Missing sides and composite figures

Work backwards from the measurement you know

A rectangle has area 48 cm² and width 6 cm. Its equal rows satisfy length × 6 = 48, so the length is 8 cm.

A different rectangle has perimeter 30 cm and width 6 cm. Opposite sides are equal. Half the perimeter, 15 cm, is one length plus one width. The length is 15-6 = 9 cm.

Area and perimeter give different equations. Read the unit and the quantity before choosing one.

A square with area 49 cm² has side 7 cm because 7 × 7 = 49. A square with perimeter 36 cm has side 36÷4 = 9 cm. Dividing an area by four would not find a side length.

Check a proposed side by putting it back into the original relationship. For a square’s area, find two equal factors. For its perimeter, add four equal sides. The number that works for one measurement may fail for the other.

Check your understanding 1 Keep the area relationship

A rectangle covers 13.5 m² and is 3 m wide. Find its length.

Missing length in metres

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The length is 4.5 m: 13.5÷3 = 4.5. Check 4.5 × 3 = 13.5 m². Halving 13.5 would treat the area as a perimeter and would not preserve the given surface.

Split a combined surface into rectangles

An L-shaped surface comes from a 10 cm by 8 cm rectangle with a 4 cm by 3 cm rectangle removed at its top right corner.

One surface, different useful splits

The outer boundary includes the inward corner. All six boundary lengths are available in the figure data.

View figure data

Lengths around the outside of the L-shaped surface

Side	Length (cm)	Direction
AB	10	right
BC	5	up
CD	4	left
DE	3	up
EF	6	left
FA	8	down

Lengths around the outside of the L-shaped surface

Side	Length (cm)	Direction
AB	10	right
BC	5	up
CD	4	left
DE	3	up
EF	6	left
FA	8	down

Subtract the missing corner’s area from the enclosing rectangle:

10 × 8 − 4 × 3 = 80 − 12 = 68 cm².

Alternatively split the L into a bottom rectangle 10 cm by 5 cm and an upper-left rectangle 6 cm by 3 cm. Then 50 + 18 = 68 cm². The two parts fill the L with no overlap and no gap.

Trace the outside for perimeter

For the same L-shape, trace A → B → C → D → E → F → A. The perimeter is 10 + 5 + 4 + 3 + 6 + 8 = 36 cm.

The removed corner creates new exposed sides. It is not correct to subtract the removed rectangle’s whole perimeter from the large rectangle’s perimeter. The shared joins inside a split figure also do not belong to the outside boundary.

Here, the two new exposed lengths replace equal lengths from the old top and right sides. The perimeter remains 36 cm even though the area has decreased. A cut in a different position can change the perimeter.

Check your understanding 2 Choose the surface and the boundary separately

Use the L-shaped figure. How much paper covers its surface? How much ribbon follows its outside edge? Enter the area and perimeter in their labelled fields. Use the figure data if you need the side lengths.

L-shaped area in square centimetres

L-shaped perimeter in centimetres

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The paper covers 68 cm². The ribbon follows 36 cm of outside edge. The two measurements use different operations and different units even though they describe the same shape.

Check your understanding 3 Use the named measurement

A rectangle has perimeter 42 m and length 13 m. What is its width?

Width in metres

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8 m. Check 13 + 8 + 13 + 8 = 42 m.

Check your understanding 4 Change where the piece is removed

A 10 cm by 8 cm card has a 4 cm by 3 cm rectangular hole cut wholly inside it, rather than removed from a corner. What is the remaining area? What is the length of all the exposed edges, including the hole?

Remaining area, exposed edge length and explanation of the hole's effect

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The area is still 80 − 12 = 68 cm². The outside edge is 36 cm, and the hole adds 14 cm of exposed edge. The total exposed edge length is 50 cm. The position of a removed piece can change boundaries without changing the removed area.

For any combined figure, mark its given lengths, derive missing lengths and decide whether the question asks for covering area, the outside boundary only, or all exposed edges.