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pretraining/mathematica/geometry/solids/10525.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{2.419,-1.751,-7.406\}$ has radius $0.35$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $1.54$ 
+Volume: $0.18$

pretraining/mathematica/geometry/solids/11445.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.078 & 0.501 & 0.399 \\
+ 0.15 & 0.864 & 0.314 \\
+ 0.262 & 0.869 & 0.037 \\
+ 0.491 & 0.04 & 0.207 \\
+ 0.538 & 0.392 & 0.406 \\
+ 0.073 & 0.955 & 0.095 \\
+\end{array}
+\right)$. 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: $1.32$ 
+Surface Area: $0.71$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/1264.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.148 & 0.433 & 0.311 \\
+ 0.894 & 0.955 & 0.523 \\
+ 0.576 & 0.94 & 0.379 \\
+ 0.849 & 0.666 & 0.869 \\
+ 0.936 & 0.074 & 0.305 \\
+ 0.824 & 0.182 & 0.861 \\
+ 0.748 & 0.255 & 0.041 \\
+\end{array}
+\right)$. 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: $1.7$ 
+Volume: $0.14$ 
+Solid Angle: $0.89$

pretraining/mathematica/geometry/solids/14807.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.804 & 0.537 & 0.022 \\
+ 0.744 & 0.603 & 0.933 \\
+ 0.173 & 0.68 & 0.658 \\
+ 0.597 & 0.495 & 0.956 \\
+ 0.525 & 0.035 & 0.372 \\
+ 0.556 & 0.752 & 0.201 \\
+ 0.121 & 0.509 & 0.62 \\
+ 0.038 & 0.02 & 0.735 \\
+ 0.132 & 0.516 & 0.414 \\
+\end{array}
+\right)$. 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:
+Solid Angle: $0.79$ 
+Volume: $0.15$ 
+Surface Area: $1.72$

pretraining/mathematica/geometry/solids/17544.txt

@@ -0,0 +1,37 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -1.618 & -1. \\
+ 0. & -1.618 & 0. \\
+ 0. & 1.618 & -1. \\
+ 0. & 1.618 & 0. \\
+ -0.951 & -1.309 & -1. \\
+ -0.951 & -1.309 & 0. \\
+ -0.951 & 1.309 & -1. \\
+ -0.951 & 1.309 & 0. \\
+ 0.951 & -1.309 & -1. \\
+ 0.951 & -1.309 & 0. \\
+ 0.951 & 1.309 & -1. \\
+ 0.951 & 1.309 & 0. \\
+ -1.539 & -0.5 & -1. \\
+ -1.539 & -0.5 & 0. \\
+ -1.539 & 0.5 & -1. \\
+ -1.539 & 0.5 & 0. \\
+ 1.539 & -0.5 & -1. \\
+ 1.539 & -0.5 & 0. \\
+ 1.539 & 0.5 & -1. \\
+ 1.539 & 0.5 & 0. \\
+ 1.376 & 0. & 0.851 \\
+ 0.425 & -1.309 & 0.851 \\
+ 0.425 & 1.309 & 0.851 \\
+ -1.114 & -0.809 & 0.851 \\
+ -1.114 & 0.809 & 0.851 \\
+ -0.851 & 0. & 1.376 \\
+ -0.263 & -0.809 & 1.376 \\
+ -0.263 & 0.809 & 1.376 \\
+ 0.688 & -0.5 & 1.376 \\
+ 0.688 & 0.5 & 1.376 \\
+\end{array}
+\right)$. Determine the GeneralizedDiameter.
+Answer:
+$3.49$

pretraining/mathematica/geometry/solids/19117.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.422 & 0.007 & 0.391 \\
+ 0.974 & 0.636 & 0.035 \\
+ 0.939 & 0.262 & 0.779 \\
+ 0.096 & 0.051 & 0.721 \\
+ 0.042 & 0.667 & 0.841 \\
+ 0.048 & 0.076 & 0.18 \\
+ 0.375 & 0.636 & 0.62 \\
+\end{array}
+\right)$. 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: $3.51$ 
+Surface Area: $2.05$ 
+Volume: $0.18$

