The area of a trapezium (also known as a trapezoid in some countries) is a fundamental concept in geometry. A trapezium is a four-sided shape (quadrilateral) with at least one pair of parallel sides. Understanding how to calculate the area of a trapezium is essential for solving problems related to area in the 11+ exam.

Key Concepts

1. Definition of a Trapezium

A trapezium has:

  • Two parallel sides (called the bases).
  • Two non-parallel sides.

2. Formula for Area

The area (A) of a trapezium can be calculated using the formula:

A = \frac{1}{2} \times (a + b) \times h

Where:

  • ( a ) = length of one base
  • ( b ) = length of the other base
  • ( h ) = height (the perpendicular distance between the two bases)

3. Visualising the Trapezium

To better understand the trapezium:

  • Draw a trapezium and label the bases ( a ) and ( b ), and the height ( h ).
  • Identify the parallel sides and note how the height connects these bases.

4. Calculating Area

  • To find the area, add the lengths of the two bases, multiply by the height, and then divide by 2.

5. Real-World Applications

Understanding the area of a trapezium is useful in real-life scenarios such as calculating land area, designing buildings, and more.

Practice Questions on Area of a Trapezium

Easy Level

  1. What is the area of a trapezium with bases 6 cm and 4 cm and a height of 5 cm?
  2. If one base is 8 cm and the other base is 10 cm with a height of 4 cm, what is the area?
  3. A trapezium has bases 5 m and 3 m and a height of 2 m. What is its area?
  4. What is the height of a trapezium if the area is 20 cm², and the bases are 5 cm and 3 cm?
  5. Find the area of a trapezium with bases of length 12 cm and 8 cm and a height of 3 cm.
  6. If a trapezium has a base of 10 m, a top base of 6 m, and a height of 2 m, what is its area?
  7. What is the area of a trapezium with bases measuring 7 cm and 5 cm and a height of 4 cm?
  8. If the bases of a trapezium are 15 cm and 10 cm, and the height is 6 cm, what is the area?
  9. Calculate the area of a trapezium where one base is 9 m, the other is 5 m, and the height is 3 m.
  10. A trapezium has an area of 30 cm², with bases of 10 cm and 8 cm. What is its height?
  11. Find the area of a trapezium with bases of 2.5 cm and 4.5 cm and a height of 2 cm.
  12. What is the area of a trapezium with bases measuring 3 cm and 7 cm and a height of 5 cm?
  13. If one base of a trapezium is 11 cm and the other is 9 cm with a height of 4 cm, what is the area?
  14. A trapezium has bases of 14 m and 6 m. If its height is 3 m, what is its area?
  15. Calculate the area of a trapezium where the bases are 10 cm and 12 cm, and the height is 4 cm.
  16. If a trapezium has an area of 40 cm², with bases measuring 8 cm and 12 cm, what is its height?
  17. What is the area of a trapezium with bases 5 m and 3 m and a height of 6 m?
  18. A trapezium has bases measuring 6 cm and 10 cm. If the height is 5 cm, what is its area?
  19. If the bases of a trapezium are 4 m and 10 m and the height is 8 m, what is the area?
  20. What is the area of a trapezium with bases of 3.5 m and 4.5 m and a height of 2 m?

Medium Level

  1. Calculate the area of a trapezium with bases of 15 cm and 5 cm and a height of 4 cm.
  2. A trapezium has bases measuring 20 m and 10 m, with a height of 6 m. What is its area?
  3. If a trapezium has an area of 50 cm² with bases of 12 cm and 8 cm, what is the height?
  4. Find the area of a trapezium with bases measuring 9 m and 7 m and a height of 5 m.
  5. If the bases of a trapezium are 11 cm and 13 cm, and the height is 4 cm, what is its area?
  6. A trapezium has bases of 6 cm and 10 cm. If its area is 32 cm², what is its height?
  7. What is the area of a trapezium with bases of 18 cm and 12 cm and a height of 5 cm?
  8. If a trapezium has bases measuring 25 m and 15 m and a height of 4 m, what is its area?
  9. Calculate the area of a trapezium with bases of 8 cm and 16 cm and a height of 3 cm.
  10. A trapezium has an area of 36 cm² and bases measuring 6 cm and 10 cm. What is its height?
  11. Find the area of a trapezium with bases measuring 5 m and 3 m and a height of 10 m.
  12. If the bases of a trapezium are 20 cm and 30 cm and the height is 10 cm, what is its area?
  13. A trapezium has bases of 14 m and 6 m. If its area is 80 m², what is its height?
  14. Calculate the area of a trapezium with bases measuring 3.5 cm and 6.5 cm and a height of 4 cm.
  15. If a trapezium has bases of 7 cm and 11 cm, and the height is 6 cm, what is its area?
  16. A trapezium has bases measuring 10 m and 4 m. If its area is 30 m², what is its height?
  17. What is the area of a trapezium with bases of 9 m and 5 m and a height of 3 m?
  18. A trapezium has bases measuring 12 cm and 18 cm. If the height is 4 cm, what is its area?
  19. If a trapezium has an area of 64 cm² and bases of 16 cm and 12 cm, what is its height?
  20. Calculate the area of a trapezium with bases of 10 cm and 20 cm and a height of 6 cm.

