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This page sets out the method for area of compound shapes in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Area of compound shapes topic page.
When you see an L-shape, T-shape or a rectangle with a triangle on top, draw one dotted line to split it.
When the shape looks like a rectangle with a corner missing, subtract.
Draw a line that makes rectangles you know.
Find each area, then add.
An L-shape can be a large rectangle minus a small rectangle.
Both methods should give the same area.
The full width is the sum of the pieces along that side.
Work out a missing side before you multiply.
Every length should be in the same unit before you multiply.
The area is in square units.
A side of 3 cm is not 3 cm².
Only products of lengths are areas.
Write “rectangle 6 by 4” and “rectangle 2 by 3”.
The marker can then award a method mark for each piece.
An L-shape is two rectangles. You can split it with one extra line. A 9cm by 5cm rectangle with a 4cm by 3cm rectangle attached, and no overlap, has area 9× 5+4× 3=45+12=57cm^2. Draw the split, write both areas, then add. Another split of the same L can give different side lengths, but the total area must match. If your two ways disagree, one of the side lengths has been read from the wrong edge.
Missing sides are often found by adding or subtracting lengths that line up. If the bottom of an L is 11cm and the top step is 6cm, the overhang is 11-6=5cm. Label that 5cm before you look up a formula. Area only uses the lengths of the pieces you chose. A length that is the outside perimeter is not automatically a side of one of your rectangles.
Method: 1. Split the L into rectangles, find each area, then add. 2. Rectangles 9cm by 5cm and 4cm by 3cm give 45+12=57cm^2. 3. Find a missing edge by subtracting aligned lengths, such as 11-6=5cm.
When a smaller shape is cut from a larger one, the shaded area is the difference. Start with the full rectangle 10× 6=60. The cut-out is 4× 2=8. The remaining area is 60-8=52cm^2. Subtraction is doing the cutting. If you add 60+8 you are gluing the small rectangle on, which is the opposite of a hole.
The cut-out does not have to be a rectangle. A circular hole of radius 2cm has area π× 2^2=4π. Take that away from the outer shape. A semicircular bite uses 1/2π r^2. Check the diagram for the words cut out, removed, or a dashed hole. Those words mean subtract. The words attached, joined, or on the side of mean add.
Method: 1. Find the outer area, find the hole, then subtract. 2. A 10 by 6 rectangle minus a 4 by 2 hole is 60-8=52cm^2. 3. A circular hole of radius 2cm removes π× 2^2=4π.
A rectangle with a triangle on top is a house shape. Find the rectangle and the triangle separately. For a 12cm by 7cm rectangle and a triangle of base 12cm and perpendicular height 5cm, the areas are 12× 7=84 and 1/2× 12× 5=30. The compound area is 84+30=114cm^2. The base of the triangle is the top of the rectangle, so it is already drawn. The 5cm is the extra height above that side.
Do not multiply 12 by 7+5 and then halve everything. That would treat the whole house as a triangle. Do not use 7+5=12 as the triangle height. Only the 5cm is perpendicular height of the triangle. If the triangle is slanted, drop a perpendicular to the base and use that length. A sloping roof edge is not the height unless it is marked as perpendicular.
Method: 1. Work out the rectangle and the triangle separately, then add. 2. Rectangle 12× 7=84 plus triangle 1/2× 12× 5=30 gives 114cm^2. 3. Use only the triangle height of 5cm, not 7+5.
A running track end, or a rectangle with a curved end, is a rectangle plus a semicircle. A rectangle 14cm by 8cm with a semicircle on one short end has the diameter along the 8cm side, so the radius is 4cm. The rectangle area is 14× 8=112. The semicircle area is 1/2π× 4^2=1/2π× 16=8π. The total is 112+8π, which is 137.1cm^2 to 1 decimal place.
