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.

trapezoid solve 1

Angles of a Trapezoid

Using the trapezoid above, we draw BB' parallel to CD.

trapezoid solve angles


Using the triangle ABB', we use the cosine rule to write

c
2 = a2 + (d-b)2 - 2 a (d - b) cos(∠BAD)

cos(∠BAD) = (a
2 + (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

a
2 = c2 + (d-b)2 - 2 c (d - b) cos(∠BB'A)

cos(∠BB'A) = (c
2 + (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



trapezoid height area


h = a cos (∠BAD)


area = (1/2)(b + d) h

Diagonals of a Trapezoid



trapezoid diagonals


Use cosine rule in triangles DAB and BCD to write:

BD
2 = a2 + d2 - 2 a d cos (∠ BAD)

BD = √ (a2 + d2 - 2 a d cos (∠ BAD))


CA
2 = c2 + d2 - 2 c d cos (∠ CDA)

CA = √ (c2 + d2 - 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.



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