pretraining/mathematica/geometry/solids/21217.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $5.132$ has its base centered at$\{7.02,9.794,3.486\}$ and its tip is at $\{2.676,1.63,0.94\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Volume: $264.49$ 
+Centroid: $\{5.93,7.75,2.85\}$ 
+Surface Area: $258.09$

pretraining/mathematica/geometry/solids/22090.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.758 & 0.79 & 0.106 \\
+ 0.834 & 0.03 & 0.451 \\
+ 0.118 & 0.973 & 0.391 \\
+ 0.134 & 0.83 & 0.169 \\
+ 0.896 & 0.745 & 0.584 \\
+ 0.065 & 0.475 & 0.545 \\
+ 0.079 & 0.087 & 0.767 \\
+ 0.2 & 0.395 & 0.322 \\
+\end{array}
+\right)$. 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: $1.5$ 
+Surface Area: $1.94$ 
+Volume: $0.16$

pretraining/mathematica/geometry/solids/25597.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.875 & 0.949 & 0.804 \\
+ 0.737 & 0.634 & 0.425 \\
+ 0.307 & 0.301 & 0.889 \\
+ 0.046 & 0.113 & 0.903 \\
+ 0.882 & 0.443 & 0.27 \\
+ 0.256 & 0.461 & 0.26 \\
+ 0.362 & 0.884 & 0.868 \\
+ 0.798 & 0.097 & 0.038 \\
+\end{array}
+\right)$. 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: $1.91$ 
+Solid Angle: $0.58$ 
+Volume: $0.14$

pretraining/mathematica/geometry/solids/29730.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.968 & 0.973 & 0.645 \\
+ 0.143 & 0.208 & 0.833 \\
+ 0.538 & 0.428 & 0.707 \\
+ 0.398 & 0.325 & 0.653 \\
+ 0.792 & 0.02 & 0.24 \\
+\end{array}
+\right)$. 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: $0.04$ 
+Volume: $0.01$ 
+Surface Area: $0.9$

pretraining/mathematica/geometry/solids/31364.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.792 & 0.462 & 0.901 \\
+ 0.584 & 0.115 & 0.216 \\
+ 0.256 & 0.659 & 0.117 \\
+ 0.005 & 0.386 & 0.849 \\
+\end{array}
+\right)$. 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: $1.12$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/33511.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.834 & 0.474 & 0.893 \\
+ 0.561 & 0.41 & 0.892 \\
+ 0.683 & 0.912 & 0.249 \\
+ 0.588 & 0.426 & 0.347 \\
+ 0.178 & 0.13 & 0.808 \\
+ 0.872 & 0.965 & 0.964 \\
+ 0.647 & 0.367 & 0.736 \\
+ 0.02 & 0.891 & 0.518 \\
+ 0.067 & 0.833 & 0.404 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Solid Angle: $1.9$ 
+Surface Area: $1.8$

pretraining/mathematica/geometry/solids/34677.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.57 & 0.245 & 0.074 \\
+ 0.965 & 0.688 & 0.072 \\
+ 0.135 & 0.392 & 0.889 \\
+ 0.048 & 0.5 & 0.411 \\
+ 0.298 & 0.981 & 0.37 \\
+\end{array}
+\right)$. 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: $0.79$ 
+Surface Area: $1.28$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/35476.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.622 & 0.033 & 0.163 \\
+ 0.406 & 0.991 & 0.15 \\
+ 0.707 & 0.631 & 0.228 \\
+ 0.886 & 0.279 & 0.922 \\
+ 0.356 & 0.264 & 0.431 \\
+ 0.624 & 0.725 & 0.138 \\
+ 0.914 & 0.22 & 0.757 \\
+ 0.396 & 0.257 & 0.289 \\
+ 0.558 & 0.813 & 0.122 \\
+ 0.694 & 0.48 & 0.723 \\
+\end{array}
+\right)$. 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: $0.93$ 
+Surface Area: $1.21$ 
+Volume: $0.08$

