Surface area and volume of similar solids

key notes :

Definition of Similar Solids

  • Two solids are similar if they have the same shape but different sizes.
  • Corresponding angles are equal, and corresponding sides are proportional.

Scale Factor (k)

  • The scale factor is the ratio of corresponding linear dimensions (lengths, widths, or heights) of two similar solids.
  • If the scale factor between two similar solids is k, then all corresponding linear measurements (edges, radii, heights) are in the ratio 1:k.

Surface Area Ratio

  • The ratio of the surface areas of two similar solids is the square of the scale factor.
  • Formula:

Surface Area of Solid 1 / Surface Area of Solid 2 = k2

  • If the scale factor is k, then:

Surface Area of larger solid = k2 × Surface Area of smaller solid

Volume Ratio

  • The ratio of the volumes of two similar solids is the cube of the scale factor.
  • Formula:

Volume of Solid 1 / Volume of Solid 2 = k3

  • If the scale factor is k, then:

Volume of larger solid = k3 × Volume of smaller solid

Example Calculation

If two similar spheres have a scale factor of 2, then:

  • Their surface area ratio is 22 = 4.
  • Their volume ratio is 23 = 8.

Key Formulas Summary

Scale Factor (k): Ratio of corresponding lengths.

Surface Area Ratio: k2.

Volume Ratio: k3.

Learn with an example

The figures below are similar.

What is the volume of the smaller triangular prism?

V2 = _____ cubic millimetres

Find the cube of the ratio of the corresponding dimensions:

(a/v)3 = (8/6)3 = (4/3)3 = 64/27

Find the ratio of the volumes:

V1/V2 = 64/V2

Use these two ratios to set up a proportion and solve for V2.

64/27 = 64/V2

64/27 ( 27V2 ) = 64/V2 ( 27V2 ) Multiply both sides by 27V2

64V2 = 64 · 27 Simplify

64V2 = 1,728 Simplify

64V2÷ 64 = 1,728 ÷ 64 Divide both sides by 64

V2= 27

The volume of the smaller triangular prism is 27 cubic millimetres.

The figures below are similar.

What is the surface area of the smaller cylinder?

S2 = _______ square centimetres

Find the square of the ratio of the corresponding dimensions:

(a/b)2 = (4/2)2 = (2/1)1 = 4/1

Find the ratio of the surface areas:

S1/S2 = 196/S2

Use these two ratios to set up a proportion and solve for S2.

4/1 = 196/S2

4/1 ( S2 ) = 196/S2 ( S2 )

4S2=196 · 1 Simplify

4S2= 196 Simplify

4S2÷ 4=196 ÷ 4 Divide both sides by 4

S2= 49

The surface area of the smaller cylinder is 49 square centimetres.

Let’s practice!