Why do you add the lengths of the bases together?

Question

Why do you add the lengths of the bases together to solve for the volume of a trapezoid?

Answer 1

Draw a diagonal to "cut" the trapezoid into two triangles

Both have a height of "h"

One has a base b₁, and area of 0.5b₁h

The other has a base b₂, and area of 0.5b₂h

The area of the trapezoid is 0.5b₁h + 0.5b₂h, or 0.5h(b₁ + b₂)

QED

Answer 2

It is the AREA of the trapezoid, not the volume. Volume applies to 3-D figures like boxes, cubes, circles, cones, and prisms , of which trapezoidal prisms are a part...

positions the trapezoid with longer base on the bottom...

A = (1/2) height ( top + bottom) = (1/2) height ( base1 + base 2), where base1 < base2

PROOF:

breaks the trapezoid into a rectangle and two right triangles...

the area of the rectangle is base1 * height

the overhang is then (Base2-base1) which is evenly divided between

the two right triangles. So the base measure of one of these right triangles

is (1/2) (base2-base1)...

so then, the area of each such right triangle is (1/2) (1/2)(base2 - base1) * height

Since there are 2 such right triangles, doubling this area cancels ONE of the factors

of 1/2...

the total area is then:

(1/2)(base2-base1)*height + base1*height =

(1/2) base2 * height - (1/2) base1 * height + base1*height =

(1/2) base2*height - (1/2) base1*height + 1* base1*height

the bolded quantities combine:

(1/2) base 2 * height + 1/2 * base1 * height =

(1/2) height ( base2 + base1) <-- factors out 1/2*height

[end of proof]