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pretraining/mathematica/geometry/solids/10250.txt

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+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -\frac{1}{\sqrt{2}} & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6-1024 \text{$\#$1}^5+256 \text{$\#$1}^4+1408 \text{$\#$1}^3-608 \text{$\#$1}^2-96 \text{$\#$1}+47\&,1\right] \\
+ 0 & \frac{1}{\sqrt{2}} & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6-1024 \text{$\#$1}^5+256 \text{$\#$1}^4+1408 \text{$\#$1}^3-608 \text{$\#$1}^2-96 \text{$\#$1}+47\&,1\right] \\
+ 0 & -\sqrt{1+\frac{1}{\sqrt{2}}} & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ 0 & \sqrt{1+\frac{1}{\sqrt{2}}} & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{\sqrt{2}} & 0 & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6-1024 \text{$\#$1}^5+256 \text{$\#$1}^4+1408 \text{$\#$1}^3-608 \text{$\#$1}^2-96 \text{$\#$1}+47\&,1\right] \\
+ \frac{1}{\sqrt{2}} & 0 & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6-1024 \text{$\#$1}^5+256 \text{$\#$1}^4+1408 \text{$\#$1}^3-608 \text{$\#$1}^2-96 \text{$\#$1}+47\&,1\right] \\
+ -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{2}}} & \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{2}}} & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6+1024 \text{$\#$1}^5+256 \text{$\#$1}^4-1408 \text{$\#$1}^3-608 \text{$\#$1}^2+96 \text{$\#$1}+47\&,6\right] \\
+ \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{2}}} & -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{2}}} & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6+1024 \text{$\#$1}^5+256 \text{$\#$1}^4-1408 \text{$\#$1}^3-608 \text{$\#$1}^2+96 \text{$\#$1}+47\&,6\right] \\
+ \frac{1}{2}+\frac{1}{\sqrt{2}} & \frac{1}{2} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\sqrt{1+\frac{1}{\sqrt{2}}} & 0 & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ \sqrt{1+\frac{1}{\sqrt{2}}} & 0 & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2} \sqrt{2+\sqrt{2}} & -\frac{1}{2} \sqrt{2+\sqrt{2}} & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2} \sqrt{2+\sqrt{2}} & \frac{\sqrt{2+\sqrt{2}}}{2} & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ \frac{\sqrt{2+\sqrt{2}}}{2} & -\frac{1}{2} \sqrt{2+\sqrt{2}} & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ \frac{\sqrt{2+\sqrt{2}}}{2} & \frac{\sqrt{2+\sqrt{2}}}{2} & \frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2} & -\frac{1}{2}-\frac{1}{\sqrt{2}} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ \frac{1}{2} & \frac{1}{2}+\frac{1}{\sqrt{2}} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ \frac{1}{2} & -\frac{1}{2}-\frac{1}{\sqrt{2}} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2} & \frac{1}{2}+\frac{1}{\sqrt{2}} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ \frac{1}{2}+\frac{1}{\sqrt{2}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2}-\frac{1}{\sqrt{2}} & \frac{1}{2} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2}-\frac{1}{\sqrt{2}} & -\frac{1}{2} & -\frac{1}{2} \sqrt{-1-\sqrt{2}+\sqrt{5+\frac{7}{\sqrt{2}}}} \\
+ -\frac{1}{2} \sqrt{1-\frac{1}{\sqrt{2}}} & -\frac{1}{2} \sqrt{1+\frac{1}{\sqrt{2}}} & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6+1024 \text{$\#$1}^5+256 \text{$\#$1}^4-1408 \text{$\#$1}^3-608 \text{$\#$1}^2+96 \text{$\#$1}+47\&,6\right] \\
+ \frac{1}{2} \sqrt{1-\frac{1}{\sqrt{2}}} & \frac{1}{2} \sqrt{1+\frac{1}{\sqrt{2}}} & \text{Root}\left[512 \text{$\#$1}^8-512 \text{$\#$1}^6+1024 \text{$\#$1}^5+256 \text{$\#$1}^4-1408 \text{$\#$1}^3-608 \text{$\#$1}^2+96 \text{$\#$1}+47\&,6\right] \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$10+6 \sqrt{3}$

