Trapezoid area: use parallel bases and perpendicular height
Identify the two bases, distinguish height from a sloping side, and check the area with a worked example.
Method
A trapezoid area calculation needs the two parallel sides, called bases, and the perpendicular distance between them. The drawing may be rotated or skewed; the bases are not necessarily the sides that look horizontal. Height meets the lines containing the bases at a right angle. A sloping side is not the height unless that side is itself perpendicular to both bases.
Add the base lengths and divide by two to find their average, also the trapezoid’s midline length. Multiply that length by the perpendicular height to obtain area. Convert all three measurements to the same length unit first. Length in centimetres produces area in square centimetres; an area conversion uses a squared length factor, not the ordinary length factor.
- Parallel basesa = 14 cm; b = 8 cm
- Average base length(14 + 8) ÷ 2 = 11 cm
- Perpendicular height and area11 cm × 4 cm = 44 cm²
Identify the two bases, distinguish height from a sloping side, and check the area with a worked example.
Worked example
Consider an isosceles trapezoid with bases 14 cm and 8 cm, height 4 cm and equal sloping sides of 5 cm. Each horizontal offset is 3 cm, forming a 3–4–5 right triangle. The midline is 11 cm and the area is 44 cm². This example’s symmetry explains the offsets; do not assume equal offsets or use this shortcut for an arbitrary trapezoid.
In Trapezoid / trapezium, enter first parallel side 14, second parallel side 8 and perpendicular height 4. The results are area 44 square units and midline 11 units, meaning cm² and cm here. Substituting the 5 cm sloping side for height gives 55 cm², an incorrect result 25% too large. Sliding the upper base sideways while preserving both bases and height does not change the area.
(14 + 8) ÷ 2 = 11 cm
11 cm × 4 cm = 44 cm²
11 cm × 5 cm = 55 cm² ≠ 44 cm²
(55 − 44) ÷ 44 × 100 = 25%
- Correct height: 4 cm
- 44 cm²11 cm × 4 cm = 44 cm²
- Wrong substitution: 5 cm sloping side
- 55 cm²11 cm × 5 cm = 55 cm²
Both bases are 14 cm and 8 cm. The correct height is 4 cm. Using the 5 cm sloping side as height deliberately gives an incorrect area for this shape.
Checks and limits
The formula describes a flat shape with straight, parallel bases. It does not give the perimeter, which needs both remaining side lengths, or the volume of a three-dimensional object. Curved or nonparallel boundaries need another model. A sketch alone is not a reliable measurement: confirm the right-angle height. For unit checking, 44 cm² equals 0.0044 m², not 0.44 m²; these are geometry results, not an installation specification.
Related calculators
Source: OpenStax — Prealgebra 2e