# 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 Trapezoid

Using the trapezoid above, we draw BB' parallel to CD. Using the triangle ABB', we use the cosine rule to write
c2 = a2 + (d-b)2 - 2 a (d - b) cos(?BAD)
cos(?BAD) = (a2 + (d-b)2 - c2) / (2 a (d - b) )
?BAD = arccos ( (a2 + (d-b)2 - c2) / (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
a2 = c2 + (d-b)2 - 2 c (d - b) cos(?BB'A)
cos(?BB'A) = (c2 + (d-b)2 - a2) / (2 c (d - b) )
?CDA = ?BB'A = arccos ( (c2 + (d-b)2 - a2) / (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
ABC = 180° - BAD and DCB = 180° - CDA

## Height and Area of a Trapezoid area = (1/2)(b + d) h

## Diagonals of a Trapezoid Use cosine rule in triangles DAB and BCD to write:
BD2 = a2 + d2 - 2 a d cos (? BAD)
BD = √ (a2 + d2 - 2 a d cos (? BAD))

CA2 = c2 + d2 - 2 c d cos (? CDA)
CA = √ (c2 + d2 - 2 c d cos (? CDA))

## More References and Links to Geometry

Geometry 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.