pretraining/mathematica/geometry/solids/11780.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.111 & 0.564 & 0.258 \\
+ 0.823 & 0.432 & 0.9 \\
+ 0.967 & 0.259 & 0.694 \\
+ 0.061 & 0.806 & 0.686 \\
+ 0.276 & 0.505 & 0.105 \\
+ 0.799 & 0.97 & 0.664 \\
+ 0.097 & 0.488 & 0.874 \\
+ 0.039 & 0.091 & 0.566 \\
+\end{array}
+\right)$. 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: $1.94$ 
+Solid Angle: $2.95$

pretraining/mathematica/geometry/solids/12777.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.646 & 0.018 & 0.976 \\
+ 0.076 & 0.262 & 0.276 \\
+ 0.696 & 0.672 & 0.723 \\
+ 0.912 & 0.255 & 0.961 \\
+ 0.158 & 0.241 & 0.029 \\
+ 0.193 & 0.994 & 0.838 \\
+ 0.514 & 0.233 & 0.374 \\
+ 0.703 & 0.9 & 0.327 \\
+\end{array}
+\right)$. 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: $2.09$ 
+Volume: $0.19$ 
+Solid Angle: $1.08$

pretraining/mathematica/geometry/solids/12924.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.697 & 0.709 & 0.18 \\
+ 0.878 & 0.578 & 0.617 \\
+ 0.546 & 0.863 & 0.643 \\
+ 0.589 & 0.903 & 0.8 \\
+ 0.999 & 0.27 & 0.669 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.01$ 
+Solid Angle: $0.16$ 
+Surface Area: $0.5$

pretraining/mathematica/geometry/solids/13104.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.038 & 0.608 & 0.345 \\
+ 0.708 & 0.634 & 0.044 \\
+ 0.5 & 0.001 & 0.629 \\
+ 0.321 & 0.976 & 0.578 \\
+ 0.918 & 0.101 & 0.725 \\
+\end{array}
+\right)$. 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: $0.98$ 
+Surface Area: $1.41$

pretraining/mathematica/geometry/solids/13587.txt

@@ -0,0 +1,31 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -0.5 & -0.5 & -1.207 \\
+ -0.5 & 0.5 & -1.207 \\
+ 0. & -0.707 & 1.207 \\
+ 0. & 0.707 & 1.207 \\
+ 0.5 & -0.5 & -1.207 \\
+ 0.5 & 0.5 & -1.207 \\
+ -0.707 & 0. & 1.207 \\
+ 0.707 & 0. & 1.207 \\
+ 0.5 & 1.207 & -0.5 \\
+ 0.5 & 1.207 & 0.5 \\
+ -0.5 & 1.207 & -0.5 \\
+ -0.5 & 1.207 & 0.5 \\
+ -1.207 & 0.5 & -0.5 \\
+ -1.207 & 0.5 & 0.5 \\
+ -1.207 & -0.5 & -0.5 \\
+ -1.207 & -0.5 & 0.5 \\
+ -0.5 & -1.207 & -0.5 \\
+ -0.5 & -1.207 & 0.5 \\
+ 0.5 & -1.207 & -0.5 \\
+ 0.5 & -1.207 & 0.5 \\
+ 1.207 & -0.5 & -0.5 \\
+ 1.207 & -0.5 & 0.5 \\
+ 1.207 & 0.5 & -0.5 \\
+ 1.207 & 0.5 & 0.5 \\
+\end{array}
+\right)$. Determine the Circumdiameter.
+Answer:
+$2.8$

pretraining/mathematica/geometry/solids/16600.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.82 & 0.548 & 0.783 \\
+ 0.722 & 0.697 & 0.369 \\
+ 0.594 & 0.11 & 0.809 \\
+ 0.724 & 0.636 & 0.831 \\
+ 0.51 & 0.799 & 0.324 \\
+\end{array}
+\right)$. 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: $0.46$ 
+Volume: $0.01$ 
+Solid Angle: $2.05$

