# Solve a Trapezoid Given its Bases and Legs

A trapezoid with bases b and d (d > b), legs a and c, and AD and BC are parallel is shown below . Calculate all its angles and its height h.

## Angles of a TrapezoidUsing the trapezoid above, we draw BB' parallel to CD. Using the triangle ABB', we use the cosine rule to write c ^{2} = a^{2} + (d-b)^{2} - 2 a (d - b) cos(∠BAD)
cos(∠BAD) = (a ^{2} + (d-b)^{2} - c^{2}) / (2 a (d - b) )
^{2} + (d-b)^{2} - c^{2}) / (2 a (d - b) ) ) In the same figure, ∠BB'A and ∠CDA have the same size. Using the same triangle, we use the cosine rule again to write a ^{2} = c^{2} + (d-b)^{2} - 2 c (d - b) cos(∠BB'A)
cos(∠BB'A) = (c ^{2} + (d-b)^{2} - a^{2}) / (2 c (d - b) )
^{2} + (d-b)^{2} - a^{2}) / (2 c (d - b) ) )In the given trapezoid, AD and BC are parallel. Hence the pairs of angles BAD and ABC and CDA and DCB are supplementary. Hence
## Height and Area of a Trapezoid
## Diagonals of a TrapezoidUse cosine rule in triangles DAB and BCD to write: BD ^{2} = a^{2} + d^{2} - 2 a d cos (∠ BAD)
^{2} + d^{2} - 2 a d cos (∠ BAD)) CA ^{2} = c^{2} + d^{2} - 2 c d cos (∠ CDA)
^{2} + d^{2} - 2 c d cos (∠ CDA))
## More References and Links to GeometryGeometry Tutorials, Problems and Interactive Applets.Trapezoid Area Calculator. Calculator to calculate the area of a trapezoid given the bases and the height. Trapezoid Calculator and Solver. An easy to use online calculator to solve trapezoid problems. The area, the angles and the diagonals of a Trapezoid are calculated given its 4 sides. |