Calculators

Geometry

Plane

a = top side (base)
b = bottom side (base)
c = left leg
d = right leg
h = height, or altitude
∠A, ∠B, ∠C, ∠D = inner angles
A = area = ½(a + b)h
P = perimeter = a + b + c + d
m = midsegment = median = (a + b) / 2

Calculate the attributes of a trapezoid from known measures. Trapezoid calculations include side lengths, inner angles, height or altitude, perimeter, midsegment or median, and area. A trapezoid is also known as a trapezium.

A trapezoid is a two-dimensional 4-sided shape with one pair of parallel sides. The parallel sides are the top and bottom sides of the trapezoid and the non-parallel sides are the legs. In this calculator we have the shapes oriented so the top side a and the bottom side b are the parallel sides (bases).

The "inclusive" definition of a trapezoid also includes 4-sided shapes with two pairs of parallel sides. So square, rectangle, rhombus and parallelogram shapes are considered special cases of trapezoids. The "exclusive" definition of a trapezoid excludes shapes with two pair of parallel sides and includes only 4-sided shapes with one pair of parallel lines.

Units: Note that units of length do not affect calculations. You can select any unit and the length, angle and area calculations will remain the same.

Types of Trapezoids

Scalene Trapezoid

A scalene trapezoid has no special symmetry: the two non-parallel legs are not equal (c ≠ d) and inner angles are not equal. One property of all trapezoids, when the parallel lines are at the top and bottom, is that the 2 adjacent inner angles on the same side are supplementary, adding up to 180°. So in this trapezoid image ∠A + ∠B = 180° and ∠C + ∠D = 180°.

Area: A = ½(a + b)h

Isosceles Trapezoid

An isosceles trapezoid has equal-length legs (c = d) and equal base angles (A = D and B = C). It has a vertical line of symmetry through the midpoints of the two parallel sides.

Area: A = ½(a + b)h

Right Trapezoid

A right trapezoid has two adjacent right angles (A = B = 90°), which means the left leg c is equal to the height h. The right leg d connects the top and bottom parallel sides at oblique angles C and D.

Area: A = ½(a + b)h

Note that the formula for area A = ½(a + b)h is the same for all three types of trapezoids. The formula includes the two parallel sides and the perpendicular height between them. Each type of trapezoid has its own set of formulas for calculating all other measures.

Trapezoid Formulas

Area: A

A = ½(a + b)h

Perimeter: P

P = a + b + c + d

Midsegment or median: m

m = (a + b) / 2

Inner Angles: ∠A, ∠B, ∠C, ∠D

• ∠A + ∠B = 180° (co-interior angles on left leg c)
• ∠C + ∠D = 180° (co-interior angles on right leg d)
• for any trapezoid, 0° < ∠A < 180° and 0° < ∠D < 180°

Height or : h

h = c sin(∠A) = d sin(∠D)

Constraint

b = a + c cos(∠A) + d cos(∠D)

Trapezoid Calculations:

The following formulas are derived from the trapezoid formulas above. We use them in this calculator to find the answers to the calculation you select.

• Find A | Given a, b, h
  Given the two parallel sides and height, calculate the area and midsegment
  • A = ½(a + b)h
  • m = (a + b) / 2
• Find a | Given A, b, h
  Given area, base and height, calculate the top base and midsegment
  • a = (2A / h) - b
  • m = (a + b) / 2
• Find b | Given A, a, h
  Given area, top base and height, calculate the base and midsegment
  • b = (2A / h) - a
  • m = (a + b) / 2
• Find h | Given A, a, b
  Given area, top and bottom bases, calculate the height and midsegment
  • h = 2A / (a + b)
  • m = (a + b) / 2
• Find d, h, angles, m, P, A | Given a, b, c, ∠A
  Given sides a, b, c and angle A, calculate all remaining properties
  • h = c sin(∠A)
  • ∠B = 180° - ∠A
  • d = √(h² + (b - a - c cos(∠A))²)
  • ∠D = atan2(h, b - a - c cos(∠A))
  • ∠C = 180° - ∠D
  • A = ½(a + b)h
  • P = a + b + c + d
  • m = (a + b) / 2
• Find c, h, angles, m, P, A | Given a, b, d, ∠D
  Given sides a, b, d and angle D, calculate all remaining properties
  • h = d sin(∠D)
  • ∠C = 180° - ∠D
  • c = √(h² + (b - a - d cos(∠D))²)
  • ∠A = atan2(h, b - a - d cos(∠D))
  • ∠B = 180° - ∠A
  • A = ½(a + b)h
  • P = a + b + c + d
  • m = (a + b) / 2
• Find h, angles, m, P, A | Given a, b, c, d
  Given all four sides, calculate the height, angles, perimeter, midsegment and area
  • x = ((b-a)² + c² - d²) / (2(b-a))
  • h = √(c² - x²)
  • ∠A = arccos(x / c)
  • ∠D = arccos((b - a - x) / d)
  • ∠B = 180° - ∠A
  • ∠C = 180° - ∠D
  • A = ½(a + b)h
  • P = a + b + c + d
  • m = (a + b) / 2
• Find c, d, h, m, P, A | Given a, b, ∠A, ∠D
  Given the two parallel bases and two base angles, calculate all remaining properties
  • h = (b - a) / (cos(∠A)/sin(∠A) + cos(∠D)/sin(∠D))
  • c = h / sin(∠A)
  • d = h / sin(∠D)
  • ∠B = 180° - ∠A
  • ∠C = 180° - ∠D
  • A = ½(a + b)h
  • P = a + b + c + d
  • m = (a + b) / 2

References

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 322, 2003.

Weisstein, Eric W. "Trapezoid." From MathWorld--A Wolfram Web Resource. Trapezoid.