pretraining/mathematica/geometry/solids/16861.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.893 & 0.105 & 0.518 \\
+ 0.396 & 0.486 & 0.82 \\
+ 0.197 & 0.077 & 0.996 \\
+ 0.596 & 0.614 & 0.337 \\
+ 0.518 & 0.775 & 0.46 \\
+ 0.779 & 0.714 & 0.281 \\
+ 0.33 & 0.155 & 0.685 \\
+ 0.657 & 0.479 & 0.337 \\
+ 0.742 & 0.355 & 0.872 \\
+\end{array}
+\right)$. 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: $1.13$ 
+Solid Angle: $1.02$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/19538.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.471 & 0.04 & 0.231 \\
+ 0.611 & 0.581 & 0.222 \\
+ 0.123 & 0.646 & 0.785 \\
+ 0.415 & 0.814 & 0.578 \\
+ 0.029 & 0.927 & 0.203 \\
+ 0.235 & 0.109 & 0.667 \\
+ 0.965 & 0.418 & 0.951 \\
+ 0.528 & 0.029 & 0.408 \\
+ 0.846 & 0.273 & 0.024 \\
+ 0.082 & 0.999 & 0.626 \\
+\end{array}
+\right)$. 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: $2.32$ 
+Solid Angle: $2.51$ 
+Volume: $0.24$

pretraining/mathematica/geometry/solids/20784.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.98 & 0.975 & 0.313 \\
+ 0.823 & 0.678 & 0.087 \\
+ 0.633 & 0.202 & 0.982 \\
+ 0.969 & 0.636 & 0.134 \\
+ 0.975 & 0.89 & 0.677 \\
+ 0.93 & 0.955 & 0.929 \\
+ 0.275 & 0.257 & 0.277 \\
+ 0.068 & 0.976 & 0.545 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.23$ 
+Solid Angle: $1.96$ 
+Surface Area: $2.26$

pretraining/mathematica/geometry/solids/2364.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.334 & 0.834 & 0.957 \\
+ 0.883 & 0.788 & 0.307 \\
+ 0.756 & 0.759 & 0.992 \\
+ 0.051 & 0.693 & 0.474 \\
+ 0.795 & 0.842 & 0.584 \\
+ 0.836 & 0.538 & 0.604 \\
+\end{array}
+\right)$. 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: $0.57$ 
+Volume: $0.04$ 
+Surface Area: $0.93$

pretraining/mathematica/geometry/solids/23885.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.765 & 0.834 & 0.127 \\
+ 0.889 & 0.107 & 0.118 \\
+ 0.894 & 0.46 & 0.876 \\
+ 0.929 & 0.842 & 0.175 \\
+ 0.821 & 0.858 & 0.331 \\
+ 0.83 & 0.652 & 0.937 \\
+\end{array}
+\right)$. 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: $0.87$ 
+Volume: $0.02$ 
+Solid Angle: $1.21$

pretraining/mathematica/geometry/solids/26674.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.563 & 0.956 & 0.352 \\
+ 0.36 & 0.608 & 0.813 \\
+ 0.319 & 0.967 & 0.44 \\
+ 0.055 & 0.953 & 0.826 \\
+ 0.312 & 0.165 & 0.318 \\
+ 0.752 & 0.682 & 0.335 \\
+ 0.619 & 0.084 & 0.762 \\
+\end{array}
+\right)$. 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: $1.33$ 
+Solid Angle: $1.15$ 
+Volume: $0.09$

pretraining/mathematica/geometry/solids/27862.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.51 & 0.784 & 0.346 \\
+ 0.108 & 0.009 & 0.191 \\
+ 0.628 & 0.471 & 0.427 \\
+ 0.902 & 0.802 & 0.774 \\
+ 0.186 & 0.264 & 0.602 \\
+ 0.27 & 0.82 & 0.825 \\
+ 0.008 & 0.872 & 0.141 \\
+\end{array}
+\right)$. 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: $1.59$ 
+Solid Angle: $3.56$

