Pies, cakes, pizzas; so many foods we eat neatly lend themselves to mathematics, because they are models of circles. A circle has an interior, its area. You can divide circles into pieces. Pizza slice pieces are sectors. Bits cut off by connecting any two points on the circle are segments. Since both sectors and segments are part of a circle's interior, both have area. We can calculate their area using formulas.

A **circle** is the set of all points equidistant from a given point on a plane. It encloses a measurable area.

*[show circle A with center point A and interior highlighted/shaded]*

The **area of a circle** is always calculated using the known relationship of π between a circle's radius, *r*, (or diameter, *d*) and its circumference:

- A = π
*r*^2 - A = π(
*d*/2)^2

When you take any two radii of the circle, the area between the radii is a **sector**:

*[show Circle A with 1/4th sector formed from Points R and P {radii RA and PA} highlighted]*

A **chord** is a line created by connecting any two points on the circle, without worrying about the center. A chord creates an area called a segment:

*[show Circle A with a segment formed by connecting Points R and P]*

The areas of both segments and sectors can be calculated in square units of whatever linear measurement you are given.

For Pi Day, your Math Club is celebrating with pies. You make or buy an 8" blueberry pie. You can quickly calculate the area of the pie, using either formula:

- A = π
*r*^2 - A = π x 4"^2
- A = 3.14159265 x 16"^2
- A = 50.265482"^2
- A = π(
*d*/2)^2 - A = π(8"/2)^2
- A = 3.14159265 x 16"^2
- A = 50.265482"^2

Buyers of whole pies can be charged your Math Club's usual cost, $0.31 cents a square inch, but what if someone wants only ¼ of the pie, or even a 1/16th of a pie? To calculate the area of a fractional part of the pie, you can set up a modified version of the area formula for the whole pie:

- A = (fractional part) x π
*r*^2 - A = ¼ x π x r^2

Or

- A = 1/16 x π x r^2

But you can *also* use a general formula, based on the central angle of the slice, represented by *n°* or ϴ:

- A =
*n°*/360° x π x*r*^2 - A = ϴ/360° x π x
*r*^2

The Math Club has 9.5" diameter apple pies to sell. You cut them into 45° slices. What is the area of each slice?

- A =
*n°*/360° x π x*r*^2 - A = 45°/360° x π x 4.75"^2
- A = 0.125 x 3.14159265 x 22.5625"^2
- A = 8.860273"^2

Some folks only have pocket change. What is the area of a 10° slice?

- A =
*n°*/360° x π x*r*^2 - A = 10°/360° x π x 4.75"^2
- A = 0.027777 x 3.14159265 x 22.5625"^2
- A = 1.968894"^2

What would a 10° slice cost, at $0.31 a square inch? Only $0.61. Such a deal!

To calculate areas of segments, you first have to know the area of the sector. You may think of the sector as a triangle and a segment put together.

To find the area of the segment, you find the area of the sector. Then you find the area of the isosceles triangle at the central angle. Subtract the area of the isosceles triangle from the area of the sector and you have the area of the segment:

A(segment) = A(sector) - A(triangle)

*[insert drawing based on Figure 1]*

If you know the radius, r, of the circle and you know the central angle, ϴ, in radians of the sector that contains the segment, you can use this formula to calculate the area, A, of only the segment:

- A = ½ x r^2(ϴ - sin(ϴ)

If you know the radius, r, of the circle and you know the central angle, ϴ, in degrees of the sector that contains the segment, you can use this formula to calculate the area, A, of only the segment:

- A = ½ × r^2 × ((π/180) ϴ - sin ϴ)

For example, take those 9.5" pies again. A sector (slice) of pie with a central angle of 60° or π/3 radians for a pie with a radius of 9.5" would plug into the formulas like this:

- A = ½ x r^2(ϴ - sinϴ)
- A = ½ x 9.5^2(π/3 - sin(π/3)
- A = 0.5 x 90.25 x 0.18117214
- A = 8.1753931 square inches
- A = ½ × r^2 × ((π/180) ϴ - sin ϴ)
- A = 0.5 x 9.5^2 x ((π/180)60 - sin60)
- A = 0.5 x 90.25 x (0.01745329251 x 60 - 0.86602540378)
- A = 0.5 x 90.25 x 0.1811721468
- A = 8.1753931 square inches

You can be as precise as you wish (and as precise as π will allow) in your calculations. Taking the final answer to the seventh decimal place is generally not necessary, but we wanted you to see that both methods return the same value to a very high degree of precision (100-millionths).

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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