Isosceles Trapezoid Calculator

An isosceles trapezoid is a trapezoid (quadrilateral with one pair of parallel sides) whose two non-parallel sides (the legs) are equal in length. This single extra symmetry — equal legs — unlocks several beautiful properties: equal base angles, equal diagonals, and a constructible height from the leg + base difference. This tutorial covers the four defining and derived properties of isosceles trapezoids, three worked examples (area, height, diagonal), and how to spot them in problems.

The four key properties

An isosceles trapezoid ABCD with parallel bases AB (longer, b₁) and CD (shorter, b₂), and equal legs AD = BC = leg:

1. The two legs are equal. AD = BC. This is the defining condition.
2. Base angles are equal in pairs. ∠A = ∠B (the two angles on the longer base), and ∠C = ∠D (the two angles on the shorter base).
3. The two diagonals are equal in length. AC = BD. This is a famously useful property in proofs.
4. It has a line of symmetry. The perpendicular bisector of either base is the axis of symmetry, and the figure is symmetric about it.

Conversely: ANY ONE of (1), (2), (3), or (4) implies all the others. Each is equivalent to the isosceles definition.

The height formula

If you know both bases and the leg length, you can compute the height:

h = √(leg² − ((b₁ − b₂) / 2)²)

Where this comes from: drop perpendiculars from C and D (the shorter base) straight down to AB (the longer base). By the symmetry of the isosceles trapezoid, these perpendiculars land at points that divide the longer base into three pieces: two equal "overhangs" of length (b₁ − b₂)/2 on each side, and a middle piece of length b₂ (directly under the shorter base).

Each overhang forms a right triangle with the leg as hypotenuse. By the Pythagorean theorem:

leg² = ((b₁ − b₂)/2)² + h², so h = √(leg² − ((b₁−b₂)/2)²).

Worked example 1 — area

Isosceles trapezoid with b₁ = 12, b₂ = 8, leg = 5.

h = √(5² − ((12−8)/2)²) = √(25 − 4) = √21 ≈ 4.58.

Area = ½ × (b₁ + b₂) × h = ½ × 20 × √21 = 10√21 ≈ 45.83.

Perimeter = b₁ + b₂ + 2 × leg = 12 + 8 + 10 = 30.

The diagonal formula

For an isosceles trapezoid:

Diagonal = √(leg² + b₁ × b₂)

Both diagonals are equal in length, given by this formula. Derivation: pick a diagonal, say AC. It connects vertex A on the longer base to vertex C on the shorter base. Using the right triangle formed by drawing AC plus a perpendicular from C down to AB, we get a right triangle with one leg being the horizontal distance from A to the foot (which equals the overhang plus b₂, totaling (b₁+b₂)/2) and the other leg being h. The hypotenuse — the diagonal — has length √(((b₁+b₂)/2)² + h²). Expanding and using h² = leg² − ((b₁−b₂)/2)² gives the simplified formula above.

Worked example 2 — diagonal from sides

For the same trapezoid (b₁ = 12, b₂ = 8, leg = 5):

Diagonal = √(5² + 12 × 8) = √(25 + 96) = √121 = 11.

Both diagonals equal 11. Verify by the other formula: diagonal² = ((b₁+b₂)/2)² + h² = 10² + 21 = 121. ✓

Why are the base angles equal?

By symmetry. Drop the axis of symmetry — the perpendicular bisector of the longer base. This axis passes through the midpoint of the shorter base too (because the trapezoid is symmetric about it). The two legs are reflections of each other across this axis, so they make equal angles with their respective bases.

Formally: ∠A and ∠B are reflections of each other across the axis, so ∠A = ∠B. Same for ∠C and ∠D.

The two pairs (∠A = ∠B) and (∠C = ∠D) are NOT equal to each other in general — they are supplementary (sum to 180°) because of the parallel-base co-interior rule.

The "cyclic" property — isosceles trapezoids inscribe in circles

An isosceles trapezoid can be inscribed in a circle — meaning all four vertices lie on a single circle. (This is the same property as a "cyclic quadrilateral".) The cyclic property follows from the equal diagonals and equal base angles.

Other trapezoids cannot be inscribed in circles unless they are isosceles. The connection between isosceles trapezoids and cyclic quadrilaterals is a deep result that often appears in olympiad-style geometry problems.

Worked example 3 — finding leg from given height + bases

Isosceles trapezoid with b₁ = 14, b₂ = 6, height h = 6. Find the leg length.

Rearranging the height formula: leg² = h² + ((b₁ − b₂)/2)² = 6² + 4² = 36 + 16 = 52.
leg = √52 = 2√13 ≈ 7.21.

Area = ½ × 20 × 6 = 60. Perimeter = 14 + 6 + 2 × 7.21 ≈ 34.42.

Special cases

• When b₁ = b₂: the "trapezoid" becomes a parallelogram (and specifically a rectangle if also isosceles + has right base angles). Technically excluded from the strict "trapezoid has exactly one pair of parallel sides" definition.
• When b₂ = 0: the trapezoid collapses to an isosceles triangle. The two legs become two equal sides of the triangle meeting at the apex (where the b₂ side disappears).
• When the legs are perpendicular to the bases: the isosceles trapezoid becomes a rectangle. Both legs are vertical, both bases horizontal.

Recognizing isosceles trapezoids in problems

Any ONE of these is sufficient to conclude isosceles:

• Legs are equal.
• Base angles are equal in pairs.
• Diagonals are equal.
• The trapezoid is cyclic (can be inscribed in a circle).
• It has a line of symmetry.

Test problems often provide ONE of these as given and expect you to derive the others.

Common mistakes

• Using the slanted leg as height. The height is the perpendicular distance between the bases, NOT the leg length. Use h = √(leg² − ((b₁−b₂)/2)²).
• Assuming all four angles are equal. Only the pairs of base angles are equal (∠A = ∠B and ∠C = ∠D). Opposite angles are not equal (they're supplementary instead).
• Confusing isosceles trapezoid with general trapezoid. A general trapezoid has independent leg lengths and angles. The "diagonal = √(leg² + b₁ × b₂)" formula applies ONLY to the isosceles case.
• Forgetting both diagonals are equal. Some students compute one diagonal and miss that the other is the same — a quick property check.