pretraining/mathematica/geometry/solids/29361.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.985 & 0.498 & 0.113 \\
+ 0.063 & 0.783 & 0.62 \\
+ 0.653 & 0.366 & 0.279 \\
+ 0.419 & 0.486 & 0.173 \\
+ 0.01 & 0.986 & 0.193 \\
+ 0.565 & 0.363 & 0.347 \\
+ 0.967 & 0.903 & 0.452 \\
+ 0.029 & 0.777 & 0.169 \\
+ 0.528 & 0.261 & 0.829 \\
+ 0.525 & 0.948 & 0.479 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $1.06$ 
+Volume: $0.16$ 
+Surface Area: $1.79$

pretraining/mathematica/geometry/solids/30977.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.458 & 0.575 & 0.345 \\
+ 0.752 & 0.929 & 0.61 \\
+ 0.153 & 0.971 & 0.842 \\
+ 0.908 & 0.319 & 0.254 \\
+ 0.35 & 0.119 & 0.909 \\
+ 0.975 & 0.661 & 0.23 \\
+ 0.875 & 0.924 & 0.744 \\
+ 0.526 & 0.345 & 0.429 \\
+ 0.023 & 0.894 & 0.957 \\
+\end{array}
+\right)$. 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: $0.14$ 
+Solid Angle: $3.15$ 
+Surface Area: $1.82$

pretraining/mathematica/geometry/solids/33016.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.844 & 0.257 & 0.299 \\
+ 0.595 & 0.749 & 0.784 \\
+ 0.177 & 0.447 & 0.384 \\
+ 0.727 & 0.812 & 0.543 \\
+ 0.482 & 0.889 & 0.144 \\
+\end{array}
+\right)$. 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: $0.51$ 
+Surface Area: $0.86$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/33845.txt

@@ -0,0 +1,6 @@
+Problem:
+A cylinder with radius $4.49$ is around the line from $\{7.092,-4.86,7.739\}$ to $\{5.145,-1.525,-5.018\}$. Estimate the cylinder's surface area, volume, and centroid.
+Answer:
+Centroid: $\{6.12,-3.19,1.36\}$ 
+Surface Area: $502.64$ 
+Volume: $844.04$

pretraining/mathematica/geometry/solids/33977.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.044 & 0.168 & 0.957 \\
+ 0.452 & 0.563 & 0.018 \\
+ 0.05 & 0.96 & 0.905 \\
+ 0.609 & 0.036 & 0.035 \\
+ 0.016 & 0.419 & 0.036 \\
+ 0.944 & 0.752 & 0.932 \\
+ 0.846 & 0.18 & 0.845 \\
+ 0.924 & 0.658 & 0.459 \\
+ 0.545 & 0.138 & 0.766 \\
+\end{array}
+\right)$. 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: $1.52$ 
+Volume: $0.39$ 
+Surface Area: $3.08$

pretraining/mathematica/geometry/solids/35454.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.299 & 0.068 & 0.264 \\
+ 0.691 & 0.055 & 0.241 \\
+ 0.746 & 0.444 & 0.861 \\
+ 0.223 & 0.555 & 0.376 \\
+ 0.286 & 0.127 & 0.139 \\
+ 0.902 & 0.769 & 0.172 \\
+ 0.732 & 0.869 & 0.962 \\
+ 0.231 & 0.834 & 0.01 \\
+ 0.926 & 0.578 & 0.718 \\
+\end{array}
+\right)$. 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: $2.2$ 
+Surface Area: $1.99$ 
+Volume: $0.19$

pretraining/mathematica/geometry/solids/37501.txt

@@ -0,0 +1,12 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & 0 & -\sqrt{\frac{2}{3}} \\
+ 0 & 0 & \sqrt{\frac{2}{3}} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & 0 \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & 0 \\
+ \frac{1}{\sqrt{3}} & 0 & 0 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$9$

pretraining/mathematica/geometry/solids/40000.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.526 & 0.384 & 0.478 \\
+ 0.153 & 0.181 & 0.253 \\
+ 0.187 & 0.646 & 0.649 \\
+ 0.071 & 0.896 & 0.564 \\
+ 0.527 & 0.364 & 0.118 \\
+\end{array}
+\right)$. 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: $0.6$ 
+Volume: $0.02$ 
+Solid Angle: $1.3$

