Combined area under a graph

Question

Working on getting distance under a graph (velocity/time)

imagine a section which is comprised of a square with a right angle triangle stacked on top of it.

Calculating area of square + area of triangle = total area under graph and distance travelled is one way.

for example base (time) is 10 seconds. Square section has a value of 10 so area is 100 m distance

object accelerates in same 10 seconds to 20 so area is (10 x 10)/2 = 50 m. So total distance is 150m

I have seen that by adding the (height of the square +height of triangle) x base : is same value

(10 + 20) = 30 x 10 / 2 = 150 m

could someone show the math theory behind this?

Answer 1

Here is the shape you describe.

Take 2 of these shapes and rotate one and stack it on top of the other:

Now you have a rectangle with side length A+C and width B so the area is (A+C)*B but to get back to the original shape you need to divide by 2 so Area = 1/2(A+C)*B