Trapezoid Angle Calculator: Find All Four Interior Angles
Enter any known angle or choose the "From sides and height" mode to solve every interior angle of a trapezoid. The calculator uses the supplementary-angle rule (co-interior angles on a transversal sum to 180 degrees) and basic trigonometry, then shows each step of the working so you can follow along.
Formula
Worked example
A trapezoid has a left leg c = 5, right leg d = 5, and height h = 4. Then sin(alpha) = 4/5 = 0.8, so alpha = arcsin(0.8) = 53.13 deg and beta = 180 - 53.13 = 126.87 deg. Because both legs are equal, delta = alpha = 53.13 deg and gamma = 126.87 deg. Sum = 53.13 + 126.87 + 53.13 + 126.87 = 360 deg.
Trapezoid angle fundamentals
A trapezoid (also called a trapezium in British English) is a quadrilateral with exactly one pair of parallel sides, called the bases. The other two sides are the legs. Because the two bases are parallel, each leg acts as a transversal cutting them, and the two co-interior angles on each transversal are supplementary: alpha + beta = 180 degrees on the left leg, and delta + gamma = 180 degrees on the right leg. Since every quadrilateral has interior angles that sum to 360 degrees, knowing any one angle lets you determine its supplement, and knowing two independent angles determines all four.
How to use this calculator
Two modes are available. In Known angles mode, enter any one base angle and choose the trapezoid type. For a general trapezoid without extra information the calculator applies the supplementary rule; for an isosceles trapezoid it sets both base angles equal; for a right trapezoid it fixes one leg at 90 degrees. In Sides and height mode, enter the two base lengths, both leg lengths, and the perpendicular height. The calculator uses the relationship h = c * sin(alpha) to find alpha from arcsin(h/c), and h = d * sin(delta) to find delta from arcsin(h/d), then derives all four angles. The Show your work panel lists every arithmetic step with your actual numbers substituted.
Special trapezoid types
- Isosceles trapezoid: both legs are equal in length, so the base angles on each side are equal: alpha = delta and beta = gamma. The shape is symmetric about the perpendicular bisector of the two bases.
- Right trapezoid: one leg is perpendicular to both bases, giving two right angles (90 degrees) on that side. At least two of the four angles are always 90 degrees.
- Acute trapezoid: both base angles (alpha and delta) are less than 90 degrees, which means both top angles exceed 90 degrees.
- General trapezoid: no special constraints on side lengths; only the supplementary rule applies.
Formulas and worked example
From a known angle: beta = 180 - alpha. In sides-and-height mode the leg forms a right triangle with the height and the horizontal offset, so sin(alpha) = h / c, giving alpha = arcsin(h / c). Similarly delta = arcsin(h / d). For example, a trapezoid with legs c = d = 5 and height h = 4 gives sin(alpha) = 4/5 = 0.8, so alpha = arcsin(0.8) = 53.13 degrees. Because the legs are equal this is isosceles, so all four angles are 53.13, 126.87, 53.13, and 126.87 degrees, summing to exactly 360 degrees.
Trapezoid types by angle properties
| Type | Alpha | Beta | Delta | Gamma | Sum |
|---|---|---|---|---|---|
| General | Any (1-179) | 180 - alpha | Any (1-179) | 180 - delta | 360 |
| Isosceles | alpha | 180 - alpha | alpha | 180 - alpha | 360 |
| Right | 90 | 90 | 90 | 90 | 360 |
| Acute | < 90 | > 90 | < 90 | > 90 | 360 |
How the four interior angles behave for each special trapezoid type.
Frequently asked questions
What are the four angles of a trapezoid?
A trapezoid has four interior angles, typically labeled alpha, beta, delta, and gamma. Alpha and beta share the left leg and are supplementary (they sum to 180 degrees). Delta and gamma share the right leg and are also supplementary. All four together always sum to 360 degrees, the same as any other quadrilateral.
How do you find the angles of a trapezoid from its sides?
If you know the two leg lengths and the perpendicular height, use trigonometry: the left base angle alpha = arcsin(height / left leg length), and the right base angle delta = arcsin(height / right leg length). The top angles follow immediately: beta = 180 - alpha and gamma = 180 - delta.
Do opposite angles in a trapezoid add up to 180 degrees?
Not in general. In a trapezoid the supplementary pairs are the two angles on the same leg, not the angles at opposite corners. Specifically, alpha + beta = 180 degrees (left leg) and delta + gamma = 180 degrees (right leg). Opposite angles (like alpha and gamma) only sum to 180 degrees if the trapezoid also happens to be a parallelogram or cyclic, which is a special case.
What makes a trapezoid isosceles, and how does it affect the angles?
An isosceles trapezoid has legs of equal length. As a result, the two base angles are equal (alpha = delta) and the two top angles are equal (beta = gamma). This symmetry also means the trapezoid can be inscribed in a circle (it is a cyclic quadrilateral), unlike a general trapezoid.
Can a trapezoid have a right angle?
Yes. A right trapezoid has one leg perpendicular to both parallel bases, giving two right angles (90 degrees) on that side. The other two angles can be any pair of supplementary angles between 1 and 179 degrees, as long as both sides of the leg are 90 degrees each.
How is a trapezoid different from a parallelogram?
A parallelogram has two pairs of parallel sides, while a trapezoid has exactly one pair. In a parallelogram, opposite angles are equal and consecutive angles are supplementary. In a trapezoid, only the angles sharing a leg are supplementary; there is no requirement for opposite angles to be equal (unless it is the special isosceles case).