Geometry & Measures
Grade 4-5Area of Compound Shapes
Compound shape questions build directly on the area of rectangles, triangles and semicircles, but the real skill being tested is spotting how to split or combine shapes correctly before calculating anything. This lesson covers splitting an L-shape into rectangles, finding a missing length before calculating area, subtracting a cut-out corner, and combining a rectangle with a triangle or semicircle.
What you need to know
- Split the shape into rectangles, triangles and other basic shapes.
- Find any missing lengths using the fact that opposite sides must match up.
- Add the parts, or subtract a cut-out from a larger shape.
- Work in one set of units throughout.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Splitting an L-shape into rectangles
An L-shape can be split into two rectangles with a single straight line, and its total area is the sum of the two rectangle areas. Any position for the splitting line works, as long as both resulting pieces are rectangles.
An L-shape splits into a 6 cm by 10 cm rectangle and a 4 cm by 6 cm rectangle. Its total area is (6 × 10) + (4 × 6) = 60 + 24 = 84 cm².
Finding a missing length first
Some L-shape questions leave one length unlabelled, which must be found before the area can be calculated, usually by adding or subtracting the lengths of two shorter sides that make up a longer one.
If the top of an L-shape is 8 cm and a step of 4 cm is added below it, the full height on the other side is x = 8 + 4 = 12 cm.
Subtracting a cut-out corner
When a shape is formed by cutting a smaller rectangle or triangle from a larger rectangle, find the area of the large rectangle, then subtract the area of the piece removed.
A 10 cm by 8 cm rectangle has a 4 cm by 4 cm rectangular notch removed from one corner, so the remaining area is (10 × 8) - (4 × 4) = 80 - 16 = 64 cm².
Combining a rectangle with a triangle or semicircle
When a triangle or semicircle is attached to a rectangle, find the area of each part separately and add them together. A semicircle attached along one edge has a diameter equal to the length of that edge.
A rectangle measuring 8 cm by 5 cm has a triangular roof of height 3 cm on top of the 8 cm edge. The total area is (8 × 5) + (½ × 8 × 3) = 40 + 12 = 52 cm².
Worked examples
Three exam-style questions, fully solved
An L-shape splits into a 6 cm by 10 cm rectangle and a 4 cm by 6 cm rectangle. Find its total area.
Easy- 1.Find the area of the first rectangle: 6 × 10
- 2.Find the area of the second rectangle: 4 × 6
- 3.Add the two areas together: 60 + 24
Answer: 84 cm²
The diagram shows a 10 cm by 8 cm rectangle with a 4 cm by 4 cm rectangular notch cut from one corner. Find the area of the remaining shape.
Medium- 1.Find the area of the large rectangle: 10 × 8 = 80 cm²
- 2.Find the area of the cut-out rectangle: 4 × 4 = 16 cm²
- 3.Subtract the cut-out area from the large rectangle: 80 - 16
Answer: 64 cm²
The diagram shows a shape made from an 8 cm by 6 cm rectangle, with a semicircle of diameter 8 cm attached to the top. Find the total area, leaving your answer in terms of π.
Hard- 1.Find the area of the rectangle: 8 × 6 = 48 cm²
- 2.Find the radius of the semicircle: 8 ÷ 2 = 4 cm
- 3.Find the area of the semicircle: ½ × π × 4² = 8π cm²
- 4.Add the two areas together: 48 + 8π
Answer: 48 + 8π cm²
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
The diagram shows an L-shape. Work out its area.
The diagram shows an L-shape. Work out its area.
The diagram shows an L-shape. Work out its area.
The diagram shows an L-shape. Work out its perimeter.
A rectangle 10 cm by 6 cm has a smaller rectangle 3 cm by 2 cm removed from one corner. Work out the area that is left.
A shape is made from a rectangle 8 cm by 6 cm with a semicircle of diameter 6 cm added to one short end. Work out the total area, leaving your answer in terms of π.
Avoid these
The mistakes students make most often
Splitting an L-shape into rectangles that overlap or leave a gap, rather than two rectangles that together make up the whole shape exactly once.
Forgetting to find a missing length first when one side of an L-shape is unlabelled, and instead trying to calculate area with an incomplete set of measurements.
Adding the cut-out area instead of subtracting it when a corner has been removed from a rectangle.
Using the full diameter instead of the radius when finding the area of a semicircle, forgetting that area = ½ × π × radius².
Forgetting to leave the answer in terms of π when the question asks for it, and instead giving a rounded decimal.
FAQ
Questions students and parents ask
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