pretraining/mathematica/geometry/solids/36128.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.691 & 0.926 & 0.885 \\
+ 0.706 & 0.259 & 0.052 \\
+ 0.625 & 0.355 & 0.706 \\
+ 0.168 & 0.745 & 0.599 \\
+ 0.983 & 0.71 & 0.252 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.07$ 
+Surface Area: $1.19$ 
+Solid Angle: $0.67$

pretraining/mathematica/geometry/solids/36311.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.527 & 0.204 & 0.399 \\
+ 0.644 & 0.628 & 0.82 \\
+ 0.959 & 0.711 & 0.035 \\
+ 0.425 & 0.115 & 0.752 \\
+ 0.965 & 0.658 & 0.775 \\
+ 0.683 & 0.278 & 0.188 \\
+ 0.962 & 0.22 & 0.012 \\
+ 0.406 & 0.675 & 0.106 \\
+\end{array}
+\right)$. 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: $1.55$ 
+Solid Angle: $4.46$

pretraining/mathematica/geometry/solids/36390.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.317 & 0.356 & 0.994 \\
+ 0.056 & 0.33 & 0.019 \\
+ 0.369 & 0.947 & 0.913 \\
+ 0.913 & 0.252 & 0.013 \\
+ 0.398 & 0.664 & 0.958 \\
+ 0.116 & 0.74 & 0.394 \\
+ 0.615 & 0.965 & 0.278 \\
+ 0.989 & 0.627 & 0.807 \\
+\end{array}
+\right)$. 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: $1.43$ 
+Surface Area: $2.45$

pretraining/mathematica/geometry/solids/36657.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.649 & 0.288 & 0.538 \\
+ 0.769 & 0.6 & 0.5 \\
+ 0.094 & 0.95 & 0.668 \\
+ 0.542 & 0.773 & 0.974 \\
+ 0.934 & 0.253 & 0.962 \\
+ 0.197 & 0.019 & 0.924 \\
+ 0.069 & 0.168 & 0.281 \\
+ 0.731 & 0.113 & 0.952 \\
+\end{array}
+\right)$. 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: $4.41$ 
+Surface Area: $1.95$ 
+Volume: $0.18$

pretraining/mathematica/geometry/solids/38253.txt

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

pretraining/mathematica/geometry/solids/38367.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.529 & 0.084 & 0.883 \\
+ 0.773 & 0.312 & 0.189 \\
+ 0.55 & 0.073 & 0.319 \\
+ 0.93 & 0.015 & 0.754 \\
+ 0.955 & 0.77 & 0.706 \\
+ 0.928 & 0.138 & 0.215 \\
+\end{array}
+\right)$. 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:
+Surface Area: $1.05$ 
+Volume: $0.06$ 
+Solid Angle: $0.82$

pretraining/mathematica/geometry/solids/38640.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.35 & 0.648 & 0.65 \\
+ 0.894 & 0.639 & 0.972 \\
+ 0.965 & 0.376 & 0.864 \\
+ 0.729 & 0.1 & 0.111 \\
+ 0.338 & 0.926 & 0.995 \\
+\end{array}
+\right)$. 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: $0.44$ 
+Surface Area: $1.$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/40361.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $3.285$ has its base centered at$\{6.029,4.026,4.665\}$ and its tip is at $\{6.181,9.543,6.577\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Volume: $66.02$ 
+Surface Area: $103.08$ 
+Centroid: $\{6.07,5.4,5.14\}$

