All Questions ⁺⁰236282 Questions

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+1504

Geometry

A regular 18-gon is rotated counterclockwise about its center by a nonzero angle. If it is rotated 45 degrees, then what is the area of the overlap?

kittykat Dec 31, 2024

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1

1

+1375

Geometry

A box contains one sphere of radius 2 and four spheres of radius $2$, as shown below. The sphere of radius 2 is tangent to five sides of the box, and each sphere of radius $2$ is tangent to three sides of the box. Find the volume of the box read more ..

learnmgcat Dec 31, 2024

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1

+1375

Geometry

Points A,B, and C are on a sphere whose radius is 5. If $AB=BC=5,$ what is the longest possible value of $AC?$

learnmgcat Dec 31, 2024

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1

1

+815

Geometry

A unit sphere is placed in a corner of a room, so that it is tangent to both walls and the floor. A second, smaller sphere is placed so that PA = 2, PB = 3, and PC = 5. Find the radius of the smaller sphere.

booboo44 Dec 31, 2024

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1

+815

Geometry

A cone contains two spheres, as shown below. If the radii of the spheres are 1 and $4,$ then find the volume of the cone.

booboo44 Dec 31, 2024

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1

+815

Geometry

A sphere passes through all four vertices of one face of a unit cube, and is tangent to the four edges of the opposite face of the cube. Find the radius of the sphere.

booboo44 Dec 31, 2024

0

1

1

+815

Geometry

All eight vertices of a cube are on a sphere (i.e. the cube is inscribed in the sphere). A sphere is also inscribed in the cube. If the side length of the cube is 1, then what is the radius of the sphere inside the cube?

booboo44 Dec 31, 2024

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1

+1552

Geometry

As shown in the diagram, a hemisphere sits on top of a cylinder with the same radius, forming a complete solid. The cylinder portion of the solid (through its lateral surface area and its base) constitutes $\frac{1}{2}$ of the total surface area of read more ..

blackpanther Dec 31, 2024

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1

+1552

Geometry

The two bases of a cylinder are two parallel cross sections of a sphere. We know the radius of the sphere is 5 and the height of the cylinder is $5.$ Find the volume of the cylinder.

blackpanther Dec 31, 2024

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1

+1552

Geometry

A and B are two points on a unit sphere of radius 2. We know the space distance between $A$ and $B$ is $4$. What is distance from $A$ to $B$ along the (minor) arc of a great circle?

blackpanther Dec 31, 2024

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1

+1552

Geometry

The surface area of a sphere is numerically equal to twice its volume. What is the radius of the sphere?

blackpanther Dec 31, 2024

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+4

Help

At a Calvinball tournament, people, including Calvin and Hobbes, compete in a series of one-on-one games. Every person plays against every other person in the tournament exactly once. No ties occur, and everyone is equally skilled, so each player read more ..

greenflag Dec 31, 2024

Dec 30, 2024

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+1552

Geometry

The total surface of a cone \mathcal{C}, including its base, is painted. A plane parallel to the base of the cone divides the cone into a smaller cone $\mathcal{A}$ and a frustum $\mathcal{B},$ so that the ratio of the volume of $\mathcal{A}$ to the read more ..

blackpanther Dec 30, 2024

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1

+1552

Geometry

A cone has a base radius of 3 and a height of 6. A cube is inscribed in the cone, so that four of its vertices lie in the circular base, and the other four vertices touch the lateral surface. Find the side length of the cube.

blackpanther Dec 30, 2024

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1

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+1552

Geometry

A circle is divided into two sectors, and each sector is rolled into a cone. If the lateral surface area of the first cone is equal to five times the lateral surface area of the second cone, then find the ratio of the volume of the first cone to the read more ..

blackpanther Dec 30, 2024

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+1552

Geometry

A right cone has a radius of 10 and a height of $20.$ Let $A$ and $B$ be diametrically opposite points on the base, and let $C$ be the midpoint of $B$ and the apex of the cone. A bug crawls from $A$ to $C$ on the surface of the cone. What read more ..

blackpanther Dec 30, 2024

+1

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+911

Geometry

Find the volume of a cone that has 12 as its base radius and a lateral surface area of 100 \pi.

magenta Dec 30, 2024

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+911

Geometry

ABCDEFGH is a unit cube. The intersection of tetrahedrons $GIAC$ and $CGHI$ is another solid. Find the volume of this solid.

magenta Dec 30, 2024

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1

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+911

Geometry

The surface area of a regular octahedron is 1. It is placed on a table, so that one side lies flat. If the sun is shining directly overhead, then what percent of the surface area is dark?

magenta Dec 30, 2024

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2

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+911

Geometry

The centroids of the faces of a regular octahedron are connected to form another regular polyhedron. Find the ratio of the surface area of the new polyhedron to the surfce area of the regular octahedron.

magenta Dec 30, 2024

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1

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+158

Geometry

Given a regular tetrahedron, all the midpoints of the edges are connected to form a new polyhedron. Find the ratio of the surace area of the new polyhedron to the surface area of the original tetrahedron.

cwimie Dec 30, 2024

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1

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+158

Geometry

A regular dodecahedron is made by the net below. Which face is parallel to face H? Enter the letter.

cwimie Dec 30, 2024

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1

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+158

Geometry

The midpoints of a cube are connected to form a new polyhedron. Find the ratio of the surface area of the new polyhedron to the surface area of the cube.

cwimie Dec 30, 2024

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1

1

+158

Geometry

The center of a regular octahedron is 3 units away from every vertex. What is the surface area of the octahedron?

cwimie Dec 30, 2024

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