Isosceles Trapezoid Area Calculator

Enter the two parallel bases and the leg length of an isosceles trapezoid to get its area (A = ½(a+b)h), height, perimeter, and base angle.

Quick Facts

Area formula
A = ½(a+b)h
Half the sum of the two parallel bases, times the height.
Height from legs
h = √(c² − ((a−b)/2)²)
Follows from the Pythagorean theorem using the equal leg length.
Perimeter
P = a + b + 2c
Sum of both bases plus the two equal legs.

Your Results

Calculated
Area
-
A = ½(a+b)h, in square units
Height
-
Perpendicular distance between the bases
Perimeter
-
a + b + 2c
Base Angle
-
Angle between a leg and the longer base

Ready

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

Formula and Method for Isosceles Trapezoid Area

Area of an Isosceles Trapezoid:

A = ½ × (a + b) × h

a, b = the two parallel bases; h = the perpendicular height between them

An isosceles trapezoid is a quadrilateral with one pair of parallel sides (the bases, a and b) and two non-parallel sides (the legs, c) that are equal in length. Because the legs are equal, the trapezoid is symmetric about the perpendicular bisector of its bases, and the two base angles on each base are equal to each other. This calculator finds the area, height, perimeter, and base angle from the two bases and the leg length.

How the calculation works

If you only know the leg length rather than the height, you first need to find the height using the Pythagorean theorem. Dropping a perpendicular from each end of the shorter base to the longer base splits the trapezoid into a rectangle in the middle and two congruent right triangles on the sides. Each right triangle has a horizontal leg of (a − b) / 2 and a hypotenuse equal to the trapezoid's leg length, c, so the vertical leg — the trapezoid's height — is h = √(c² − ((a − b) / 2)²). Once h is known, the area follows directly from A = ½(a + b)h, and the base angle at the longer base is θ = arccos(((a − b) / 2) / c). The perimeter is simply P = a + b + 2c, since both legs are the same length.

Common mistakes

  • Confusing the legs with the height: the leg length (c) is the slanted side, not the vertical height (h). Using c directly in the area formula instead of h will overstate the area.
  • Base order: enter the longer base as a and the shorter base as b; the formula ((a − b) / 2) assumes a ≥ b.
  • Impossible dimensions: the leg length must be longer than half the difference between the bases, (a − b) / 2, or the trapezoid cannot physically close.
  • Mixing units: keep the bases and leg length in the same unit before entering them; area comes out in square units of whatever length unit you chose.

Real-world applications

  • Architecture and construction use trapezoidal cross-sections for roof trusses, retaining walls, and window and door openings.
  • Civil engineering uses the isosceles trapezoid shape for canal and channel cross-sections, road embankments, and dam faces.
  • Woodworking and fabrication use it for tabletops, lampshades, and other symmetric tapered panels.
  • Land surveying uses trapezoid area formulas to estimate the area of trapezoid-shaped parcels and plots.

Frequently Asked Questions

What is the formula for the area of an isosceles trapezoid?
The area equals half the sum of the two parallel bases times the height: A = ½(a+b)h. If you only know the leg length, first find the height using h = √(c² − ((a−b)/2)²), which comes from the Pythagorean theorem applied to the right triangle formed by the leg, the height, and half the difference of the bases.
How do I find the height of an isosceles trapezoid from its leg length?
Drop a perpendicular from each end of the shorter base down to the longer base. This creates two congruent right triangles, each with a horizontal leg of (a−b)/2 and a hypotenuse equal to the trapezoid's leg length c. By the Pythagorean theorem, the height is h = √(c² − ((a−b)/2)²).
What are the base angles of an isosceles trapezoid?
Because the two legs are equal, the two angles along the longer base are equal to each other, and the two angles along the shorter base are equal to each other. The base angle at the longer base can be found with θ = arccos(((a−b)/2) / c).
What if the leg length is too short for the given bases?
The leg length must be greater than half the difference between the two bases, (a−b)/2, or the trapezoid cannot close because the leg would not reach from the shorter base down to the longer base. If your inputs fail this check, increase the leg length or reduce the difference between the bases.

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