When two rectangles are similar, square the ratio of corresponding lengths to get the ratio of their areas. To find the area of the border, subtract the inner rectangle’s area from the outer rectangle’s area.
The length ratio describes the two rectangles. The border is what remains between them, so find the two area shares before calculating the border’s share.
- Check that the rectangles are similar.
- Square both terms of the length ratio.
- Subtract the inner area share from the outer area share.
Worked example | An inner-to-outer length ratio of 3:5
A rectangular sign has a similar rectangular cut-out. The ratio of corresponding lengths of the cut-out to the sign is 3:5. The outer rectangle has an area of 200 cm².
Square the terms to get an inner-to-outer area ratio of 9:25. The inner rectangle takes 9/25 of 200 cm², or 72 cm². The border area is 200 − 72 = 128 cm².
Find the border-to-inner ratio
Use the same 25 area units for the outer rectangle and 9 for the inner rectangle. The border takes 25 − 9 = 16 units. The border-to-inner ratio is 16:9.
The border covers 16/25, or 64%, of the outer rectangle. You can answer that percentage question from the length ratio alone. The actual outer area is needed only when the question asks for an area in square units.
Work backwards from the border area
An inner rectangle and an outer rectangle are similar, with corresponding lengths in the ratio 2:3. The border area is 45 cm². Find the outer area.
Answer. The inner and outer areas are in the ratio 4:9, so the border takes five units. Each unit is 45 ÷ 5 = 9 cm². The outer area is 81 cm².
Frequently asked questions
What if I only know one pair of corresponding lengths?
That pair gives the scale factor when similarity is given or established. For rectangles that may be different shapes, check the second pair of sides before squaring the ratio.
Does the cut-out have to be centred?
Its position does not affect the remaining area, provided the cut-out lies entirely within the outer rectangle.
Official information
SEAB, 2027 SEC G3 Mathematics syllabus (K310), Geometry and Measurement, ratios of areas of similar figures. Syllabus checked 11 November 2026. The examples are original.