Area of Parallelograms, Trapezia and Compound Shapes
The area of a parallelogram is base x perpendicular height, using the same idea as a rectangle but with a height measured at right angles to the base rather than the slanting side, while the area of a trapezium is area = 1/2 x (a + b) x h, where a and b are the lengths of the two parallel sides and h is the perpendicular distance between them. Compound shapes made from several of these pieces are found by splitting the shape into rectangles, triangles, parallelograms or trapezia, finding each area, then adding or subtracting.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a parallelogram, identify the base and the perpendicular height (not the slanted side), then use area = base x height.
- For a trapezium, identify the two parallel sides (call them a and b) and the perpendicular distance (height) between them.
- Use the formula area = 1/2 x (a + b) x h for a trapezium, adding the two parallel sides together before halving.
- For a compound shape, split it into rectangles, triangles, parallelograms or trapezia using straight lines.
- Find the area of each separate piece using the correct formula, then add the pieces together, or subtract if part of the shape has been removed.
- Give the final answer with the correct area unit (cm^2 or m^2), and check the answer is a sensible size for the shape drawn.
Worked example
A trapezium has parallel sides of 6 cm and 10 cm, and a perpendicular height of 4 cm. Find its area.
- Identify a = 6, b = 10 and h = 4.
- Add the parallel sides: a + b = 6 + 10 = 16.
- Use the formula: area = 1/2 x 16 x 4.
- Calculate: 1/2 x 16 = 8, then 8 x 4 = 32.
- The area of the trapezium is 32 cm^2.
Practice questions
Try each question, then tap to reveal the answer.
Q1Find the area of a parallelogram with base 9 cm and height 5 cm.Show answer
Answer: 9 x 5 = 45 cm^2
Q2Find the area of a trapezium with parallel sides 5 cm and 7 cm, and height 4 cm.Show answer
Answer: 1/2 x (5 + 7) x 4 = 1/2 x 12 x 4 = 24 cm^2
Q3A parallelogram has an area of 60 cm^2 and a base of 12 cm. Find its height.Show answer
Answer: Height = 60 / 12 = 5 cm
Q4Find the area of a trapezium with parallel sides 8 cm and 12 cm, and height 6 cm.Show answer
Answer: 1/2 x (8 + 12) x 6 = 1/2 x 20 x 6 = 60 cm^2
Q5A compound shape is a rectangle 10 cm by 5 cm with a 4 cm by 3 cm rectangle cut from one corner. Find the remaining area.Show answer
Answer: (10 x 5) - (4 x 3) = 50 - 12 = 38 cm^2
Q6Find the area of a parallelogram with base 15 cm and height 8 cm.Show answer
Answer: 15 x 8 = 120 cm^2
Q7A trapezium has an area of 45 cm^2, and parallel sides of 4 cm and 6 cm. Find its height.Show answer
Answer: 45 = 1/2 x (4 + 6) x h, so 45 = 5h and h = 9 cm
Q8An L-shaped compound shape is a 12 cm by 8 cm rectangle with a 5 cm by 3 cm rectangle removed from one corner. Find its area.Show answer
Answer: (12 x 8) - (5 x 3) = 96 - 15 = 81 cm^2
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A parallelogram has a base of 11 cm and a perpendicular height of 7 cm. Calculate its area, giving the correct unit.
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A trapezium-shaped field has parallel sides of 40 m and 65 m, with a perpendicular distance of 24 m between them. (a) Find the area of the field. (b) The farmer wants to divide the field into two equal areas using a straight line. Work out the area of each half.
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Free printable worksheet
Want more practice on paper? Download the area of parallelograms, trapezia and compound shapes worksheet pack - 6 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 8 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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