Adding and Subtracting Areas

Finding areas of complex shapes can be daunting at times. Many times people try to formulate strange and unnecessarily complicated formulas for these areas.

Instead of making up rather complex and unnecessary formulas for these shapes, we can sometimes use simpler shapes within the complex shape in order to find the resulting area. We do this by adding areas. For example, let's take a look at the following example.

To solve the question, we must take into account that the trapezoid can be split into two simpler shapes, a triangle and a rectangle.

The area of the rectangle is 3 x 5 = 15.

The area of the triangle is 0.5 x 2 x 3 = 3.

Adding these up, we find the total area of the trapezoid to be 18.

We can use another method to find this area, subtracting areas. In this case, we can remove the area of FCE from the area of ABEF to find the area of ABEC.

Adding and subtracting areas can be very hard in some cases. For example, if you have a sphere inscribed in a octrahedron, how can you find the volume which is part of the octahedron but not the sphere?

Well there is a relatively simple idea to approach this seemingly complex problem. First, you have to find the volume of the sphere, and we will subtract this off from the volume of the octahedron. The volume of a octahedron is just the volume of two individual tetrahedrons, the latter of which can be easily computed. In the end, we get a formula; however, try to derive this yourself for more practice. After doing this, we just subtract the two quantities and arrive at the final answer.