pretraining/mathematica/geometry/solids/43511.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.564 & 0.717 & 0.132 \\
+ 0.919 & 0.744 & 0.38 \\
+ 0.775 & 0.442 & 0.737 \\
+ 0.618 & 0.584 & 0.243 \\
+ 0.978 & 0.487 & 0.29 \\
+\end{array}
+\right)$. 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: $0.38$ 
+Volume: $0.01$ 
+Surface Area: $0.38$

pretraining/mathematica/geometry/solids/45315.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.94 & 0.936 & 0.468 \\
+ 0.876 & 0.031 & 0.181 \\
+ 0.695 & 0.69 & 0.858 \\
+ 0.225 & 0.847 & 0.758 \\
+ 0.375 & 0.818 & 0.179 \\
+ 0.843 & 0.268 & 0.723 \\
+ 0.606 & 0.106 & 0.822 \\
+ 0.501 & 0.632 & 0.023 \\
+ 0.423 & 0.826 & 0.886 \\
+ 0.323 & 0.543 & 0.789 \\
+\end{array}
+\right)$. 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: $0.19$ 
+Solid Angle: $1.52$ 
+Surface Area: $2.01$

pretraining/mathematica/geometry/solids/45374.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.737 & 0.05 & 0.373 \\
+ 0.936 & 0.681 & 0.562 \\
+ 0.081 & 0.185 & 0.115 \\
+ 0.966 & 0.369 & 0.944 \\
+ 0.574 & 0.528 & 0.887 \\
+ 0.756 & 0.988 & 0.514 \\
+ 0.732 & 0.644 & 0.276 \\
+ 0.866 & 0.2 & 0.077 \\
+ 0.418 & 0.302 & 0.738 \\
+\end{array}
+\right)$. 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: $0.18$ 
+Solid Angle: $3.03$

pretraining/mathematica/geometry/solids/46793.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.298 & 0.023 & 0.643 \\
+ 0.023 & 0.851 & 0.305 \\
+ 0.502 & 0.812 & 0.002 \\
+ 0.615 & 0.967 & 0.13 \\
+ 0.853 & 0.335 & 0.978 \\
+ 0.882 & 0.544 & 0.587 \\
+ 0.169 & 0.608 & 0.596 \\
+ 0.11 & 0.46 & 0.422 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.69$ 
+Solid Angle: $0.8$ 
+Volume: $0.12$

pretraining/mathematica/geometry/solids/46878.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.938 & 0.353 & 0.817 \\
+ 0.9 & 0.938 & 0.056 \\
+ 0.153 & 0.031 & 0.261 \\
+ 0.046 & 0.989 & 0.828 \\
+\end{array}
+\right)$. 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: $0.75$ 
+Volume: $0.15$

pretraining/mathematica/geometry/solids/47195.txt

@@ -0,0 +1,12 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & 0.707 \\
+ 0. & -0.707 & 0. \\
+ 0. & 0.707 & 0. \\
+ -0.707 & 0. & 0. \\
+ 0.707 & 0. & 0. \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$8.$

pretraining/mathematica/geometry/solids/48556.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.86 & 0.338 & 0.71 \\
+ 0.807 & 0.068 & 0.551 \\
+ 0.179 & 0.161 & 0.855 \\
+ 0.37 & 0.475 & 0.937 \\
+ 0.806 & 0.518 & 0.198 \\
+ 0.888 & 0.605 & 0.979 \\
+ 0.075 & 0.147 & 0.347 \\
+ 0.772 & 0.311 & 0.208 \\
+ 0.591 & 0.006 & 0.898 \\
+\end{array}
+\right)$. 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: $1.64$ 
+Volume: $0.14$ 
+Solid Angle: $5.3$

