Finding the area of a trapezoid

A trapezoid has one pair of parallel sides. Its area comes from the average length of those parallel bases multiplied by the perpendicular distance between them.

The formula

Use A = ½(b₁ + b₂)h. Here b₁ and b₂ are the lengths of the parallel sides, and h is the perpendicular height between them. You can read the same formula as A = ((b₁ + b₂) ÷ 2) × h: average the bases, then multiply by height.

Worked example

Suppose the parallel bases are 8 inches and 14 inches, and the perpendicular height is 5 inches. Add the bases: 8 + 14 = 22. Divide by 2 to get the average base length, 11. Multiply by the height: 11 × 5 = 55 square inches.

Choose the perpendicular height

The height is not necessarily the slanted side of the trapezoid. It is the shortest perpendicular distance from one parallel base to the other. A right angle marker identifies it directly. If a slanted leg is labeled but no perpendicular height is given, do not substitute that leg unless the diagram or geometry shows it is perpendicular.

A second way to see the formula

Place two congruent trapezoids together to form a parallelogram whose base is b₁ + b₂ and whose height is h. The parallelogram's area is (b₁ + b₂)h. One trapezoid is half of that combined shape, so divide by 2.

Check units and common errors

• Area uses square units: square feet, square centimeters, and so on.
• The bases are the parallel sides, even if they are drawn at an angle on the page.
• Use the perpendicular height, not a sloping leg.
• Do not forget the one-half. It averages the two base lengths.
• Keep units consistent before substituting into the formula.

Exam shortcut

If the bases are 10 and 18 and the height is 7, the average base is 14, so the area is 14 × 7 = 98 square units. The shortcut of adding the bases, dividing by two, then multiplying by height is often easier to track than inserting every number at once.

If a problem gives the area and asks for a missing dimension, rearrange the same relationship. For example, h = 2A ÷ (b₁ + b₂). Check that the result has the expected unit and that doubling the area before dividing is consistent with the original one-half factor.

Use the average of the parallel bases

A trapezoid’s area is one-half the sum of its parallel bases times its perpendicular height: A = ½(b₁ + b₂)h. The two bases are the parallel sides, which may be the shorter and longer edges in a diagram. The height is the perpendicular distance between them, not the slanted length of a nonparallel side. Mark the pair of parallel sides before choosing values.

If the bases are 8 cm and 14 cm and the perpendicular height is 5 cm, A = ½(8 + 14)(5) = ½(22)(5) = 55 cm². The expression averages the base lengths, then multiplies by height: average base = 11 cm, and 11 × 5 = 55 cm². This provides a quick reasonableness check against a rectangle 11 cm wide and 5 cm tall.

Why the formula works

Two congruent copies of a trapezoid can be arranged to make a parallelogram with base length b₁ + b₂ and height h. The parallelogram’s area is (b₁ + b₂)h, so one trapezoid has half that area. This explains the one-half factor and shows why the bases are added before multiplying by height.

If one parallel base is unknown, rearrange the formula only after substituting the known area and height. For example, if A = 60, h = 6, and b₁ = 8, then 60 = ½(8 + b₂)6 = 3(8 + b₂). Dividing by 3 gives 20 = 8 + b₂, so b₂ = 12. Substitute back to verify: half of 20 times 6 is 60.

Distinguish height from a slanted side

In an isosceles trapezoid, the nonparallel sides may look symmetric, but they are not the height unless they meet a base at a right angle. If only a slanted side and base lengths are provided, the height may need to be found from a right triangle formed by dropping perpendiculars from the shorter base. Do not use a diagonal or slanted leg as h without evidence that it is perpendicular.

Area uses square units because it measures a two-dimensional region. A result of 55 cm is a length; the correct unit is 55 cm². Convert all lengths to the same unit first. If one base is in meters and another in centimeters, convert before adding.

• Identify the two parallel bases.
• Use perpendicular distance between the bases for h.
• Add the bases, multiply by h, then divide by 2.
• For a missing value, solve algebraically and substitute to check.
• Report area in square units and keep units consistent.

Solve for a missing base using the average-base form

Because A = ½(b₁ + b₂)h, the area equals the average of the parallel bases multiplied by the perpendicular height. If the area is 84 square meters, height is 7 meters, and one base is 10 meters, solve 84 = ½(10 + b₂)7. Divide by 3.5 to get 24 = 10 + b₂, so the missing base is 14 meters. Substitute to check that the average base is 12 and 12×7 = 84.

If the height is missing, rearrange to h = 2A/(b₁+b₂). Use the perpendicular distance between the bases, not the slanted side. A result that is larger than both bases may be possible for height; use the drawing and units to judge whether it is plausible rather than relying on a fixed shape proportion.

Apply the distinction carefully

The two bases of a trapezoid are the parallel sides; the height is the perpendicular distance between them, not the slanted length of a leg. The formula A = ½(b₁ + b₂)h averages the base lengths and multiplies by the height. For bases 9 cm and 15 cm with height 4 cm, the average base is 12 cm and the area is 48 square centimeters. If a diagram supplies a diagonal leg instead of the height, use the right triangle formed by dropping a perpendicular to find the height when the needed side information is available. Keep the units squared and check that swapping the two bases leaves the answer unchanged. A rectangle is a special case when both bases are equal.