Volume of Similar Solids

Understanding Similar Solids

Similar solids are three-dimensional shapes that have the same shape but different sizes. Their corresponding linear dimensions are proportional.

Key Concepts

• Linear Scale Factor: The ratio of corresponding linear dimensions (e.g., heights, radii) between similar solids.
• Volume Scale Factor: The cube of the linear scale factor, which relates the volumes of similar solids.

 

Volume of Similar Solids

If two solids are similar with a linear scale factor of $k$, then their volumes are related by the volume scale factor $k^3$.

 

Formula

If the volume of the smaller solid is $V_1$ and the volume of the larger solid is $V_2$, then:

$$V_2 = k^3 \times V_1$$

 

Examples

Example 1: Pyramids

Two similar pyramids have heights in the ratio 1:3. If the volume of the smaller pyramid is 20 cm³, find the volume of the larger pyramid.

 

Example 2: Spheres

Two similar spheres have radii in the ratio 2:5. If the volume of the smaller sphere is 32 cm³, find the volume of the larger sphere.