Hard Level

  1. A trapezium has bases of 15 m and 25 m. If the area is 200 m², what is the height?
  2. If a trapezium has bases measuring 18 cm and 22 cm, and the height is 5 cm, what is its area?
  3. Calculate the height of a trapezium with an area of 75 cm² and bases measuring 10 cm and 30 cm.
  4. What is the area of a trapezium with bases of 8 m and 12 m and a height of 7 m?
  5. A trapezium has an area of 96 cm². If one base is 12 cm, what is the other base if the height is 8 cm?
  6. If a trapezium has bases of 10 m and 30 m, and the area is 100 m², what is the height?
  7. Calculate the area of a trapezium with bases measuring 14 m and 16 m and a height of 4 m.
  8. What is the height of a trapezium with an area of 48 cm² and bases of 8 cm and 12 cm?
  9. If a trapezium has bases measuring 25 cm and 35 cm, and the area is 300 cm², what is the height?
  10. A trapezium has an area of 150 m². If one base is 20 m and the other is 40 m, what is the height?
  11. Calculate the area of a trapezium where the bases are 3 m and 9 m, and the height is 6 m.
  12. If a trapezium has bases of 10 m and 20 m, and the area is 90 m², what is the height?
  13. What is the area of a trapezium with bases measuring 40 m and 60 m and a height of 5 m?
  14. A trapezium has bases of 25 cm and 35 cm. If its area is 300 cm², what is the height?
  15. If a trapezium has an area of 200 cm² and bases measuring 20 cm and 30 cm, what is the height?
  16. Calculate the area of a trapezium with bases measuring 4 m and 10 m and a height of 8 m.
  17. What is the height of a trapezium with an area of 120 cm² and bases of 15 cm and 25 cm?
  18. If a trapezium has bases of 18 cm and 30 cm, and the area is 240 cm², what is the height?
  19. A trapezium has an area of 150 m². If one base is 20 m and the other is 40 m, what is the height?
  20. Calculate the area of a trapezium where the bases are 12 m and 18 m, and the height is 4 m.

Answers and Explanations

Easy Level

  1. A = \frac{1}{2} \times (6 + 4) \times 5 = 25 \text{ cm}^2
  2. A = \frac{1}{2} \times (8 + 10) \times 4 = 36 \text{ cm}^2
  3. A = \frac{1}{2} \times (5 + 3) \times 2 = 8 \text{ m}^2
  4. h = \frac{20}{(5 + 3)} = 2.5 \text{ cm}
  5. A = \frac{1}{2} \times (12 + 8) \times 3 = 30 \text{ cm}^2
  6. A = \frac{1}{2} \times (10 + 6) \times 2 = 16 \text{ m}^2
  7. A = \frac{1}{2} \times (7 + 5) \times 4 = 24 \text{ cm}^2
  8. A = \frac{1}{2} \times (15 + 10) \times 6 = 75 \text{ cm}^2
  9. A = \frac{1}{2} \times (9 + 5) \times 3 = 21 \text{ m}^2
  10. h = \frac{30}{(10 + 8)} = 2.5 \text{ cm}
  11. A = \frac{1}{2} \times (2.5 + 4.5) \times 2 = 7 \text{ cm}^2
  12. A = \frac{1}{2} \times (3 + 7) \times 5 = 25 \text{ cm}^2
  13. A = \frac{1}{2} \times (11 + 9) \times 4 = 80 \text{ cm}^2
  14. A = \frac{1}{2} \times (14 + 6) \times 3 = 30 \text{ m}^2
  15. h = \frac{20}{(10 + 4)} = 1.67 \text{ m}
  16. A = \frac{1}{2} \times (9 + 5) \times 3 = 21 \text{ cm}^2
  17. h = \frac{5 \times (4 + 6)}{(4 + 6)} = 5 \text{ cm}
  18. A = \frac{1}{2} \times (6 + 10) \times 4 = 32 \text{ m}^2
  19. A = \frac{1}{2} \times (10 + 4) \times 6 = 84 \text{ m}^2
  20. A = \frac{1}{2} \times (3.5 + 4.5) \times 2 = 8 \text{ m}^2

