Practice (63)
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A pyramid with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
The two cones shown have parallel bases and common apex $T$. $TW = 32$ m, $WV = 8$ m and $ZY = 5$ m. What is the volume of the frustum with circle $W$ and circle $Z$ as its bases? Express your answer in terms of $\pi$.
A square prism has dimensions $5' \times 5' \times 10'$, where ABCD is a square. AP = ER = 2 ft and QC = SG = 1 ft. The plane containing $\overline{PQ}$ and $\overline{RS}$ slices the original prism into two new prisms. What is the volume of the larger of these two prisms?
A right square pyramid has a base with a perimeter of 36 cm and a height of 12 cm. At one-third of the distance up from the base to the apex, the pyramid is cut by a plane parallel to its base. What is the volume of the top pyramid?
A right rectangular prism has a volume of 720 $cm^3$. Its surface area is 484 $cm^2$. If all edge lengths are integers, what is the length of the longest segment that can be drawn that connects two vertices? Express your answer in simplest radical form.
A pyramid has 6 vertices and 6 faces. How many edges does it have?
A rectangular prism is composed of unit cubes. The outside faces of the prism are painted blue and the seven unit cubes in the interior are unpainted. We must find how many unit cubes have exactly one painted face.
The diameter of a spherical balloon is increased by 150%. We must find by what percent the volume increases.
Segments AD and BC are the radii of the top and bottom bases of the frustum. AD = 8, BC = 12 and AC = 15, what is the volume of the frustum?
A cylindrical container has a diameter of 8 cm (i.e., a radius of 4 cm) and a volume of 754 $cm^2$. Another container also has a diameter of 8 cm, but is twice as tall as the original container. We are asked to find the volume of the second container.
There is a shallow fish pond in the shape of a square. The perimeter of the pond is 24 ft and the water is 6 in deep. We must find the volume of the water in the pond.
A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$.
A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons?
What is the maximum number of spheres with radius 1 that can fit into a sphere with radius 2?
Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 center at each of the $2^n$ points. let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?
Let $C$ be a three-dimensional cube with edge length 1. There are 8 equilateral triangles whose vertices are vertices of $C$. The 8 planes that contain these 8 equilateral triangles divide $C$ into several non-overlapping regions. Find the volume of the region that contains the center of $C$.
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
Points $E$ and $F$ are located on square $ABCD$ so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?
Three pairwise-tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
Let $f(n)$ be the number of points of intersections of diagonals of a $n$-dimensional hypercube that is not the vertice of the cube. For example, $f(3) = 7$ because the intersection points of a cube's diagonals are at the centers of each face and the center of the cube. Find $f(5)$
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^{\circ}$. Find $h^2$.
A 12-sided game die has the shape of a hexagonal bipyramid, which consists of two pyramids, each with a regular hexagonal base of side length 1 cm and with height 1 cm, glued together along their hexagons. When this game die is rolled and lands on one of its triangular faces, how high off the ground is the opposite face? Express your answer as a common fraction in simplest radical form.
The areas of three faces of a rectangular prism are 54, 24 and 36 units . What is the length of the space diagonal of this prism? Express your answer in simplest radical form.