Skip to content

Area of a trapezoid

Measurement · level 4, lesson 3 of 7

A question to start

A flower bed has a straight front edge 10 m long and a straight back edge 4 m long, parallel to the front, 3 m behind it. Its two ends slope in. It is a trapezoid.

Its neighbour's flower bed is a plain rectangle, 7 m by 3 m. Which bed needs more soil to cover it?

They need exactly the same. Notice that 7 is halfway between 4 and 10. A trapezoid always has the same area as the rectangle whose length is the average of its two parallel sides. This lesson shows why, in three different ways.

The idea

What a trapezoid is

A trapezoid is a four-sided shape with (at least) one pair of parallel sides. The two parallel sides are called its bases. We usually call their lengths \(a\) and \(b\). The height \(h\) is the distance between the two bases, measured at a right angle to them. The other two sides, the legs, can slope in any way, and their lengths are not needed for the area. (In Britain this shape is called a trapezium. Geometry's quadrilateral family explains how it is related to the others: every parallelogram counts as a trapezoid too.)

Way 1: two copies make a parallelogram

This is the same trick that worked for triangles. Make an exact copy of the trapezoid, turn it a half turn, and fit it against the original along one leg.

abh → ba ab

Why do they make a parallelogram? Turning the copy a half turn keeps every side pointing in the same direction. The copy's top is its long base \(b\) and its bottom is its short base \(a\). Along the bottom we now have \(b\) then \(a\), and along the top \(a\) then \(b\): both lines are \(a + b\) long, and both are parallel. The two outside legs are a leg of the original and the same leg of the copy, turned, so they are equal and parallel too. That is a parallelogram with base \(a + b\) and height \(h\).

Its area is \((a + b) \times h\). The trapezoid is half of it:

\[\text{area of a trapezoid} = \tfrac12 (a + b) \times h = \frac{a + b}{2} \times h\]

Way 2: cut it into two triangles

Draw one diagonal (a line from one corner to the opposite corner). It cuts the trapezoid into two triangles.

ab blue: ½ × b × h green: ½ × a × h
  • The lower triangle has base \(b\). Its top corner is on the top line, so its height is \(h\). Its area is \(\tfrac12 b h\).
  • The upper triangle has base \(a\) (turn the page upside down to see it standing on \(a\)). Its third corner is on the bottom line, so its height is also \(h\). Its area is \(\tfrac12 a h\).

Together: \(\tfrac12 a h + \tfrac12 b h = \tfrac12 (a + b) h\), the same formula. Two completely different cuts give the same answer, which is a good sign the formula is right.

Way 3: the average width

The formula \(\frac{a + b}{2} \times h\) can be read as "the average of the two bases, times the height". Here is why that makes sense. Mark the middle point of each sloping leg, and through each one draw a line straight up and down, at a right angle to the bases. Near each leg, this cuts a small right triangle off the bottom corner, and leaves an empty triangle of exactly the same shape above it, at the top. Turn each small triangle a half turn about the leg's middle point and it drops into the empty space. The shape becomes a rectangle, as tall as the trapezoid. Its length is the distance between the two upright lines, which is the length of the line joining the middles of the legs, the line halfway up. That halfway line is longer than \(a\) and shorter than \(b\), and it is exactly halfway between them: their average. (For the flower bed it is 7 m: the bed is 6 m wider at the front than at the back, and halfway back it is 3 m wider than the back, \(4 + 3 = 7\).)

So a trapezoid is, for area, just a rectangle of length "average base". That answers the flower beds: the average of 4 m and 10 m is 7 m, so the trapezoid's area is \(7 \times 3 = 21\) m², the same as the 7 m by 3 m rectangle.

One formula, three shapes

The trapezoid formula quietly contains two others.

  • If the two bases are equal (\(a = b\)), the trapezoid is a parallelogram, and \(\tfrac12 (b + b) h = \tfrac12 \times 2b \times h = b h\). That is the parallelogram rule.
  • If the top base shrinks to nothing (\(a = 0\)), the top side becomes a single point, and the trapezoid becomes a triangle: \(\tfrac12 (0 + b) h = \tfrac12 b h\). That is the triangle rule.

So the trapezoid formula is the "big" one, and the other two are special cases of it.

Worked examples

1. The flower bed. \(a = 4\) m, \(b = 10\) m, \(h = 3\) m.

Area \(= \tfrac12 (4 + 10) \times 3 = \tfrac12 \times 14 \times 3 = 7 \times 3 = 21\) m².