pretraining/mathematica/geometry/solids/49645.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.992 & 0.514 & 0.659 \\
+ 0.882 & 0.427 & 0.472 \\
+ 0.365 & 0.212 & 0.1 \\
+ 0.971 & 0.698 & 0.049 \\
+ 0.322 & 0.506 & 0.94 \\
+ 0.257 & 0.379 & 0.4 \\
+ 0.215 & 0.838 & 0.673 \\
+\end{array}
+\right)$. 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: $1.51$ 
+Solid Angle: $1.08$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/50327.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.607 & 0.03 & 0.62 \\
+ 0.263 & 0.679 & 0.514 \\
+ 0.314 & 0.535 & 0.732 \\
+ 0.319 & 0.804 & 0.144 \\
+ 0.407 & 0.073 & 0.827 \\
+ 0.88 & 0.234 & 0.527 \\
+ 0.85 & 0.022 & 0.881 \\
+ 0.752 & 0.22 & 0.422 \\
+ 0.596 & 0.398 & 0.94 \\
+ 0.992 & 0.817 & 0.851 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.15$ 
+Solid Angle: $3.23$ 
+Surface Area: $1.66$

pretraining/mathematica/geometry/solids/50666.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.311 & 0.593 & 0.846 \\
+ 0.562 & 0.891 & 0.403 \\
+ 0.236 & 0.301 & 0.393 \\
+ 0.57 & 0.152 & 0.033 \\
+ 0.8 & 0.868 & 0.693 \\
+ 0.385 & 0.744 & 0.114 \\
+ 0.881 & 0.498 & 0.87 \\
+ 0.883 & 0.861 & 0.495 \\
+ 0.525 & 0.01 & 0.808 \\
+\end{array}
+\right)$. 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: $2.07$ 
+Surface Area: $1.85$ 
+Volume: $0.18$

pretraining/mathematica/geometry/solids/51435.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.122 & 0.202 & 0.606 \\
+ 0.328 & 0.04 & 0.846 \\
+ 0.369 & 0.551 & 0.742 \\
+ 0.202 & 0.446 & 0.824 \\
+ 0.024 & 0.216 & 0.805 \\
+ 0.47 & 0.307 & 0.22 \\
+ 0.526 & 0.336 & 0.954 \\
+ 0.482 & 0.39 & 0.013 \\
+ 0.704 & 0.593 & 0.165 \\
+ 0.666 & 0.344 & 0.312 \\
+\end{array}
+\right)$. 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: $1.11$ 
+Volume: $0.07$ 
+Solid Angle: $3.2$

pretraining/mathematica/geometry/solids/52900.txt

@@ -0,0 +1,5 @@
+Problem:
+An ellipsoid centered at $\{-6.805,9.328,4.149\}$ has radii $\{3.566,8.49,8.808\}$. Estimate the ellipsoid's surface area and volume.
+Answer:
+Volume: $1117.$ 
+Surface Area: $604.41$

pretraining/mathematica/geometry/solids/53433.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.278 & 0.193 & 0.549 \\
+ 0.371 & 0.394 & 0.502 \\
+ 0.613 & 0.391 & 0.765 \\
+ 0.339 & 0.087 & 0.749 \\
+\end{array}
+\right)$. 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: $0.$ 
+Solid Angle: $0.15$ 
+Surface Area: $0.18$

pretraining/mathematica/geometry/solids/54657.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{-7.777,-7.704,-6.222\}$ has radius $0.728$. Estimate the sphere's surface area and volume.
+Answer:
+Volume: $1.62$ 
+Surface Area: $6.66$

pretraining/mathematica/geometry/solids/54683.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & -1. & 0. \\
+ 0. & 1. & 0. \\
+ -0.289 & -0.5 & 0.816 \\
+ -0.289 & 0.5 & 0.816 \\
+ 0.577 & 0. & 0.816 \\
+ -0.866 & -0.5 & 0. \\
+ -0.866 & 0.5 & 0. \\
+ 0.866 & -0.5 & 0. \\
+ 0.866 & 0.5 & 0. \\
+\end{array}
+\right)$. Determine the Circumcenter.
+Answer:
+$\{0.,0.,0.\}$

