Circumference of a Circle

Key Ideas

• the circumference is the perimeter of a circle

• as you increase the number of sides of a regular polygon, it becomes more and more like a circle

Try the gadget below to see for yourself. Simply enter a value in the box, then press "enter" or click somewhere outside the box

Let's see how we can make use of that.

We'll start with a 4 sided regular polygon (aka square) inscribed in a circle. The only "number" we're given is the radius, R. We want to try to find the perimeter of the square in terms of R.

Since a square has 4 sides, we can make 4 triangles by connecting 4 radii to the 4 corners of the square.

Now, if we could find the length of side AB, we could just multiply it times 4 because there are 4 sides, and then we would have our perimeter. But alas, it won't be so easy. In order to find sides of a triangle, we need right triangles so we can use sine and cosine. So we need to chop triangle AEB into two right triangles.

Now we have a right triangle AEF. If we could figure out ∠AEF. we could use sine to solve for side AF.

Let's figure out ∠AEF. Well, a full rotation around a circle is 360°. But since we divided that into the 4 angles formed by our 4 triangles, each triangle will have the angle (360° ÷ 4). But then we cut that triangle in half, so

• ∠AEF = (360° ÷ 4) ÷ 2, which is the same thing as

• ∠AEF = (180° ÷ 4) (we're going to leave it like this to show something later)

So, now we can use sine. Recall

Sine of an Angle =

so

Sin(180° ÷ 4) =

But AF is half of AB, and AB is a side of the square, which we'll call s. This means

• AF = (1/2)s

And we can see that

• AE = R

Substituting into our sine equation, we get

Sin(180° ÷ 4) =

Multiplying both sides by 2R, we get

2Rsin(180° ÷ 4) = s

or

s = 2Rsin(180° ÷ 4)

Woohoo, we found a side. But since we have 4 sides, we need to multiply times 4 to find the perimeter.

P = 4 x 2Rsin(180° ÷ 4)

Rearranging

P = 2 x 4sin(180° ÷ 4) x R

Now let's try the same thing for a six sided regular polygon (hexagon)

∠AEF = (360° ÷ 6) ÷ 2, which is the same thing as

∠AEF = (180° ÷ 6)

Sin(180° ÷ 6) =

Multiplying both sides by 2R, we get

2Rsin(180° ÷ 6) = s

or

s = 2Rsin(180° ÷ 6)

Woohoo, we found a side. But since we have 6 sides, we need to multiply times 6 to find the perimeter.

P = 6 x 2Rsin(180° ÷ 6)

Rearranging

P = 2 x 6sin(180° ÷ 6) x R

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Now let's see what we have so far. For regular polygons

See the pattern? Indeed, we could make the table as long as we want