Isosceles trapezoid area and perimeter formulas with worked examples

For an isosceles trapezoid with base lengths a and b, perpendicular height h, and equal leg length c, the area is S = (a + b) × h ÷ 2 and the perimeter is P = a + b + 2c. The height and the sloping leg are different measurements.

What is an isosceles trapezoid?

An isosceles trapezoid has parallel bases and equal angles at each base. For the usual trapezoid with unequal bases, its non-parallel sides, called legs, are equal. In trapezoid ABCD with AB parallel to CD, the legs are AD and BC.

Properties and ways to recognize it

• The legs are equal: AD = BC.
• The angles at each base are equal: angle A = angle B, and angle C = angle D.
• The diagonals are equal: AC = BD.
• The four vertices lie on a circle.

To prove that a quadrilateral is an isosceles trapezoid, first establish that the intended bases are parallel. Then prove equal angles at one base or equal diagonals. Equal sides alone do not establish the required shape: a quadrilateral may instead be a parallelogram. Compare the properties and formulas of a parallelogram.

Area formula

S = (a + b) × h ÷ 2

Add the bases, multiply by the perpendicular distance between them, and divide by two. This is the general trapezoid area formula; equal legs are not required to use it. If the lengths are in centimeters, the area is in square centimeters (cm²).

If the area and both bases are known, rearrange the formula to find the height: h = 2S ÷ (a + b).

Perimeter formula

P = a + b + 2c

Perimeter is the distance around all four sides. For an isosceles trapezoid, the two legs contribute 2c. To find one leg from the perimeter, use c = (P − a − b) ÷ 2. Keep all lengths in the same unit.

Five worked examples

1. Find the perimeter

The bases are 23 cm and 15 cm, and each leg is 7 cm.

P = 23 + 15 + 2 × 7 = 52 cm.

2. Find the leg length

The perimeter is 56 cm, and the bases are 18 cm and 10 cm. Subtract the bases, then split the remaining length equally between the legs.

c = (56 − 18 − 10) ÷ 2 = 14 cm.

3. Find the area

The bases are 12 cm and 18 cm, and the height is 8 cm.

S = (12 + 18) × 8 ÷ 2 = 120 cm².

4. Find the height

The area is 128 cm², and the bases are 12 cm and 20 cm.

h = 2 × 128 ÷ (12 + 20) = 8 cm.

5. Find the bases from a ratio, then calculate area

The height is 56 cm. The longer base is 24 cm longer than the shorter one, and the shorter base is two-fifths of the longer base.

The bases represent 5 parts and 2 parts, so their difference is 3 parts. One part is 24 ÷ 3 = 8 cm. The longer base is 5 × 8 = 40 cm; the shorter base is 2 × 8 = 16 cm.

S = (40 + 16) × 56 ÷ 2 = 1,568 cm².

Before calculating, check that you have the perpendicular height for area and the leg length for perimeter. Label the answer with a length unit for perimeter or a square unit for area.