0.577 \\
+ -1.633 & 0. & 0. \\
+ -1.633 & 0.816 & -0.577 \\
+ -1.633 & 0.816 & 0.577 \\
+ -0.816 & -1.633 & -0.577 \\
+ -0.816 & -1.633 & 0.577 \\
+ -0.816 & -0.816 & -1.155 \\
+ -0.816 & -0.816 & 0. \\
+ -0.816 & -0.816 & 1.155 \\
+ -0.816 & 0. & -1.732 \\
+ -0.816 & 0. & -0.577 \\
+ -0.816 & 0. & 0.577 \\
+ -0.816 & 0. & 1.732 \\
+ -0.816 & 0.816 & -1.155 \\
+ -0.816 & 0.816 & 0. \\
+ -0.816 & 0.816 & 1.155 \\
+ -0.816 & 1.633 & -0.577 \\
+ -0.816 & 1.633 & 0.577 \\
+ 0. & -1.633 & 0. \\
+ 0. & -0.816 & -1.732 \\
+ 0. & -0.816 & -0.577 \\
+ 0. & -0.816 & 0.577 \\
+ 0. & -0.816 & 1.732 \\
+ 0. & 0. & -1.155 \\
+ 0. & 0. & 1.155 \\
+ 0. & 0.816 & -1.732 \\
+ 0. & 0.816 & -0.577 \\
+ 0. & 0.816 & 0.577 \\
+ 0. & 0.816 & 1.732 \\
+ 0. & 1.633 & 0. \\
+ 0.816 & -1.633 & -0.577 \\
+ 0.816 & -1.633 & 0.577 \\
+ 0.816 & -0.816 & -1.155 \\
+ 0.816 & -0.816 & 0. \\
+ 0.816 & -0.816 & 1.155 \\
+ 0.816 & 0. & -1.732 \\
+ 0.816 & 0. & -0.577 \\
+ 0.816 & 0. & 0.577 \\
+ 0.816 & 0. & 1.732 \\
+ 0.816 & 0.816 & -1.155 \\
+ 0.816 & 0.816 & 0. \\
+ 0.816 & 0.816 & 1.155 \\
+ 0.816 & 1.633 & -0.577 \\
+ 0.816 & 1.633 & 0.577 \\
+ 1.633 & -0.816 & -0.577 \\
+ 1.633 & -0.816 & 0.577 \\
+ 1.633 & 0. & 0. \\
+ 1.633 & 0.816 & -0.577 \\
+ 1.633 & 0.816 & 0.577 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$120.$

pretraining/mathematica/geometry/solids/70413.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.942 & 0.026 & 0.867 \\
+ 0.738 & 0.574 & 0.044 \\
+ 0.125 & 0.568 & 0.203 \\
+ 0.753 & 0.57 & 0.282 \\
+\end{array}
+\right)$. 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: $0.04$ 
+Surface Area: $0.71$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/70513.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.111 & 0.004 & 0.158 \\
+ 0.454 & 0.242 & 0.176 \\
+ 0.235 & 0.594 & 0.295 \\
+ 0.341 & 0.625 & 0.742 \\
+ 0.056 & 0.838 & 0.564 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $0.68$ 
+Solid Angle: $0.29$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/71243.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.317 & 0.77 & 0.519 \\
+ 0.954 & 0.014 & 0.867 \\
+ 0.551 & 0.345 & 0.217 \\
+ 0.947 & 0.463 & 0.624 \\
+ 0.126 & 0.012 & 0.635 \\
+ 0.824 & 0.484 & 0.82 \\
+ 0.579 & 0.736 & 0.666 \\
+ 0.596 & 0.744 & 0.219 \\
+ 0.308 & 0.199 & 0.734 \\
+\end{array}
+\right)$. 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: $1.55$ 
+Solid Angle: $1.6$

pretraining/mathematica/geometry/solids/71837.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.064 & 0.687 & 0.305 \\
+ 0.684 & 0.916 & 0.827 \\
+ 0.449 & 0.688 & 0.866 \\
+ 0.321 & 0.887 & 0.317 \\
+ 0.259 & 0.176 & 0.756 \\
+ 0.084 & 0.007 & 0.508 \\
+ 0.735 & 0.705 & 0.715 \\
+\end{array}
+\right)$. 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: $0.86$