pretraining/mathematica/geometry/solids/43908.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.526 & 0.216 & 0.839 \\
+ 0.116 & 0.941 & 0.924 \\
+ 0.092 & 0.004 & 0.9 \\
+ 0.007 & 0.99 & 0.864 \\
+ 0.382 & 0.526 & 0.102 \\
+ 0.419 & 0.53 & 0.121 \\
+ 0.881 & 0.013 & 0.655 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Solid Angle: $4.84$ 
+Surface Area: $1.74$

pretraining/mathematica/geometry/solids/43929.txt

@@ -0,0 +1,22 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -1 & 0 & -\sqrt{\sqrt{3}-1} \\
+ -\frac{1}{2} & -\frac{\sqrt{3}}{2} & -\sqrt{\sqrt{3}-1} \\
+ -\frac{1}{2} & \frac{\sqrt{3}}{2} & -\sqrt{\sqrt{3}-1} \\
+ 0 & -1 & 0 \\
+ 0 & 1 & 0 \\
+ \frac{1}{2} & -\frac{\sqrt{3}}{2} & -\sqrt{\sqrt{3}-1} \\
+ \frac{1}{2} & \frac{\sqrt{3}}{2} & -\sqrt{\sqrt{3}-1} \\
+ 1 & 0 & -\sqrt{\sqrt{3}-1} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & \sqrt{\frac{2}{3}} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & \sqrt{\frac{2}{3}} \\
+ \frac{1}{\sqrt{3}} & 0 & \sqrt{\frac{2}{3}} \\
+ -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\
+ -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
+ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\
+ \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
+\end{array}
+\right)$. Determine the Centroid.
+Answer:
+$\left\{0,0,-\frac{\sqrt{\frac{34388823}{2}+12170346 \sqrt{3}-90 \sqrt{807276361713 \sqrt{3}-1281344819375}}}{15068}\right\}$

pretraining/mathematica/geometry/solids/44813.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.921 & 0.883 & 0.667 \\
+ 0.53 & 0.651 & 0.203 \\
+ 0.592 & 0.844 & 0.445 \\
+ 0.408 & 0.005 & 0.1 \\
+ 0.833 & 0.449 & 0.599 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.01$ 
+Solid Angle: $0.17$ 
+Surface Area: $0.69$

pretraining/mathematica/geometry/solids/45114.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.086 & 0.448 & 0.635 \\
+ 0.561 & 0.393 & 0.359 \\
+ 0.18 & 0.11 & 0.601 \\
+ 0.289 & 0.217 & 0.397 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.18$ 
+Volume: $0.$ 
+Surface Area: $0.23$

pretraining/mathematica/geometry/solids/45125.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.108 & 0.985 & 0.388 \\
+ 0.714 & 0.079 & 0.836 \\
+ 0.697 & 0.032 & 0.998 \\
+ 0.91 & 0.529 & 0.47 \\
+ 0.268 & 0.912 & 0.806 \\
+ 0.33 & 0.852 & 0.21 \\
+ 0.669 & 0.14 & 0.201 \\
+ 0.146 & 0.543 & 0.303 \\
+ 0.798 & 0.875 & 0.384 \\
+ 0.091 & 0.868 & 0.695 \\
+\end{array}
+\right)$. 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: $2.03$ 
+Solid Angle: $1.77$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/45461.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -1 & -\frac{\sqrt{\frac{3}{2}}}{2} \\
+ 0 & 1 & -\frac{\sqrt{\frac{3}{2}}}{2} \\
+ -\frac{1}{\sqrt{3}} & -1 & \frac{1}{2 \sqrt{6}} \\
+ -\frac{1}{\sqrt{3}} & 1 & \frac{1}{2 \sqrt{6}} \\
+ -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} & \frac{5}{2 \sqrt{6}} \\
+ -\frac{1}{2 \sqrt{3}} & \frac{1}{2} & \frac{5}{2 \sqrt{6}} \\
+ \frac{1}{\sqrt{3}} & 0 & \frac{5}{2 \sqrt{6}} \\
+ \frac{2}{\sqrt{3}} & 0 & \frac{1}{2 \sqrt{6}} \\
+ -\frac{\sqrt{3}}{2} & -\frac{1}{2} & -\frac{\sqrt{\frac{3}{2}}}{2} \\
+ -\frac{\sqrt{3}}{2} & \frac{1}{2} & -\frac{\sqrt{\frac{3}{2}}}{2} \\
+ \frac{\sqrt{3}}{2} & -\frac{1}{2} & -\frac{\sqrt{\frac{3}{2}}}{2} \\
+ \frac{\sqrt{3}}{2} & \frac{1}{2} & -\frac{\sqrt{\frac{3}{2}}}{2} \\
+\end{array}
+\right)$. Determine the Circumcenter.
+Answer:
+$\{0,0,0\}$

