Perimeter and Area of Compound Shapes
A compound shape is made by joining together, or cutting away, two or more basic shapes such as rectangles. Finding its perimeter and area means first working out any side lengths that are not given directly, then splitting the shape into simpler rectangles (or treating it as a full rectangle with a piece removed), calculating each part separately, and combining the results.
Before you start
Make sure you're comfortable with these topics first:
Method
- Redraw or annotate the shape, marking on every length you are given and labelling any missing lengths with a letter.
- Use the fact that the total length along one side of a rectilinear compound shape must match the total length along the opposite side, to work out missing horizontal and vertical lengths.
- Split the shape into two or more rectangles with an imaginary straight line, or treat it as a large rectangle with one or more smaller rectangles removed.
- Calculate the area of each part separately, then add the parts together (or subtract, if a piece has been removed).
- For the perimeter, add together only the lengths of the actual outer edges of the shape, not any line you drew to help split it into rectangles.
- Give your final answer with the correct units (cm or cm^2), and check that an area found by subtraction is smaller than the area of the original full rectangle.
Worked example
A shape is formed by cutting a 5 cm by 4 cm rectangular notch from each of the two bottom corners of a 20 cm by 12 cm rectangle, leaving a bridge-shaped remainder. Calculate (a) the area and (b) the perimeter of the remaining shape.
- Find the area of the original full rectangle: 20 x 12 = 240 cm^2.
- Find the area of one notch, 5 x 4 = 20 cm^2, and since two identical notches are removed, their total area is 20 x 2 = 40 cm^2.
- Subtract the notches from the full rectangle: 240 - 40 = 200 cm^2. This is the answer to (a).
- For the perimeter, work around the outline: the top edge stays 20 cm, and each tall side edge shortens to 12 - 4 = 8 cm (there are two of these).
- The bottom edge between the two notches shortens to 20 - 5 - 5 = 10 cm, and each notch adds two extra edges of 5 cm and 4 cm, contributing (5 + 4) x 2 = 18 cm in total.
- Add every edge together: 20 + 8 + 8 + 10 + 18 = 64 cm. This is the answer to (b).
Practice questions
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Q1A compound shape is made from a 9 cm by 6 cm rectangle with a 3 cm by 2 cm rectangle removed from one corner. Find its area.Show answer
Answer: 48 cm^2 ((9 x 6) - (3 x 2) = 54 - 6)
Q2An L-shaped room is formed from an 8 m by 5 m rectangle with a 3 m by 2 m rectangle removed from one corner. Find its perimeter.Show answer
Answer: 26 m. Removing a rectangular notch from a corner does not change the total perimeter of a rectangle, so it is the same as the perimeter of the original 8 m by 5 m rectangle: 2 x (8 + 5) = 26 m.
Q3A shape is made from two rectangles placed side by side: one 6 cm by 4 cm and one 3 cm by 4 cm, joined along their 4 cm sides to form a longer rectangle. Find the total area.Show answer
Answer: 36 cm^2 ((6 x 4) + (3 x 4) = 24 + 12)
Q4A cross-shaped tile is made from a 12 cm by 4 cm rectangle and a 4 cm by 12 cm rectangle overlapping at their centres, where the overlapping square is 4 cm by 4 cm. Find the total area of the cross.Show answer
Answer: 80 cm^2 ((12 x 4) + (12 x 4) - (4 x 4), since the overlapping square would otherwise be counted twice)
Q5A patio is L-shaped, made from a 6 m by 4 m rectangle with a 2 m by 1 m rectangle removed from a corner. Find its area.Show answer
Answer: 22 m^2 ((6 x 4) - (2 x 1) = 24 - 2)
Q6State the two steps you should always take before calculating the area of a compound shape made of several rectangles.Show answer
Answer: Find any missing side lengths using the measurements you are given, then split the shape into simple rectangles (or treat it as a full rectangle with a piece removed).
Q7A shape is formed from a 10 cm by 7 cm rectangle with a 4 cm by 3 cm rectangle removed from the middle of one long side, not from a corner. Explain why the perimeter is not the same as the original rectangle's perimeter.Show answer
Answer: Removing a notch from the middle of a side, rather than from a corner, adds two extra edges to the outline where the notch is cut, so the perimeter increases instead of staying the same.
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
A shape is made from a 14 cm by 9 cm rectangle with a 5 cm by 3 cm rectangle removed from one corner. Calculate the area of the remaining shape.
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A shape is formed by cutting a 3 cm by 2 cm rectangular notch from each of the two bottom corners of a 16 cm by 10 cm rectangle, leaving a bridge-shaped remainder. Calculate the perimeter of the remaining shape.
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Free printable worksheet
Want more practice on paper? Download the perimeter and area of compound shapes worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 30 of 11+ Maths Workbook 2, the whole course as one free printable PDF.
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