Geometry: Circumference and Area: Pi Anyone?

Circumference and Area: Pi Anyone?

One of the unusual things about circles is that they are all similar to each other. I mean this in the official sense. It's not just that circles have the same shape but that everything is proportional as well. It is from proportionalities that the following postulate is presented.

• Postulate 17.3: The ratio of the circumference of a circle to the length of its diameter is a unique positive constant.

This constant of proportionality is represented by the Greek letter π (pronounced Pi, as in key lime). The reason that a Greek letter is used is because π is an irrational number. It cannot be written as neatly as 3 or 3.1, or 22/7, even though these numbers are approximations to π. It is impossible to write down the precise value of π.

If you denote the circumference of a circle by C and the length of a diameter of the circle by d, then C = πd. Because the length of a diameter of a circle is twice the length of the radius, if you let r denote the length of the radius of the circle, you have d = 2r and C = 2πr.

Now that you know how to compute the circumference of a circle, it is time to revisit arcs. Remember that there are two measurements associated with arcs: degree measurement and length. It turns out that the lengths of arcs can be found using a proportionality. The ratio of the degree measure m of an arc to 360º (the degree measure of a circle) is the same as the ratio of the length l of the arc to the circumference. The equation is given by

• ^m/_360º = ¹/_C = ¹/_2πr.

Rearranging this equation to solve for the arc length, you get

• 1 = ^m/_360ºC.

Let's do some calculations.

• Example 1: Find the circumference of the circle and the length of ˆABC in Figure 17.6.

Figure 17.6The circle with radius 5 inches, and m∠AOC = 125º.

• Solution: To find the circumference of the circle, just use the equation that relates the circumference of a circle to the radius of the circle:
• C = 2πr = 2π(5) = 10π inches.
• To find the length of ˆABC , notice that the central angle has measure 125º. Using the equation for arc length, you have:
• 1 = ^m/_360º C = ^125º/_360º(10π) = ^125π/₃₆ inches

Now consider the problem of finding the area A of a circle whose radius has length r. The formula for this is A = πr2. You can find the area of a sector using proportions, just as you found the length of an arc. If the area of the entire circle is πr2, the area of a sector with degree measure θ is found by evaluating

• A_sector = ^θπr2/₃₆₀.
• Example 2: Find the area of a circle whose radius has a length of 15 inches.
• Solution: You can solve this problem easily using the equation for area: A = πr² = π(15)² = 225π inches²
• Example 3: Find the area of the sector with a radius length of 4 inches and central angle measuring 120º.
• Solution: Substituting into our equation, you see that the area is
• ^δπr2/₃₆₀ = ^120π(4)2/₃₆₀ = ^16π/₃ inches².