pretraining/mathematica/geometry/solids/4697.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.944 & 0.299 & 0.66 \\
+ 0.805 & 0.91 & 0.753 \\
+ 0.146 & 0.313 & 0.878 \\
+ 0.779 & 0.89 & 0.884 \\
+ 0.531 & 0.734 & 0.756 \\
+\end{array}
+\right)$. 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: $0.65$ 
+Solid Angle: $0.2$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/49755.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.663 & 0.852 & 0.895 \\
+ 0.484 & 0.261 & 0.844 \\
+ 0.89 & 0.254 & 0.968 \\
+ 0.418 & 0.832 & 0.957 \\
+ 0.372 & 0.207 & 0.071 \\
+ 0.653 & 0.908 & 0.022 \\
+\end{array}
+\right)$. 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: $2.42$ 
+Surface Area: $1.7$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/51965.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.327 & 0.001 & 0.516 \\
+ 0.31 & 0.047 & 0.005 \\
+ 0.771 & 0.518 & 0.332 \\
+ 0.954 & 0.623 & 0.632 \\
+ 0.083 & 0.311 & 0.622 \\
+ 0.463 & 0.926 & 0.282 \\
+ 0.261 & 0.937 & 0.984 \\
+ 0.676 & 0.039 & 0.781 \\
+ 0.274 & 0.849 & 0.107 \\
+\end{array}
+\right)$. 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:
+Volume: $0.26$ 
+Solid Angle: $3.26$ 
+Surface Area: $2.39$

pretraining/mathematica/geometry/solids/53889.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.154 & 0.075 & 0.561 \\
+ 0.362 & 0.279 & 0.996 \\
+ 0.906 & 0.602 & 0.332 \\
+ 0.697 & 0.703 & 0.793 \\
+ 0.74 & 0.172 & 0.827 \\
+ 0.068 & 0.77 & 0.425 \\
+ 0.058 & 0.398 & 0.008 \\
+ 0.433 & 0.615 & 0.877 \\
+\end{array}
+\right)$. 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: $0.17$ 
+Solid Angle: $1.61$

pretraining/mathematica/geometry/solids/54215.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.174 & 0.285 & 0.822 \\
+ 0.371 & 0.642 & 0.953 \\
+ 0.823 & 0.07 & 0.016 \\
+ 0.542 & 0.156 & 0.231 \\
+ 0.961 & 0.475 & 0.92 \\
+ 0.782 & 0.589 & 0.142 \\
+ 0.993 & 0.667 & 0.973 \\
+ 0.874 & 0.077 & 0.997 \\
+ 0.438 & 0.256 & 0.016 \\
+\end{array}
+\right)$. 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: $1.42$ 
+Surface Area: $2.14$

pretraining/mathematica/geometry/solids/54669.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{7.036,4.515,-0.259\}$ has radius $0.705$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $1.47$ 
+Surface Area: $6.24$

