Similar Shapes

Understanding Similar Shapes

  • Similar shapes are shapes that have the same shape, but not necessarily the same size.
  • Two shapes are similar if their corresponding angles are equivalent and their corresponding lengths are proportional.
  • In a scale drawing, the scale factor is the ratio of any length on the model to the corresponding length on the real object. For similar shapes, the scale factor is the ratio of corresponding lengths between the two shapes.

Checking if Shapes are Similar

  • To check if shapes are similar, compare the angles and lengths of the shapes.
  • Corresponding angles in similar shapes are equal. Use a protractor to measure the angles if they’re not given.
  • Corresponding sides in similar shapes are in the same ratio. This is known as the scale factor. Divide the length of one side of the shape by the corresponding side in the other shape to check this.
  • If the ratios are equal, and the angles are the same, the shapes are similar.

Scale Factors

  • The scale factor is the ratio of corresponding side lengths between two similar shapes.
  • Multiply the lengths of the sides of a shape by the scale factor to find the lengths of the corresponding sides in a similar shape.
  • To find the scale factor from a real object to a model, divide a length on the model by the corresponding length on the real object.
  • To find the scale factor from a model to a real object, divide a length on the real object by the corresponding length on the model.

Areas of Similar Shapes

  • The area of a shape will change by the scale factor squared if the shape is made larger or smaller, i.e., if you enlarge a shape by a scale factor of 2, its area will become 4 times as large.
  • If the scale factor of the figure is k, then the area of the figure changes by the scale factor k^2.
  • To find the new area, multiply the original area by the square of the scale factor.
  • To find the scale factor from a larger area to a smaller one, take the square root of the ratio of the areas.

Volumes of Similar Shapes

  • The volume of a shape will change by the scale factor cubed if the shape is made larger or smaller, i.e., if you enlarge a shape by a scale factor of 3, its volume will become 27 times as large.
  • If the scale factor of the figure is k, then the volume of the figure changes by the scale factor k^3.
  • To find the new volume, multiply the original volume by the cube of the scale factor.
  • To find the scale factor from a larger volume to a smaller one, take the cube root of the ratio of the volumes.