Isosceles Trapezoid Calculator

Enter the two parallel bases and the leg length of an isosceles trapezoid to get its height, area, perimeter, and diagonal length.

Quick Facts

Height formula
h = √(c² − ((a−b)/2)²)
Derived from the Pythagorean theorem using half the base difference as one leg of a right triangle.
Area formula
A = (a + b)/2 × h
Average the two parallel bases, then multiply by the height.
Diagonal formula
d = √(c² + a×b)
Both diagonals are equal in length — a defining property of isosceles trapezoids.
Perimeter formula
P = a + b + 2c
Sum of both bases plus the two equal legs.

Your Results

Calculated
Area
-
A = (a+b)/2 × h, in square units
Perimeter
-
P = a + b + 2c
Height
-
Perpendicular distance between the two bases
Diagonal
-
Both diagonals are equal in an isosceles trapezoid

Ready

Enter the two bases and leg length, then press Calculate.

Formula and Method for the Isosceles Trapezoid

An isosceles trapezoid is a quadrilateral with one pair of parallel sides (the bases, of different lengths) and two non-parallel sides (the legs) that are equal in length. That leg symmetry also makes the two base angles at each base equal, and gives the shape a vertical line of symmetry. Given the longer base a, the shorter base b, and the leg length c, this calculator finds the height, area, perimeter, and diagonal length.

How the calculation works

Because the bases have different lengths, the shorter base sits centered above the longer one, leaving an equal overhang of (a − b)/2 on each side. Each leg, together with that overhang and the trapezoid's height, forms a right triangle, so the Pythagorean theorem gives the height directly: h = √(c² − ((a−b)/2)²). Once the height is known, the area is the average of the two bases times the height: A = (a + b)/2 × h. The perimeter is simply the sum of both bases plus the two equal legs: P = a + b + 2c. The diagonal — which is the same length on both sides in an isosceles trapezoid — works out to d = √(c² + a×b), found by placing the trapezoid on a coordinate grid and applying the distance formula.

Common mistakes

  • Swapping the bases: enter the longer parallel side as a and the shorter one as b — the height formula assumes a is greater than b.
  • Leg too short: if the leg length c is less than or equal to half the difference between the bases, (a−b)/2, the shape cannot close and no valid trapezoid exists (the height would require a negative number under the square root).
  • Confusing legs with diagonals: the legs are the two non-parallel sides connecting the bases directly; the diagonals connect opposite corners and are always longer than the legs.

Real-world applications

  • Architecture and design use trapezoidal cross-sections for windows, dormers, keystones, and lamp shades.
  • Civil engineering relies on the same shape for bridge cross-sections, retaining walls, and trapezoidal drainage or canal channels.
  • Manufacturing and carpentry use the leg-and-base relationship to cut symmetric trapezoidal panels, table tops, and roof trusses.
  • Land surveying computes the area of trapezoidal parcels with this exact formula when two sides run parallel.

Frequently Asked Questions

What makes a trapezoid isosceles?
A trapezoid is isosceles when its two non-parallel sides (legs) are equal in length, which also makes the two base angles at each parallel side equal and gives the shape a vertical line of symmetry.
How do you find the height of an isosceles trapezoid from its bases and leg length?
Use the Pythagorean theorem: h = √(c² − ((a−b)/2)²), where a and b are the longer and shorter parallel bases and c is the leg length. The term (a−b)/2 is the horizontal offset between the ends of the two bases.
Are the diagonals of an isosceles trapezoid equal?
Yes — both diagonals have exactly the same length, d = √(c² + a×b). Equal diagonals are a defining property of isosceles trapezoids and distinguish them from general (scalene) trapezoids.
What is the area formula for an isosceles trapezoid?
Area equals the average of the two parallel bases times the height: A = (a + b)/2 × h. For example, bases of 12 and 6 with a height of 4 give an area of (12+6)/2 × 4 = 36 square units.

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