pretraining/mathematica/geometry/solids/54801.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.327 & 0.803 & 0.44 \\
+ 0.565 & 0.793 & 0.687 \\
+ 0.325 & 0.733 & 0.9 \\
+ 0.142 & 0.603 & 0.81 \\
+ 0.373 & 0.106 & 0.545 \\
+ 0.731 & 0.147 & 0.341 \\
+ 0.8 & 0.07 & 0.601 \\
+ 0.449 & 0.712 & 0.047 \\
+ 0.825 & 0.209 & 0.621 \\
+ 0.684 & 0.651 & 0.968 \\
+\end{array}
+\right)$. 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: $1.52$ 
+Solid Angle: $3.$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/57410.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.995 & 0.514 & 0.578 \\
+ 0.534 & 0.709 & 0.931 \\
+ 0.682 & 0.022 & 0.895 \\
+ 0.87 & 0.562 & 0.181 \\
+ 0.432 & 0.213 & 0.085 \\
+ 0.005 & 0.55 & 0.686 \\
+ 0.782 & 0.193 & 0.749 \\
+ 0.609 & 0.395 & 0.003 \\
+ 0.778 & 0.91 & 0.944 \\
+ 0.641 & 0.011 & 0.818 \\
+\end{array}
+\right)$. 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: $2.02$ 
+Solid Angle: $2.75$

pretraining/mathematica/geometry/solids/57571.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.022 & 0.043 & 0.275 \\
+ 0.552 & 0.943 & 0.35 \\
+ 0.742 & 0.589 & 0.966 \\
+ 0.211 & 0.566 & 0.693 \\
+ 0.708 & 0.051 & 0.633 \\
+ 0.013 & 0.804 & 0.594 \\
+ 0.803 & 0.771 & 0.974 \\
+ 0.059 & 0.84 & 0.215 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.15$ 
+Surface Area: $2.$ 
+Solid Angle: $0.57$

pretraining/mathematica/geometry/solids/58061.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.191 & 0.17 & 0.724 \\
+ 0.284 & 0.593 & 0.997 \\
+ 0.104 & 0.809 & 0.638 \\
+ 0.689 & 0.68 & 0.902 \\
+ 0.42 & 0.745 & 0.333 \\
+ 0.227 & 0.144 & 0.675 \\
+\end{array}
+\right)$. 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: $0.85$ 
+Solid Angle: $1.45$

pretraining/mathematica/geometry/solids/5850.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.962 & 0.285 & 0.518 \\
+ 0.335 & 0.872 & 0.916 \\
+ 0.627 & 0.347 & 0.726 \\
+ 0.125 & 0.357 & 0.11 \\
+ 0.132 & 0.828 & 0.483 \\
+ 0.263 & 0.323 & 0.909 \\
+ 0.04 & 0.227 & 0.58 \\
+ 0.813 & 0.372 & 0.137 \\
+ 0.562 & 0.876 & 0.902 \\
+ 0.024 & 0.629 & 0.973 \\
+\end{array}
+\right)$. 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: $2.03$ 
+Volume: $0.19$ 
+Solid Angle: $1.1$

pretraining/mathematica/geometry/solids/58933.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.937 & 0.583 & 0.334 \\
+ 0.389 & 0.529 & 0.95 \\
+ 0.233 & 0.862 & 0.807 \\
+ 0.58 & 0.095 & 0.649 \\
+\end{array}
+\right)$. 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: $0.72$ 
+Solid Angle: $0.12$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/59499.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.818 & 0.206 & 0.509 \\
+ 0.253 & 0.661 & 0.849 \\
+ 0.896 & 0.74 & 0.629 \\
+ 0.108 & 0.867 & 0.446 \\
+ 0.33 & 0.359 & 0.056 \\
+ 0.474 & 0.902 & 0.033 \\
+ 0.416 & 0.238 & 0.95 \\
+ 0.109 & 0.588 & 0.25 \\
+ 0.186 & 0.133 & 0.544 \\
+ 0.722 & 0.505 & 0.008 \\
+ 0.389 & 0.966 & 0.24 \\
+ 0.723 & 0.686 & 0.004 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.25$ 
+Solid Angle: $2.24$ 
+Surface Area: $2.2$

