Explain how the formula for the area of a trapezoid is derived from the formula for the area of a triangle?

Question

Evelyn Arboleda · Accepted Answer

Trapezoid can be divided into two triangles by a diagonal. These triangles will have bases that correspond to trapezoid's bases and altitudes equal to trapezoid's altitude.

One of the way to explain a formula for an area of a trapezoid using a formula for a triangle can be as follows. Assume a trapezoid #ABCD# with lower base #AB# and upper base #CD# (they are parallel) and sides #AD# and #BC#. Connect vertices #A# and #C# with a diagonal. Consider triangle #Delta ABC# as having a base #AB# and an altitude from vertex #C# down to point #M# on base #AB# (#CM_|_AB#). Its area is #S_1 = 1/2*AB*CM# Consider triangle #Delta BCD# as having a base #CD# and an altitude from vertex #B# up to point #N# on base #CD# (#BN_|_CD#). Its area is #S_2 = 1/2*CD*BN# Altitudes #CM# and #BN# are equal and constitute the distance between two parallel bases #AB# and #CD#. They both are equal to the altitude of the trapezoid #h#. Therefore, we can represent areas of our two triangles as #S_1 = 1/2*AB*h# #S_2 = 1/2*CD*h# Adding them together, we get the area of the whole trapezoid: #S = S_1 + S_2 = 1/2(AB+CD)h#, which is usually represented in words as "half-sum of the bases times the altitude".

Sadie Adams · Answer

undefined The formula for the area of a trapezoid can be derived from the formula for the area of a triangle by considering the trapezoid as the difference between two triangles. The area of a triangle is given by the formula A = 1/2 * base * height.

In a trapezoid, if we extend the two non-parallel sides to meet at a point, we create two triangles and a rectangle. The area of the trapezoid is the sum of the areas of these two triangles minus the area of the rectangle.

The formula for the area of a trapezoid, therefore, can be expressed as: 
A = (1/2 * base1 * height) + (1/2 * base2 * height) - (base2 - base1) * height.

Simplifying this expression yields the formula for the area of a trapezoid:
A = (1/2) * (base1 + base2) * height.