2. Do the half first, or last. A trapezoid has bases 5 cm and 8 cm and height 6 cm.

Average base first: \((5 + 8) \div 2 = 6.5\), then \(6.5 \times 6 = 39\) cm². Or half last: \((5 + 8) \times 6 = 78\), then \(78 \div 2 = 39\) cm². Same answer; pick whichever is easier.

3. Missing height. A trapezoid has bases 9 m and 15 m, and area 72 m². How far apart are the bases?

The average base is \((9 + 15) \div 2 = 12\). Then \(12 \times h = 72\), so \(h = 6\) m.

4. Missing base. A trapezoid has area 50 cm², height 5 cm, and one base 6 cm. How long is the other base?

\(\frac{6 + b}{2} \times 5 = 50\). Divide by 5: the average base is 10. The average of 6 and \(b\) is 10, so \(6 + b = 20\) and \(b = 14\) cm. (Check: \(\tfrac12 (6 + 14) \times 5 = 10 \times 5 = 50\). ✓)

5. A ditch. A ditch has a cross-section shaped like an upside-down trapezoid: 3 m wide at the top, 1 m wide at the bottom, and 1.5 m deep. What is the area of the cross-section?

The bases are 3 m and 1 m (the depth is the height): \(\tfrac12 (3 + 1) \times 1.5 = 2 \times 1.5 = 3\) m². If the ditch is 40 m long, it holds \(3 \times 40 = 120\) cubic metres of water when full (more on volume in lesson 5).

6. A roof. One face of a roof is a trapezoid 12 m along the bottom edge, 6 m along the ridge at the top, and 4 m from bottom to top, measured up the slope at a right angle to the edges. Shingles come in bundles that each cover 3 m². How many bundles?

Area \(= \tfrac12 (12 + 6) \times 4 = 9 \times 4 = 36\) m². Bundles: \(36 \div 3 = 12\).

Try it

  1. Find the area of a trapezoid with bases 6 cm and 10 cm and height 4 cm.
  2. Find the area of a trapezoid with bases 2.5 m and 5.5 m and height 3 m.
  3. A trapezoid has bases 7 cm and 13 cm, legs of 5 cm and 8 cm, and height 4 cm. Find its area. Which numbers did you not need?
  4. A trapezoid has bases 8 m and 12 m and area 60 m². Find its height.
  5. A trapezoid has area 42 cm², height 6 cm and one base 4 cm. Find the other base.
  6. On squared paper, a trapezoid has its bottom base along the grid from \((0, 0)\) to \((8, 0)\) and its top base from \((2, 4)\) to \((5, 4)\). Find its area with the formula, then check by cutting it into a rectangle and two right triangles.
  7. Use the formula with \(a = b\) to find the area of a parallelogram with base 9 and height 4, and with \(a = 0\) to find the area of a triangle with base 9 and height 4.
Answers
  1. \(\tfrac12 (6 + 10) \times 4 = 8 \times 4 = 32\) cm².
  2. \(\tfrac12 (2.5 + 5.5) \times 3 = 4 \times 3 = 12\) m².
  3. \(\tfrac12 (7 + 13) \times 4 = 10 \times 4 = 40\) cm². The legs (5 cm and 8 cm) are not needed.
  4. Average base \((8 + 12) \div 2 = 10\), so \(10 \times h = 60\) and \(h = 6\) m.
  5. \(\frac{4 + b}{2} \times 6 = 42\), so the average base is \(42 \div 6 = 7\). Then \(4 + b = 14\) and \(b = 10\) cm.
  6. Bases 8 and 3, height 4: \(\tfrac12 (8 + 3) \times 4 = 22\) squares. Cutting: drop lines straight down from \((2, 4)\) and \((5, 4)\). The middle rectangle is 3 by 4, area 12. The left triangle has base 2 and height 4, area 4. The right triangle has base \(8 - 5 = 3\) and height 4, area 6. Total \(12 + 4 + 6 = 22\). ✓
  7. \(a = b = 9\): \(\tfrac12 (9 + 9) \times 4 = 36\), which is \(9 \times 4\). ✓ \(a = 0\), \(b = 9\): \(\tfrac12 (0 + 9) \times 4 = 18\), which is \(\tfrac12 \times 9 \times 4\). ✓

Common mistake

Adding all four sides. Only the two parallel sides go into the formula. The legs are used for the perimeter, never for the area.

Forgetting the half, or halving twice. \(\frac{a + b}{2}\) already includes the half. Either average the bases and multiply by \(h\), or add the bases, multiply by \(h\) and then halve, but not both.

Where it leads