pretraining/mathematica/geometry/solids/66114.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.274 & 0.743 & 0.957 \\
+ 0.478 & 0.105 & 0.386 \\
+ 0.435 & 0.312 & 0.147 \\
+ 0.021 & 0.147 & 0.793 \\
+ 0.955 & 0.16 & 0.318 \\
+ 0.195 & 0.567 & 0.331 \\
+\end{array}
+\right)$. 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: $1.28$ 
+Volume: $0.08$ 
+Solid Angle: $0.52$

pretraining/mathematica/geometry/solids/66854.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.208 & 0.444 & 0.742 \\
+ 0.142 & 0.209 & 0.966 \\
+ 0.095 & 0.008 & 0.669 \\
+ 0.381 & 0.206 & 0.379 \\
+\end{array}
+\right)$. 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: $0.39$ 
+Surface Area: $0.31$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/67018.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.162 & 0.941 & 0.265 \\
+ 0.404 & 0.615 & 0.93 \\
+ 0.749 & 0.223 & 0.725 \\
+ 0.455 & 0.064 & 0.063 \\
+ 0.874 & 0.282 & 0.574 \\
+ 0.951 & 0.753 & 0.451 \\
+ 0.909 & 0.267 & 0.203 \\
+ 0.687 & 0.644 & 0.81 \\
+\end{array}
+\right)$. 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: $1.79$ 
+Volume: $0.15$ 
+Solid Angle: $0.65$

pretraining/mathematica/geometry/solids/71412.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.548 & 0.039 & 0.374 \\
+ 0.69 & 0.885 & 0.286 \\
+ 0.558 & 0.655 & 0.417 \\
+ 0.254 & 0.165 & 0.942 \\
+ 0.805 & 0.742 & 0.756 \\
+ 0.884 & 0.13 & 0.562 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.22$ 
+Surface Area: $1.17$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/73725.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.022 & 0.239 & 0.698 \\
+ 0.812 & 0.361 & 0.9 \\
+ 0.724 & 0.789 & 0.101 \\
+ 0.499 & 0.938 & 0.326 \\
+ 0.889 & 0.062 & 0.846 \\
+ 0.472 & 0.093 & 0.171 \\
+ 0.049 & 0.283 & 0.674 \\
+\end{array}
+\right)$. 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: $1.73$ 
+Solid Angle: $0.83$

pretraining/mathematica/geometry/solids/73987.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.609 & 0.158 & 0.958 \\
+ 0.661 & 0.21 & 0.83 \\
+ 0.741 & 0.754 & 0.559 \\
+ 0.675 & 0.069 & 0.636 \\
+ 0.146 & 0.758 & 0.939 \\
+ 0.184 & 0.575 & 0.389 \\
+ 0.775 & 0.838 & 0.038 \\
+ 0.476 & 0.326 & 0.414 \\
+\end{array}
+\right)$. 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: $0.92$ 
+Surface Area: $1.5$

pretraining/mathematica/geometry/solids/75967.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.004 & 0.114 & 0.611 \\
+ 0.438 & 0.76 & 0.378 \\
+ 0.358 & 0.599 & 0.359 \\
+ 0.049 & 0.723 & 0.187 \\
+\end{array}
+\right)$. 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: $0.35$ 
+Solid Angle: $0.03$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/76527.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.865 & 0.638 & 0.927 \\
+ 0.206 & 0.79 & 0.515 \\
+ 0.928 & 0.637 & 0.751 \\
+ 0.48 & 0.233 & 0.639 \\
+ 0.342 & 0.583 & 0.194 \\
+ 0.19 & 0.863 & 0.636 \\
+\end{array}
+\right)$. 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: $0.88$ 
+Volume: $0.04$ 
+Solid Angle: $0.86$

pretraining/mathematica/geometry/solids/76823.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.281 & 0.985 & 0.259 \\
+ 0.909 & 0.729 & 0.362 \\
+ 0.841 & 0.592 & 0.544 \\
+ 0.986 & 0.271 & 0.186 \\
+ 0.349 & 0.857 & 0.99 \\
+ 0.727 & 0.417 & 0.225 \\
+ 0.12 & 0.137 & 0.803 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.95$ 
+Surface Area: $1.78$ 
+Volume: $0.14$