Isosceles Trapezoid Calculator

Calculate area, height, diagonals, angles, perimeter, and midsegment of an isosceles trapezoid from its parallel sides and leg.

Enter Trapezoid Dimensions

An isosceles trapezoid has two parallel sides (a and b) and two equal legs (c).

Result

Area
--
square units
Height (h) --
Area --
Perimeter --
Midsegment (median) --
Diagonal (d) --
Base angles --
Top angles --
Inradius (if exists) --

Step-by-Step Solution

A = (a + b) / 2 * h

Understanding Isosceles Trapezoids

An isosceles trapezoid is a quadrilateral with exactly one pair of parallel sides (called bases) and two equal non-parallel sides (called legs). It has a line of symmetry perpendicular to both bases. The base angles are equal, and the diagonals are equal in length.

Key Formulas

Height

Computed from the leg and the difference of the bases using the Pythagorean theorem.

h = sqrt(c² - ((b-a)/2)²)

Area

Average of the two parallel sides multiplied by the height.

A = (a + b) / 2 * h

Diagonal

Both diagonals are equal in an isosceles trapezoid.

d = sqrt(c² + a*b)

Base Angle

The angle between the longer base and a leg.

θ = arccos((b-a) / (2c))

Midsegment (Median)

A segment connecting the midpoints of the legs, parallel to both bases.

m = (a + b) / 2

Perimeter

Sum of all four sides.

P = a + b + 2c

Properties of Isosceles Trapezoids

  • The two bases are parallel but of different lengths.
  • The two legs are equal in length.
  • Base angles are equal: the two angles adjacent to the longer base are equal, as are the two angles adjacent to the shorter base.
  • The diagonals are equal in length.
  • There is a line of symmetry passing through the midpoints of both bases.
  • Co-interior angles (same-side angles) sum to 180 degrees.
  • An isosceles trapezoid is cyclic (can be inscribed in a circle).

When Is It Valid?

An isosceles trapezoid with parallel sides a and b and leg c requires that the leg is long enough to connect the two bases. Specifically, the condition is: c > |b - a| / 2. If the leg is too short, no valid trapezoid can be formed.

Special Cases

  • Rectangle: When c = h (legs are perpendicular to both bases), the isosceles trapezoid becomes a rectangle with a = b.
  • Isosceles triangle: When a = 0, the trapezoid degenerates into an isosceles triangle.
  • Square: When a = b = c, and the angles are all 90 degrees, it becomes a square.

Real-World Applications

  • Architecture: Symmetrical window and door designs.
  • Engineering: Cross-sections of symmetrical beams and channels.
  • Road design: Symmetrical road cross-sections and embankments.
  • Optics: Trapezoidal prisms and lens designs.

How to Use This Calculator

  1. Enter the shorter parallel side (a), the longer parallel side (b), and the leg length (c).
  2. Click "Calculate All Properties" to see height, area, diagonals, angles, and more.
  3. The step-by-step section shows the full derivation of each property.