Geometric Relationships in Three-Dimensional Shapes Quiz

Each face of a cube is a square.

The sum of interior angles of any polygon is 180 * (n - 2) degrees, where n is the number of sides. A tetrahedron has 4 faces, so the sum of its interior angles is 180 * (4 - 2) = 360 degrees.

The volume of a cube is calculated as side length cubed, so for a cube with side length 5 units, the volume is 5 * 5 * 5 = 125 cubic units.

A dodecahedron has 12 faces.

The surface area of a cylinder is calculated as 2πrh + 2πr^2, where r is the radius and h is the height. Substituting r = 4 and h = 6 gives 48π square units.

A cone has a circular base.

An octahedron is a polyhedron with 12 edges and 8 vertices.

A rectangular prism has 12 distinct diagonals when all its sides are different in length.

A triangular prism has 6 edges on its lateral faces and 3 edges connecting its bases, totaling 9 edges.

In a regular pyramid, all edges are of equal length.

A regular octahedron has 12 vertices, so there are 12 * (12 - 1) / 2 = 66 total diagonals, but each diagonal connects two vertices, so we divide by 2 to get the number of diagonals, which is 33. However, there are 12 edges, which are not counted as diagonals, so the total number of diagonals is 33 - 12 = 21. Since each diagonal connects two vertices, we have counted each diagonal twice, so we divide by 2 to get the final answer: 21 / 2 = 10.5, which we round to 11. So, a regular octahedron has 11 diagonals.

If the lengths of all edges of a regular tetrahedron are tripled, its volume increases by a factor of 3^3 = 27.

A sphere is not a polyhedron because it does not have flat faces.

Two tetrahedra can share up to 4 faces.

A sphere is a perfectly round geometrical object in three-dimensional space, and it does not have any vertices.

An icosahedron is a polyhedron with 20 faces and 12 vertices.

A pentagonal pyramid has 6 faces.