Area of a Trapezoid
Let's Derive the Area of a Trapezoid!
The formula for the area of a trapezoid is . There are 3 different ways we can prove this.
Deriving the formula using a parallelogram:
What is the area of a parallelogram?
How can we write the area of the parallelogram in terms of the original trapezoid from the graph above?
*Note: when you move the top half of the trapezoid, the label for side a moves. However, a is still the line segment AB
Deriving the formula using a rectangle and a triangle:
We can find the area of the trapezoid by adding the areas of the rectangle and the triangle it is made up of.
What is the area of a rectangle?
What is the area of a triangle?
What would the base of the rectangle be? What about the triangle?
*Hint: Think about how we split the trapezoid up. If you're not sure, take a look at the graph below.
If the base of the trapezoid is b and we removed a, the base of the triangle would be (b - a)
Now, try adding the formulas for the area of the rectangle and the triangle in terms of a, b, and h. See if it matches the area of a trapezoid.
Deriving the formula using two triangles:
We can use the formula for the areas of the two triangles to derive the formula of the trapezoid.
As you recalled earlier, the area of a triangle is
Add the formulas to see if it matches the area of a trapezoid.