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

Related items

Scenes

Cuboid

  • a
  • b
  • c
  • 2a
  • 2b
  • 2c
  • 3a
  • 3b
  • 3c

A cuboid is a right prism with a rectangular base. The volume of a cuboid is the product of the lengths of three edges that meet at one vertex, i.e. its height, width and length, here noted with a, b,c.

If we enlarge a cuboid by a scale factor of 2, the lengths of the edges will be doubled. Since the three factors in the volume formula of the cuboid are doubled, the volume of the enlarged cuboid is eight times larger than that of the original cuboid.

If we enlarge a cuboid by a scale factor of 3, the lengths of the edges will be tripled. Since the three factors in the volume formula of the cuboid are tripled, the volume of the enlarged cuboid is 27 times larger than that of the original cuboid.

In general, if we enlarge a cuboid by a scale factor λ, the volume increases by λ³.

Cube

  • a
  • 2a
  • 3a

A cube is a cuboid with square faces. It is one of the five Platonic solids. The volume of a cube is the product of the lengths of three edges that meet at one vertex, that is, the cube of its edge length (here noted with) a.

If we enlarge a cube by a scale factor of 2, the edge length will be doubled. Since the base of the power in the volume formula for the cube is doubled, the volume of the enlarged cube is eight times larger than that of the original cube.

If we enlarge a cube by a scale factor of 3, the edge length will be tripled. Since the base of the power in the volume formula for the cube is tripled, the volume of the enlarged cube is 27 times larger than that of the original cube.

In general, if we enlarge a cube by a scale factor λ, the volume increases by λ³.

Regular square pyramid

  • a
  • b
  • h
  • 2a
  • 2b
  • 2h
  • 3a
  • 3b
  • 3h

A regular square pyramid is a pyramid with a square base and congruent isosceles triangles as lateral faces. The volume of a regular square pyramid is one third the product of its base area (base edge squared, a²) and its height (h).

If we enlarge a regular square pyramid by a scale factor of 2, both the length of its base edge and of its height will be doubled. Since both the base of the power and the other factor in the formula for the pyramid are doubled, the volume of the enlarged pyramid is eight times larger than that of the original one.

If we enlarge a regular square pyramid by a scale factor of 3, both the length of its base edge and of its height will be tripled. Since both the base of the power and the other factor in the formula for the pyramid are tripled, the volume of the enlarged pyramid is 27 times larger than that of the original one.

In general, if we enlarge a regular square pyramid by a scale factor λ, the volume increases by λ³.

Right circular cone

  • r
  • h
  • 2r
  • 2h
  • 3r
  • 3h

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 centre 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 eight times larger than that of the original one.

If we enlarge a right circular cone by a scale factor 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.

Sphere

  • r
  • 2r
  • 3r

A sphere is the set of points in space that are at equal distance from a given point in space (the centre 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, the volume of the enlarged sphere is eight times larger than that of the original one.

If we enlarge a sphere by a scale factor 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, 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.

Related items

Perimeter, area, surface area and volume

This animation presents the formulas to calculate the perimeter and area of shapes as well as the surface area and volume of solids.

Volume of spheres (Cavalieri´s principle)

Calculating the volume of a sphere is possible using an appropriate cylinder and cone.

Volume of spheres (demonstration)

The sum of the volume of the ´tetrahedrons´ gives an approximation of the volume of the sphere.

Conic solids

This animation demonstrates various types of cones and pyramids.

Cube

This animation demonstrates the components (vertices, edges, diagonals and faces) of the cube, one of the Platonic solids.

Cuboid

A cuboid is a polyhedron with six rectangular faces.

Regular square pyramid

A regular square pyramid is a right pyramid with a square base and four triangular faces.

Volume and surface area (exercise)

An exercise about the volume and surface area of solids generated from a ´base cube´.

Volume of a tetrahedron

To calculate the volume of a tetrahedron we start by calculating the volume of a prism.

Grouping of cuboids

This animation demonstrates various types of cuboids through everyday objects.

Grouping of solids

This animation demonstrates various groups of solids through examples.

Grouping of solids 1

This animation demonstrates various groups of solids through examples.

Grouping of solids 2

This animation demonstrates various groups of solids through examples.

Grouping of solids 3

This animation demonstrates various groups of solids through examples.

Grouping of solids 4

This animation demonstrates various groups of solids through examples.

Added to your cart.