Area of a Trapezoid

The space inside a trapezoid equals the average of its two parallel sides, multiplied by the height.

A = 12(a + b)h

The two parallel sides are a and b. The height h is the straight-up distance between them.

What each symbol means

A trapezoid is a four-sided shape with exactly one pair of parallel sides (in some countries it is called a trapezium). Those two parallel sides are usually different lengths, so the shape has a wide end and a narrow end. This formula finds the space inside.

The formula is A = 12(a + b)h. In plain words: add the two parallel sides, take half of that, then multiply by the height.

What each part means

The letters a and b are the two parallel sides. The order does not matter, since you are just adding them. The height h is the straight-up distance between those two sides, measured at a right angle - not the length of a slanted edge.

Only the parallel sides go inside the brackets: The slanted sides are not used in the formula. You only add the two sides that run parallel to each other.

A worked example

Find the area of a trapezoid with parallel sides 6 cm and 10 cm and height 4 cm

  1. Start with the formula: A = 12(a + b)h.
  2. Put in the values: A = 12(6 + 10) × 4.
  3. Add the parallel sides first: 6 + 10 = 16.
  4. Take half: 12 × 16 = 8.
  5. Multiply by the height: 8 × 4 = 32.
  6. The area is 32 square centimeters.

Why the "average"?: Adding the two sides and halving them gives their average length. So a trapezoid has the same area as a rectangle whose width is the average of the two parallel sides.

Why it works (two copies make a parallelogram)

  1. Take your trapezoid, with parallel sides a (short) and b (long) and height h.
  2. Make an exact copy, flip it upside down, and slot it against the first one.
  3. The two pieces fit together into a parallelogram. Its base is a + b (the two sides join into one long side) and its height is still h.
  4. A parallelogram has area base times height, so the two trapezoids together cover (a + b) × h.
  5. One trapezoid is half of that: A = 12(a + b)h.

Practice questions

Question 1

Find the area of a trapezoid with parallel sides 4 cm and 8 cm and height 5 cm.

  1. Write the formula: A = 12(a + b)h.
  2. Substitute a = 4, b = 8, h = 5: A = 12(4 + 8) × 5.
  3. Add the parallel sides: 4 + 8 = 12.
  4. Take half: 12 × 12 = 6, then multiply by the height: 6 × 5 = 30.
  5. The area is 30 square cm.

Answer: 30 square cm

Question 2

A trapezoid has parallel sides 7 m and 11 m and height 6 m. What is its area?

  1. Start from A = 12(a + b)h.
  2. Substitute a = 7, b = 11, h = 6: A = 12(7 + 11) × 6.
  3. Add the parallel sides: 7 + 11 = 18, and half of 18 is 9.
  4. Multiply by the height: 9 × 6 = 54 square m.

Answer: 54 square m

Question 3

A trapezoid has parallel sides 5 cm and 5 cm and height 10 cm. Find its area.

  1. When both parallel sides are equal, the shape is really a rectangle, but the formula still works.
  2. Substitute into A = 12(a + b)h: A = 12(5 + 5) × 10.
  3. Add and halve: 5 + 5 = 10, and half of 10 is 5.
  4. Multiply by the height: 5 × 10 = 50 square cm.

Answer: 50 square cm

Question 4

A trapezoid has parallel sides 3 m and 9 m and height 4 m. What is its area?

  1. Use A = 12(a + b)h with a = 3, b = 9, h = 4.
  2. Add the parallel sides: 3 + 9 = 12.
  3. Take half: 12 × 12 = 6.
  4. Multiply by the height: 6 × 4 = 24 square m.

Answer: 24 square m

Question 5

A trapezoid has an area of 40 square cm, parallel sides 6 cm and 10 cm. Find its height.

  1. Start from A = 12(a + b)h with A = 40, a = 6, b = 10.
  2. Work out the bracket: 12(6 + 10) = 12 × 16 = 8, so 40 = 8 × h.
  3. Divide both sides by 8 to find the height: h = 40 ÷ 8.
  4. Work it out: h = 5 cm.

Answer: 5 cm