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Ratio of volumes of similar solids

Ratio of volumes of similar solids

This 3D scene explains the correlation between the ratio of similarity and the ratio of volume of geometric solids.

Mathematics

Keywords

volume, sphere, pyramid, cube, cuboid, right circular cone, ratio, surface, formula, radius, height, regular square pyramid, edge, motherboard, solid figure, space, similarity, középpont, geometry, solid geometry, mathematics

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Scenes

A right circular cone is a cone with a circular base in which the orthogonal projection of the apex on the bottom base coincides with the center of the base. The volume of a right circular cone is one third the product of its base area (r²π) and its height (h).

If we enlarge a right circular cone by a scale factor of 2, both the radius of the base and the height will be doubled. Since both the base of the power and the other factor in the formula for the cone the volume are doubled, the volume of the enlarged right circular cone is 8 times larger than that of the original one.

If we enlarge a right circular cone by a scale factor λ of 3, both the radius of the base and the height will be tripled. Since both the base of the power and the other factor in the formula for the cone the volume are tripled, the volume of the enlarged right circular cone is 27 times larger than that of the original one.

In general, if we enlarge a right circular cone by a scale factor λ, its volume increases by λ³.

A sphere is the set of points in space that are at equal distance from a given point in space (the center of the sphere, O). The volume of a sphere is equal to four-thirds the product of π and the cube of the radius of the sphere.

If we enlarge a sphere by a scale factor of 2, the length of its radius will be doubled. Since the base of the power in the formula for the volume of the sphere is doubled, so the volume of the enlarged sphere is 8 times larger than that of the original one.

If we enlarge a sphere by a scale factor λ of 3, the length of its radius will be tripled. Since the base of the power in the formula for the volume of the sphere is tripled, so the volume of the enlarged sphere is 27 times larger than that of the original one.

In general, if we enlarge a sphere by a scale factor λ, its volume increases by λ³.

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