pretraining/mathematica/geometry/solids/73105.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.979 & 0.234 & 0.099 \\
+ 0.303 & 0.643 & 0.636 \\
+ 0.788 & 0.873 & 0.467 \\
+ 0.918 & 0.59 & 0.627 \\
+ 0.275 & 0.084 & 0.612 \\
+ 0.459 & 0.819 & 0.103 \\
+ 0.766 & 0.187 & 0.959 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.75$ 
+Volume: $0.16$ 
+Solid Angle: $0.97$

pretraining/mathematica/geometry/solids/73142.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.91 & 0.117 & 0.666 \\
+ 0.288 & 0.147 & 0.333 \\
+ 0.489 & 0.115 & 0.256 \\
+ 0.872 & 0.929 & 0.775 \\
+ 0.809 & 0.478 & 0.902 \\
+ 0.157 & 0.581 & 0.187 \\
+ 0.208 & 0.778 & 0.597 \\
+\end{array}
+\right)$. 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: $1.52$ 
+Volume: $0.1$ 
+Solid Angle: $0.75$

pretraining/mathematica/geometry/solids/73287.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.49 & 0.06 & 0.369 \\
+ 0.263 & 0.548 & 0.127 \\
+ 0.912 & 0.984 & 0. \\
+ 0.923 & 0.857 & 0.141 \\
+ 0.484 & 0.659 & 0.996 \\
+ 0.914 & 0.686 & 0.435 \\
+ 0.749 & 0.556 & 0.19 \\
+ 0.042 & 0.079 & 0.421 \\
+ 0.496 & 0.622 & 0.11 \\
+ 0.433 & 0.114 & 0.886 \\
+\end{array}
+\right)$. 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: $2.48$ 
+Surface Area: $1.97$

pretraining/mathematica/geometry/solids/74382.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.095 & 0.477 & 0.138 \\
+ 0.346 & 0.56 & 0.881 \\
+ 0.964 & 0.402 & 0.465 \\
+ 0.542 & 0.091 & 0.022 \\
+ 0.087 & 0.316 & 0.413 \\
+ 0.018 & 0.431 & 0.501 \\
+ 0.877 & 0.432 & 0.475 \\
+ 0.192 & 0.674 & 0.958 \\
+ 0.778 & 0.216 & 0.501 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.08$ 
+Solid Angle: $1.24$ 
+Surface Area: $1.39$

pretraining/mathematica/geometry/solids/74469.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.791 & 0.203 & 0.029 \\
+ 0.926 & 0.592 & 0.058 \\
+ 0.566 & 0.103 & 0.057 \\
+ 0.319 & 0.002 & 0.535 \\
+\end{array}
+\right)$. 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: $0.01$ 
+Surface Area: $0.38$ 
+Solid Angle: $1.15$

pretraining/mathematica/geometry/solids/75003.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.552 & 0.376 & 0.73 \\
+ 0.167 & 0.133 & 0.826 \\
+ 0.572 & 0.97 & 0.612 \\
+ 0.046 & 0.991 & 0.733 \\
+ 0.168 & 0.309 & 0.174 \\
+ 0.913 & 0.481 & 0.336 \\
+ 0.749 & 0.845 & 0.286 \\
+ 0.827 & 0.679 & 0.243 \\
+ 0.888 & 0.948 & 0.713 \\
+ 0.387 & 0.178 & 0.495 \\
+\end{array}
+\right)$. 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: $4.31$ 
+Surface Area: $2.$

pretraining/mathematica/geometry/solids/75734.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.839 & 0.925 & 0.621 \\
+ 0.248 & 0.129 & 0.801 \\
+ 0.791 & 0.001 & 0.018 \\
+ 0.071 & 0.993 & 0.672 \\
+ 0.874 & 0.949 & 0.556 \\
+ 0.015 & 0.945 & 0.834 \\
+ 0.014 & 0.964 & 0.465 \\
+ 0.979 & 0.582 & 0.909 \\
+ 0.76 & 0.962 & 0.228 \\
+ 0.031 & 0.334 & 0.436 \\
+\end{array}
+\right)$. 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: $0.35$ 
+Solid Angle: $4.99$ 
+Surface Area: $2.89$