Add the semicircle because it is attached, not cut away. If the same semicircle were a window cut out of the rectangle, you would subtract 8π instead. Label the pieces on the diagram: rectangle 112, semicircle 8π. Then decide + or - from the picture. Round only after the areas have been combined. Rounding 8π first and the rectangle later is unnecessary here, but on a longer decimal you should still wait until the last line.
Method: 1. Add the rectangle area to the semicircle area when the curve is attached. 2. Rectangle 14× 8=112 and semicircle 8π give 112+8π=137.1cm^2 to 1 decimal place. 3. The diameter is the 8cm end, so the radius is 4cm.
Most compound shape questions only label some of the sides. You are expected to work out the others. In a shape with right angled corners, the opposite sides add up to the same total. If the full width is 10 m and one step in the bottom edge is 6 m, the other part of the width must be 10 - 6 = 4 m. Do these subtractions first and write each new length on the diagram.
Only use a length you were given or one you have worked out. Never measure from the picture, because exam diagrams are "not drawn accurately". Once every edge is known, you can split the shape, find the area, and also add up the perimeter. Mark each new length in pencil so you can see at a glance whether any side is still missing.
Method: 1. Compare opposite sides: the horizontal edges must total the same on the top and the bottom. 2. Do the same for the vertical edges on the left and the right. 3. Write each missing length on the sketch. 4. Split the shape, or subtract a corner, and then work out the area or perimeter.
Tip: The perimeter of an L-shape with right angle corners equals the perimeter of the smallest rectangle that fits round it.
An L-shape can be split into two rectangles, or seen as one big rectangle with a corner taken away. Both give the same area, so doing it both ways is an excellent check when you have time. Choose the way that needs the fewest calculations. A shape with a notch or a hole is usually quicker by subtraction, and a shape with a bump or a triangular roof by addition.
When you subtract, the piece taken away must be fully inside the big shape. When you add, the pieces must not overlap. Write a short plan such as "rectangle minus square" or "rectangle plus triangle" before you work anything out. The method marks in the mark scheme are given for exactly these two sub-areas.
Method: 1. Decide whether the shape is "big shape minus a piece" or "pieces added together". 2. Work out each area separately and write each one down. 3. Add or subtract as your plan says. 4. Check by splitting the shape the other way, or by estimating with rounded numbers.
A rectangle with a semicircle on one end combines two formulas. The area is the rectangle plus half of π r^2, where r is half of the diameter. The perimeter is different: it includes the curved edge, which is half of the circumference π d ÷ 2, but it does not include the diameter where the semicircle joins the rectangle, because that line is now inside the shape.
Exam questions might ask for an answer in terms of π, for example 560 + 49π, or a decimal to a given accuracy, such as 3 significant figures. Keep π as a symbol until the last line if the answer should be exact, and keep the full calculator value if you must round. A shape with a semicircle on each end of a rectangle forms a full circle of the same diameter.
Method: 1. Find the radius by halving the diameter that the semicircle sits on. 2. Area of the curved part: 1/2π r^2. Curved part of the perimeter: π r. 3. Leave out the diameter line from the perimeter where the shapes join. 4. Add the parts and give the answer exactly or rounded, as the question asks.
Tip: Two semicircles on opposite ends make one whole circle, so use π r^2 and 2π r for the curved parts together.
Many exam questions put a compound shape in a garden or a room and then ask for a cost. After finding the area, divide by the area covered by one bag, roll or tile. That number is nearly always not a whole number, and you must round up because you cannot buy part of a roll. Then multiply the whole number by the price.
If the question asks about a fence or an edging, it is the perimeter that matters, not the area. Read the question for words like "around the edge" and "covers". Write the unit with each line: square metres for area, metres for length, and pounds for cost. Marks are usually given for the area, for the division, for rounding up, and for the final cost.
Method: 1. Decide whether you need area (covering) or perimeter (edging). 2. Calculate it carefully using the missing lengths method. 3. Divide by the amount one item covers and round up to a whole number of items. 4. Multiply by the price of one item and write the answer with a pound sign.
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