We are asked to explain how we use areas of polygons to find the surface area formulas for three-dimensional figures. To answer the question, let's take a look at an example.

Example

Here is a cylinder. A cylinder is a three-dimensional figure with two parallel circular bases that are the same size.

Let's now take a look at its net.

Net of a cylinder with the radii r marked using dashed lines and the heigh labeled as h

We can see that the net consists of shapes we know — a rectangle (a polygon) and two circles. As a result, the surface area of the cylinder is the sum of the areas of the figures. Let's recall the formula for the area of a circle with radius

r .

A = π r^{2}

Now, the area of a rectangle is its length multiplied by the width. In the graph we can see that the width of the rectangle is

h .

Also, notice that the length is the circumference

C

of the bases.

Let's remember that the circumference

C

of a circle with radius

r

is twice the product of

π

and

r .

C = 2 π r

So far we have found that the

length

of the rectangle is

2 π r

and the

width

equals

h .

Let's now multiply these values. This will give us the area of the rectangle.

A = (2 π r) (h)

Lastly, we can add the areas of the figures. The result will be the surface area S.A. of our cylinder. When doing so, keep in mind that the cylinder has two circular bases, not just one.

S . A . = 2 (π r^{2}) + (2 π r) h

Summary

The sides of most three-dimensional figures we know are polygons or circles. This is why we can calculate the area of each side separately using the formulas we already know, then add them. This will give us a formula for the surface area of the figure.

Exercise

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