Isosceles Trapezoid

Isosceles Trapezoid

A trapezoid with base angles that are the same. Consequently, the legs will be congruent to each other as well.

Isosceles trapezoid with two parallel sides labeled "base" (top and bottom) and two equal sides labeled "leg" (left and right).

Note: This is US usage. In the UK this figure would be called an isosceles trapezium.

Key Formula

A = \frac{(b_1 + b_2)}{2} \cdot h

Where:

$A$ = Area of the isosceles trapezoid
$b_1$ = Length of the longer base (one parallel side)
$b_2$ = Length of the shorter base (other parallel side)
$h$ = Height (perpendicular distance between the two bases)

Worked Example

Problem: An isosceles trapezoid has bases of length 20 cm and 12 cm, and each leg is 5 cm long. Find the height and the area.

Step 1: Find the horizontal overhang on each side. Because the trapezoid is symmetric, each leg extends beyond the shorter base by the same amount.

\text{overhang} = \frac{b_1 - b_2}{2} = \frac{20 - 12}{2} = 4 \text{ cm}

Step 2: Use the Pythagorean theorem on the right triangle formed by the leg, the overhang, and the height.

h = \sqrt{\ell^2 - \text{overhang}^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = 3 \text{ cm}

Step 3: Apply the trapezoid area formula using both bases and the height.

A = \frac{(20 + 12)}{2} \cdot 3 = \frac{32}{2} \cdot 3 = 16 \cdot 3 = 48 \text{ cm}^2

Answer: The height is 3 cm and the area is 48 cm².

Another Example

This example uses a base angle and trigonometry instead of being given the leg length directly. It also demonstrates how to find the diagonal, which is a key property of isosceles trapezoids (both diagonals are equal).

Problem: An isosceles trapezoid has bases of 14 cm and 8 cm. One base angle measures 60°. Find the leg length, the height, and the diagonal length.

Step 1: Find the overhang on each side, just as before.

\text{overhang} = \frac{14 - 8}{2} = 3 \text{ cm}

Step 2: Use trigonometry with the base angle of 60° to find the leg length. In the right triangle, the overhang is adjacent to the base angle.

\cos 60° = \frac{\text{overhang}}{\ell} \implies \ell = \frac{3}{\cos 60°} = \frac{3}{0.5} = 6 \text{ cm}

Step 3: Find the height using the sine of the base angle.

h = \ell \cdot \sin 60° = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \approx 5.20 \text{ cm}

Step 4: Find a diagonal using the right triangle formed by the height and the horizontal distance from one end of the longer base to the opposite end of the shorter base. That horizontal distance is overhang + shorter base = 3 + 8 = 11 cm.

d = \sqrt{h^2 + 11^2} = \sqrt{(3\sqrt{3})^2 + 121} = \sqrt{27 + 121} = \sqrt{148} = 2\sqrt{37} \approx 12.17 \text{ cm}

Answer: The leg is 6 cm, the height is 3√3 ≈ 5.20 cm, and each diagonal is 2√37 ≈ 12.17 cm.

Frequently Asked Questions

What is the difference between a trapezoid and an isosceles trapezoid?

A trapezoid is any quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a special type of trapezoid where the two non-parallel sides (legs) are equal in length. This extra condition makes the figure symmetric about the perpendicular bisector of the bases, giving it congruent base angles and equal diagonals.

Are the diagonals of an isosceles trapezoid equal?

Yes. The diagonals of an isosceles trapezoid are always congruent. This follows from the line of symmetry that runs through the midpoints of both bases. In fact, having equal diagonals is one way to prove that a trapezoid is isosceles.

What do the angles of an isosceles trapezoid add up to?

Like every quadrilateral, the interior angles sum to 360°. Because the figure is symmetric, each pair of base angles is congruent. If one base angle is α, the adjacent angle on the same leg is 180° − α. So the four angles are α, α, 180° − α, 180° − α.

Isosceles Trapezoid vs. General Trapezoid

	Isosceles Trapezoid	General Trapezoid
Definition	Trapezoid with congruent legs	Quadrilateral with exactly one pair of parallel sides
Legs	Equal in length	Can be different lengths
Base angles	Each pair of base angles is congruent	Base angles are generally unequal
Diagonals	Equal in length	Generally unequal
Line of symmetry	One line of symmetry (perpendicular bisector of bases)	No line of symmetry in general
Area formula	A = ½(b₁ + b₂)h	A = ½(b₁ + b₂)h (same formula)

Why It Matters

Isosceles trapezoids appear frequently in geometry courses when studying quadrilateral properties, angle relationships, and proofs involving symmetry. They also show up in real-world contexts such as the cross-sections of channels, bridge supports, and handbag shapes. Understanding their special properties — equal legs, congruent base angles, equal diagonals — is essential for solving coordinate geometry problems and for proving that a given quadrilateral belongs to this category.

Common Mistakes

Mistake: Assuming all trapezoids have equal diagonals or congruent base angles.

Correction: These properties hold only for isosceles trapezoids. A general trapezoid has no such symmetry. Always verify that the legs are congruent before applying these special properties.

Mistake: Forgetting to halve the base difference when finding the overhang to compute the height.

Correction: The overhang on each side is (b₁ − b₂)/2, not (b₁ − b₂). Because the trapezoid is symmetric, the difference in base lengths is split equally on both sides.

Related Terms

Trapezoid — General category that includes isosceles trapezoids
Angle — Base angles are congruent in this shape
Leg of a Trapezoid — The two congruent non-parallel sides
Congruent — Legs and base angle pairs are congruent
Trapezium — UK term for trapezoid
Quadrilateral — Broader family of four-sided polygons
Parallelogram — Has two pairs of parallel sides, unlike a trapezoid
Rectangle — Can be viewed as a special isosceles trapezoid with equal bases