pretraining/mathematica/geometry/solids/55407.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.375 & 0.98 & 0.961 \\
+ 0.31 & 0.784 & 0.148 \\
+ 0.647 & 0.59 & 0.225 \\
+ 0.64 & 0.46 & 0.449 \\
+ 0.915 & 0.234 & 0.552 \\
+\end{array}
+\right)$. 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: $0.08$ 
+Surface Area: $0.83$ 
+Volume: $0.02$

pretraining/mathematica/geometry/solids/55630.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.352 & 0.763 & 0.247 \\
+ 0.998 & 0.68 & 0.921 \\
+ 0.005 & 0.017 & 0.454 \\
+ 0.287 & 0.802 & 0.012 \\
+ 0.673 & 0.775 & 0.103 \\
+ 0.332 & 0.002 & 0.583 \\
+ 0.356 & 0.463 & 0.843 \\
+\end{array}
+\right)$. 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: $1.76$ 
+Solid Angle: $4.58$

pretraining/mathematica/geometry/solids/5594.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.079 & 0.939 & 0.044 \\
+ 0.668 & 0.298 & 0.685 \\
+ 0.373 & 0.012 & 0.938 \\
+ 0.823 & 0.497 & 0.444 \\
+ 0.91 & 0.286 & 0.464 \\
+ 0.889 & 0.372 & 0.172 \\
+ 0.393 & 0.987 & 0.84 \\
+ 0.135 & 0.496 & 0.636 \\
+\end{array}
+\right)$. 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:
+Volume: $0.15$ 
+Surface Area: $1.96$ 
+Solid Angle: $0.48$

pretraining/mathematica/geometry/solids/57492.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.456 & 0.307 & 0.289 \\
+ 0.08 & 0.542 & 0.584 \\
+ 0.948 & 0.221 & 0.577 \\
+ 0.286 & 0.81 & 0.992 \\
+ 0.858 & 0.764 & 0.99 \\
+ 0.432 & 0.093 & 0.897 \\
+ 0.188 & 0.043 & 0.175 \\
+ 0.766 & 0.67 & 0.978 \\
+\end{array}
+\right)$. 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: $3.54$ 
+Volume: $0.14$ 
+Surface Area: $1.78$

pretraining/mathematica/geometry/solids/61137.txt

@@ -0,0 +1,27 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)} & 0 & \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)} & 0 & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & 0 & \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & 0 & -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} & \frac{1}{4} \left(1-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} & \frac{1}{4} \left(\sqrt{5}-1\right) & \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} & \frac{1}{4} \left(1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} & \frac{1}{4} \left(\sqrt{5}-1\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)} & -\frac{1}{2} & \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)} & \frac{1}{2} & \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)} & -\frac{1}{2} & -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)} & \frac{1}{2} & -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(\sqrt{5}-1\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(\sqrt{5}-3\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(\sqrt{5}-3\right) & \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(1-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+ \frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(\sqrt{5}-1\right) & \frac{1}{2} \sqrt{\frac{1}{10} \left(25-11 \sqrt{5}\right)} \\
+\end{array}
+\right)$. Determine the Incenter.
+Answer:
+$\{0,0,0\}$

pretraining/mathematica/geometry/solids/62463.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.266 & 0.75 & 0.34 \\
+ 0.966 & 0.585 & 0.399 \\
+ 0.828 & 0.885 & 0.52 \\
+ 0.566 & 0.064 & 0.663 \\
+ 0.615 & 0.649 & 0.788 \\
+ 0.185 & 0.725 & 0.519 \\
+ 0.655 & 0.888 & 0.774 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.05$ 
+Solid Angle: $1.41$ 
+Volume: $0.06$

