Scale factors between similar shapes, missing lengths, and how areas and volumes scale (Higher).
What you will learn
- Decide whether two shapes are similar
- Find missing lengths in similar triangles
- Use the area scale factor k² and the volume scale factor k³ (Higher)
Same shape, any size
Two shapes are similar if one is an enlargement of the other: the angles are equal and every length is multiplied by the same scale factor k.
Find k from one pair of matching sides, then multiply (small to big) or divide (big to small).
- Triangle ABC ~ triangle DEF, AB = 4 cm, DE = 10 cm
- k = 10 ÷ 4 = 2.5
- BC = 6 cm, so EF = 6 × 2.5 = 15 cm
- DF = 20 cm, so AC = 20 ÷ 2.5 = 8 cm
Angles do not change: in similar triangles, matching angles are equal, so a missing angle comes from the angle sum, not from the scale factor.
To test whether two rectangles are similar, divide matching sides: 4 by 3 and 10 by 7.5 give 10 ÷ 4 = 2.5 and 7.5 ÷ 3 = 2.5, the same, so they are similar. Shapes that are always similar to each other: any two squares, any two equilateral triangles, any two circles, and any two regular polygons with the same number of sides. Two rectangles or two rhombuses need not be: their side ratios or angles can differ.
Two right-angled triangles with the same angles (such as two ramps with the same slope) are similar: a ramp rising 0.5 m over 4 m and a longer one over 10 m rise 0.5 × 10 ÷ 4 = 1.25 m.
A line drawn parallel to one side of a triangle cuts off a smaller similar triangle. Compare the whole side with the whole side: AD with AB, not AD with DB.
The rest of this lesson and its 8 practice questions are locked.
In this lesson
- Same shape, any size
- Areas and volumes (Higher)
- The slip that costs the most
- Try it: First question
8 practice questions, marked as you go; about 25 minutes in all.
Key terms
Similar
In this unit: Transformations, congruence and similarity
Translations, reflections, rotations and enlargements on a grid and how to describe them fully, combined transformations and invariant points, the congruence conditions, and similar shapes with their length, area and volume scale factors.