Trapezoid: definition, types, properties and formulas

A trapezoid is a quadrilateral, a flat shape with four straight sides, that has at least one pair of parallel sides. The parallel sides are called the bases, the other two sides are the legs, and the distance between the bases is the height. Its area is half the sum of the bases times the height. In British English the same shape is called a trapezium.

What is a trapezoid?

Picture a triangle with its top sliced off by a cut parallel to the bottom. What is left is a trapezoid: a wide base, a narrower top that runs parallel to it, and two slanted legs joining them. A popcorn box seen from the side, a lampshade and many bridge supports have this shape.

In trapezoid ABCD, the long base AD is 18 and the short base BC is 6. They are parallel, which the matching arrows show. The slanted sides AB and CD are the legs, and here they are equal, so this is an isosceles trapezoid. The dashed line is the height, 8, which meets the base at a right angle.

Exactly one pair or at least one pair?

There are two definitions of a trapezoid, and textbooks don't agree:

• Exclusive definition: exactly one pair of parallel sides. Under this one, a parallelogram is not a trapezoid. This is the definition most often used in school textbooks.
• Inclusive definition: at least one pair of parallel sides. Under this one, every parallelogram, rectangle, rhombus and square is also a trapezoid. University geometry courses nearly always use it.

The formulas on this page work under both definitions. If a question asks whether a square is a trapezoid, check which definition your course uses.

Parts of a trapezoid

Types of trapezoids

Isosceles trapezoid

An isosceles trapezoid has legs of equal length. That gives it a line of symmetry through the midpoints of the two bases, and three useful facts: each pair of base angles is equal, the diagonals are equal in length, and the two legs lean in by the same amount. Trapezoid ABCD at the top of this page is isosceles.

Right trapezoid

A right trapezoid has one leg perpendicular to both bases, which makes two right angles. That straight leg is also the height, so right trapezoids are the easiest kind to work with.

Here the bottom base AB is 9, the top base DC is 5 and the straight leg AD is 3. The slanted leg BC drops 3 over a run of 9 − 5 = 4.

Properties of a trapezoid

• It has four sides and four angles, and the angles add up to 360°, like every quadrilateral.
• Angles along each leg add up to 180°. The leg crosses two parallel lines, so the two angles on the same side are same-side interior angles, which are supplementary.
• The midsegment is parallel to the bases, and its length is the average of the two bases.
• The diagonals cut each other in the same ratio as the bases. If one base is twice the other, each diagonal is split 2 : 1.
• In an isosceles trapezoid, base angles are equal, diagonals are equal, and there is one line of symmetry.

Trapezoid angles

Two facts solve almost every angle question: the angles along a leg add to 180°, and in an isosceles trapezoid the base angles are equal.

Example 1: an isosceles trapezoid

One base angle of an isosceles trapezoid is 65°. Find the other three angles.

The other angle on the same base is also 65°, because base angles are equal. Each angle at the other base shares a leg with a 65° angle, so it is 180° − 65° = 115°. The angles are 65°, 65°, 115° and 115°, which add to 360° as they should.

Example 2: angles with algebra

The two angles along one leg of a trapezoid are (3x + 10)° and (2x + 20)°. Find both angles.

So the angles are 3(30) + 10 = 100° and 2(30) + 20 = 80°.

Area of a trapezoid

The area of a trapezoid is the average of the two bases multiplied by the height:

Why does this work? Take two copies of the same trapezoid, turn one upside down and put them side by side. Together they make a parallelogram with base b₁ + b₂ and height h, so its area is (b₁ + b₂)h. One trapezoid is half of that. You can also read the formula as the midsegment times the height.

Example 3: find the area

A trapezoid has bases of 8 inches and 14 inches and a height of 5 inches. Find its area.

The area is 55 square inches.

Example 4: find the height

A trapezoid has an area of 72 cm² and bases of 10 cm and 14 cm. How tall is it?

Rearrange the formula to h = 2A ÷ (b₁ + b₂):

The height is 6 cm.

Example 5: find a missing base

A trapezoid has an area of 90 ft², a height of 6 ft and one base of 12 ft. Find the other base.

The other base is 18 ft.

Perimeter of a trapezoid

The perimeter is the distance around the outside, so add all four sides: P = b₁ + b₂ + leg₁ + leg₂. The catch is that questions often give you the height instead of a leg. Then you find the leg with the Pythagorean theorem.

Example 6: perimeter of an isosceles trapezoid

Trapezoid ABCD at the top of this page has bases 6 and 18 and a height of 8. Find its perimeter and area.

Drop a perpendicular from each end of the short base. The long base is 18 − 6 = 12 longer, and because the trapezoid is isosceles, that extra length is split evenly: 6 on each side. Each leg is the hypotenuse of a right triangle with sides 6 and 8:

The perimeter is 44 units and the area is 96 square units.

Example 7: a right trapezoid

The right trapezoid ABCD in the types section has bases 5 and 9 and height 3. The slanted leg drops 3 over a run of 4, so it is √(4² + 3²) = 5. The perimeter is 5 + 9 + 3 + 5 = 22 and the area is ½(5 + 9)(3) = 21.

Example 8: a yard in the shape of a trapezoid

A backyard is a trapezoid with parallel sides of 40 ft and 60 ft, which are 25 ft apart. One bag of grass seed covers 500 ft². How many bags do you need?

1,250 ÷ 500 = 2.5, and you can't buy half a bag, so you need 3 bags.

Midsegment of a trapezoid

The midsegment joins the midpoints of the two legs. It is parallel to both bases, and its length is their average. In trapezoid ABCD the bases are 6 and 18, so the midsegment is (6 + 18) ÷ 2 = 12. Work backward the same way: if a midsegment is 11 and one base is 7, the other base is 2 × 11 − 7 = 15.

Test yourself on the properties before moving on.

Is a square a trapezoid?

It depends on the definition. Under the inclusive definition (at least one pair of parallel sides), a square has two pairs, so it is a trapezoid, and so are rectangles, rhombuses and all other parallelograms. Under the exclusive definition (exactly one pair), none of them are. Every trapezoid is a quadrilateral, but not every quadrilateral is a trapezoid: a kite, for example, has no parallel sides.

Trapezoid vs trapezium

They are the same shape with different names. In the US and Canada it is a trapezoid. In the UK, Ireland, Australia and most other English-speaking countries it is a trapezium, and the area formula is the same. To add to the confusion, older American textbooks used "trapezium" for a quadrilateral with no parallel sides at all, so if you see the word in an American book, check what it means there.

Common mistakes

• Using a slanted leg as the height. The height must be perpendicular to the bases. Find it with the Pythagorean theorem if you need to.
• Forgetting the ½. (b₁ + b₂)h is the area of two trapezoids, not one.
• Adding the height into the perimeter. The perimeter uses the four sides. The height is only a side in a right trapezoid.
• Splitting the overhang unevenly. Only in an isosceles trapezoid does the extra length of the long base split equally on both sides.
• Mixing units. Area is in square units (in², cm², ft²), perimeter in ordinary units.
• Assuming base angles are equal. That is only true for isosceles trapezoids.

Practice with a short timed paper

Area, perimeter and angle problems with trapezoids come up in geometry and in algebra word problems. The daily Geometry, Algebra 2 & SAT Math and Pre-Algebra & Algebra 1 papers are short timed practice tests of 8 questions, with a new one every day.

Read next

Maths · GeometryPerimeter: formulas for every shape, with examplesWhat perimeter is and how to find it: formulas for squares, rectangles, triangles, polygons and circles, plus compound shapes, missing sides and examples.