pretraining/mathematica/geometry/solids/62800.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.765 & 0.151 & 0.149 \\
+ 0.921 & 0.396 & 0.944 \\
+ 0.302 & 0.188 & 0.864 \\
+ 0.759 & 0.571 & 0.572 \\
+ 0.726 & 0.572 & 0.475 \\
+ 0.431 & 0.415 & 0.374 \\
+ 0.413 & 0.442 & 0.58 \\
+ 0.685 & 0.166 & 0.563 \\
+\end{array}
+\right)$. 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: $0.05$ 
+Solid Angle: $0.59$ 
+Surface Area: $0.91$

pretraining/mathematica/geometry/solids/6361.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.642 & 0.564 & 0.672 \\
+ 0.142 & 0.301 & 0.432 \\
+ 0.485 & 0.277 & 0.334 \\
+ 0.474 & 0.319 & 0.638 \\
+ 0.103 & 0.902 & 0.734 \\
+ 0.195 & 0.638 & 0.108 \\
+ 0.548 & 0.996 & 0.042 \\
+\end{array}
+\right)$. 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: $1.77$ 
+Volume: $0.1$ 
+Surface Area: $1.26$

pretraining/mathematica/geometry/solids/64.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.49 & 0.87 & 0.745 \\
+ 0.729 & 0.141 & 0.544 \\
+ 0.328 & 0.324 & 0.917 \\
+ 0.911 & 0.896 & 0.075 \\
+ 0.778 & 0.304 & 0.956 \\
+ 0.294 & 0.609 & 0.155 \\
+\end{array}
+\right)$. 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: $1.57$ 
+Solid Angle: $1.68$

pretraining/mathematica/geometry/solids/65390.txt

@@ -0,0 +1,6 @@
+Problem:
+A cone with radius $9.218$ has its base centered at$\{0.415,1.75,6.224\}$ and its tip is at $\{7.631,5.443,4.497\}$. Estimate the cone's surface area, volume, and centroid.
+Answer:
+Centroid: $\{2.22,2.67,5.79\}$ 
+Surface Area: $625.94$ 
+Volume: $737.49$

pretraining/mathematica/geometry/solids/66007.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.348 & 0.268 & 0.312 \\
+ 0.761 & 0.442 & 0.319 \\
+ 0.683 & 0.735 & 0.022 \\
+ 0.608 & 0.963 & 0.526 \\
+ 0.187 & 0.423 & 0.91 \\
+ 0.408 & 0.539 & 0.175 \\
+ 0.748 & 0.977 & 0.499 \\
+ 0.585 & 0.193 & 0.865 \\
+\end{array}
+\right)$. 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: $0.1$ 
+Surface Area: $1.36$ 
+Solid Angle: $1.89$

pretraining/mathematica/geometry/solids/67603.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.659 & 0.59 & 0.28 \\
+ 0.653 & 0.169 & 0.62 \\
+ 0.008 & 0.47 & 0.718 \\
+ 0.584 & 0.01 & 0.364 \\
+ 0.892 & 0.064 & 0.126 \\
+ 0.179 & 0.737 & 0.635 \\
+ 0.117 & 0.577 & 0.271 \\
+ 0.656 & 0.429 & 0.988 \\
+ 0.034 & 0.063 & 0.25 \\
+ 0.853 & 0.992 & 0.439 \\
+\end{array}
+\right)$. 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: $5.99$ 
+Surface Area: $2.28$

pretraining/mathematica/geometry/solids/69303.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.038 & 0.096 & 0.83 \\
+ 0.335 & 0.986 & 0.489 \\
+ 0.362 & 0.984 & 0.242 \\
+ 0.929 & 0.363 & 0.084 \\
+ 0.095 & 0.678 & 0.358 \\
+ 0.67 & 0.211 & 0.986 \\
+ 0.35 & 0.091 & 0.903 \\
+ 0.683 & 0.897 & 0.37 \\
+ 0.747 & 0.803 & 0.408 \\
+ 0.145 & 0.51 & 0.205 \\
+\end{array}
+\right)$. 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: $0.85$ 
+Surface Area: $2.11$ 
+Volume: $0.21$