Isosceles Trapezoid: Properties, Area & Examples

TL;DR

An isosceles trapezoid is a four-sided shape with one pair of parallel sides (the bases) and two non-parallel sides (the legs) of equal length, which gives it equal base angles, equal diagonals, and a line of symmetry. This article covers its definition, properties, the area formula A= \frac{1}{2}(a+b)h, perimeter, diagonals, six worked examples, and where students go wrong.

The Only Trapezoid That Looks the Same in a Mirror

Most four-sided shapes lose their look the moment you flip them, but the isosceles trapezoid is built to be symmetric: fold it down the middle and the two halves land exactly on each other. That single act of folding is where every one of its special properties comes from — the equal legs, the equal base angles, even the equal diagonals.

Once you see the symmetry, you never have to memorise the property list — you can rebuild it by asking what folding forces to be equal.

What Is an Isosceles Trapezoid?

An isosceles trapezoid (called an isosceles trapezium in British and Indian textbooks) is a trapezoid whose two non-parallel sides, called the legs, are equal in length. A trapezoid is any quadrilateral with at least one pair of parallel sides — the bases — and the "isosceles" part adds the condition that the legs match.

The two equal legs make the shape symmetric about the vertical line joining the midpoints of the two bases. That symmetry is the defining feature: a trapezoid with unequal legs is a plain trapezoid, while equal legs promote it to isosceles.

What Are the Properties of an Isosceles Trapezoid?

Every property below is a direct consequence of the symmetry. If you can picture the fold, you can predict each one.

The legs are equal — the two non-parallel sides have the same length.
The base angles are equal. The two angles at the longer base are equal to each other (∠A=∠B), and the two at the shorter base are equal (∠D=∠C).
Co-base angles are supplementary. An angle at the bottom and the angle directly above it on the same leg add to 180° (∠A + ∠D = 180°).
The diagonals are equal in length. (AC=BD). This is unusual — most quadrilaterals have unequal diagonals.
It has exactly one line of symmetry, the perpendicular bisector of both bases.
The interior angles sum to 360°, as in every quadrilateral.

Notice the diagonals are equal but do not bisect each other — they cross but not at their midpoints. That single distinction is what keeps an isosceles trapezoid from being mistaken for a rectangle, where the diagonals are both equal and bisect each other.

Is an Isosceles Trapezoid a Parallelogram?

No. A parallelogram needs two pairs of parallel sides; an isosceles trapezoid has exactly one pair (the bases), while its legs slant inward and are not parallel.

The Area of an Isosceles Trapezoid

The area of any trapezoid — isosceles or not — depends only on the two parallel bases and the perpendicular height between them:

A=\frac{1}{2}(a + b)h,

where a and b are the lengths of the two parallel sides (bases) and h is the perpendicular distance between them.

If a problem gives you the area and asks for the height, rearrange the same formula:

h=\frac{2A}{a + b}.

The Perimeter and Diagonals

The perimeter is just the distance around, so add all four sides:

P=a + b + 2c.

Examples of Isosceles Trapezoid

Example 1: Find the area of an isosceles trapezoid with parallel sides 8 cm and 14 cm and height 5 cm

A=\frac{1}{2}(8 + 14)(5) = 55 cm².

Example 2: An isosceles trapezoid has bases 6 m and 10 m, legs of 5 m each.

The real height is calculated using the Pythagorean theorem and gives approximately 4.58 m.

Final answer: about 36.6 m².

Example 3: The area of an isosceles trapezoid is 90 cm² with bases 10 cm and 8 cm. Find the height

h=\frac{2(90)}{10 + 8} = 10 cm.

Example 4: Find the perimeter of an isosceles trapezoid with bases 7 cm and 13 cm and legs of 5 cm each

P = 7 + 13 + 2(5) = 30 cm.

Example 5: In isosceles trapezoid ABCD, one base angle ∠A measures 70°. Find the other three angles

Final answer: ∠A = ∠B = 70°, ∠C = ∠D = 110°.

Example 6: An isosceles trapezoid has perimeter 44 cm, bases 10 cm and 16 cm. Find the length of each leg

c = 9 cm.

Why the Isosceles Trapezoid Matters

Bridges and dams. A dam's cross-section is an isosceles trapezoid — wide at the base to resist water pressure.
Architecture and furniture. Lampshades, buckets, and tapered table legs are isosceles-trapezoid profiles.

Common Errors When Working With Isosceles Trapezoids

Mistake 1: Using a leg as the height

Don't do this: Treat the slanted side as if it stood straight up.

Mistake 2: Assuming the diagonals bisect each other

Mistake 3: Calling it a parallelogram

Key Takeaways

An isosceles trapezoid has one pair of parallel bases and two equal legs.
Its area is A=\frac{1}{2}(a+b)h.
The base angles are equal, co-base angles are supplementary, and the diagonals are equal but do not bisect each other.
It is not a parallelogram.

Practice These Problems to Solidify Your Understanding

Find the area of an isosceles trapezoid with bases 9 cm and 15 cm and height 6 cm.
An isosceles trapezoid has perimeter 38 m with bases 8 m and 12 m. Find each leg.
One base angle of an isosceles trapezoid is 65°. Find all four interior angles.