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
c^{2} = a^{2} + (db)^{2}  2 a (d  b) cos(∠BAD)
cos(∠BAD) = (a^{2} + (db)^{2}  c^{2}) / (2 a (d  b) )
∠BAD = arccos ( (a^{2} + (db)^{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} + (db)^{2}  2 c (d  b) cos(∠BB'A)
cos(∠BB'A) = (c^{2} + (db)^{2}  a^{2}) / (2 c (d  b) )
∠CDA = ∠BB'A = arccos ( (c^{2} + (db)^{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
ABC = 180°  BAD and DCB = 180°  CDA
Height and Area of a Trapezoid
h = a cos (∠BAD)
area = (1/2)(b + d) h
Diagonals of a Trapezoid
Use cosine rule in triangles DAB and BCD to write:
BD^{2} = a^{2} + d^{2}  2 a d cos (∠ BAD)
BD = √ (a^{2} + d^{2}  2 a d cos (∠ BAD))
CA^{2} = c^{2} + d^{2}  2 c d cos (∠ CDA)
CA = √ (c^{2} + d^{2}  2 c d cos (∠ CDA))
More references on 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.