Medium Level

  1. A = \frac{1}{2} \times (3 + 2) \times 4 = 10 \text{ cm}^2
  2. A = \frac{1}{2} \times (20 + 10) \times 6 = 90 \text{ m}^2
  3. h = \frac{50}{(12 + 8)} = 2.5 \text{ cm}
  4. A = \frac{1}{2} \times (9 + 7) \times 5 = 40 \text{ m}^2
  5. A = \frac{1}{2} \times (11 + 13) \times 4 = 48 \text{ cm}^2
  6. h = \frac{32}{(6 + 10)} = 2 \text{ cm}
  7. A = \frac{1}{2} \times (20 + 10) \times 6 = 90 \text{ m}^2
  8. A = \frac{1}{2} \times (15 + 10) \times 5 = 62.5 \text{ cm}^2
  9. h = \frac{75}{(18 + 22)} = 2.5 \text{ cm}
  10. A = \frac{1}{2} \times (30 + 40) \times 5 = 175 \text{ m}^2
  11. A = \frac{1}{2} \times (25 + 30) \times 10 = 275 \text{ cm}^2
  12. h = \frac{90}{(20 + 10)} = 3 \text{ cm}
  13. A = \frac{1}{2} \times (10 + 5) \times 4 = 30 \text{ m}^2
  14. A = \frac{1}{2} \times (10 + 10) \times 5 = 50 \text{ cm}^2
  15. A = \frac{1}{2} \times (9 + 5) \times 4 = 28 \text{ cm}^2
  16. h = \frac{36}{(10 + 20)} = 1.2 \text{ m}
  17. A = \frac{1}{2} \times (30 + 40) \times 8 = 280 \text{ cm}^2
  18. A = \frac{1}{2} \times (25 + 15) \times 5 = 100 \text{ m}^2
  19. A = \frac{1}{2} \times (20 + 30) \times 6 = 150 \text{ m}^2
  20. A = \frac{1}{2} \times (40 + 60) \times 10 = 500 \text{ cm}^2

Hard Level

  1. h = \frac{200}{(15 + 25)} = 8 \text{ m}
  2. A = \frac{1}{2} \times (18 + 22) \times 5 = 100 \text{ cm}^2
  3. h = \frac{75}{(10 + 30)} = 2.5 \text{ cm}
  4. A = \frac{1}{2} \times (8 + 12) \times 7 = 70 \text{ m}^2
  5. 25 + 30 = 5 \text{ cm}
  6. h = \frac{100}{(10 + 30)} = 2.5 \text{ cm}
  7. A = \frac{1}{2} \times (25 + 35) \times 30 = 900 \text{ cm}^2
  8. h = \frac{240}{(18 + 30)} = 4 \text{ cm}
  9. 200 \text{ cm}^2
  10. A = \frac{1}{2} \times (12 + 8) \times 6 = 120 \text{ m}^2
  11. 180 \text{ cm}^2
  12. 200 \text{ m}^2
  13. h = \frac{300}{(30 + 10)} = 10 \text{ cm}
  14. A = \frac{1}{2} \times (15 + 25) \times 5 = 100 \text{ cm}^2
  15. A = \frac{1}{2} \times (20 + 40) \times 10 = 600 \text{ cm}^2
  16. A = \frac{1}{2} \times (12 + 18) \times 4 = 60 \text{ m}^2
  17. 10 \text{ m}^2
  18. 60 \text{ cm}^2
  19. 50 \text{ m}^2
  20. 80 \text{ cm}^2

This set of questions and answers provides a comprehensive overview of the area of a trapezium relevant to the 11+ exam, covering various difficulty levels and encouraging students to develop their understanding and problem-solving skills.