Calculate Lengths, Areas, and Volume of Similar Figures - JSS3 Mathematics Lesson Note

When dealing with similar figures, certain properties are preserved even though the figures are scaled up or down. The ratios of corresponding sides (scale factor) apply to lengths, areas, and volumes of similar shapes.

Examples:

Lengths: If two figures are similar with a scale factor of 3:2, then the ratio of their corresponding lengths will also be 3:2.

Areas: The ratio of the areas of similar figures is the square of the scale factor. For example, if the scale factor between two squares is 2, the ratio of their areas is 2^2=4

Volumes: For three-dimensional shapes like cubes or spheres, the ratio of their volumes is the cube of the scale factor. If the scale factor between two cubes is 3, then the ratio of their volumes is 3^3=27

Solve Problems on Quantitative Reasoning Involving Similar Shapes

Quantitative reasoning involving similar shapes often involves applying the properties we discussed to solve practical problems. These problems can range from calculating unknown dimensions based on given scale factors to finding areas or volumes of similar figures.

Example Problem:

Given two similar triangles where the smaller triangle has sides of 3 cm, 4 cm, and 5 cm, and the larger triangle has a perimeter of 30 cm, find the lengths of its sides.

Determine the scale factor between the two triangles. In this case, since the perimeter of the larger triangle is 30 cm (which is 6 times the perimeter of the smaller triangle), the scale factor is 6.

Use the scale factor to find the lengths of the sides of the larger triangle:

Side lengths of the larger triangle: 

3×6=18 cm, 

4×6=24 cm, 

5×6=30 cm.

This process demonstrates how scale factors can be used to solve problems involving similar shapes, ensuring that proportions are maintained across dimensions.