A pizza with an 18-inch diameter is cut into 12 equal slices. What is the area of each slice? How many linear inches of crust does each one have? What is the total perimeter of each slice?
We’re going to take this one question at a time. So, we’ve cut a circle into 12 equal slices. That means that the area of our circle has been cut into 12 equal pieces.
The following transcript is provided for your convenience.
So, if I find the area of my circle, and then divide it by 12, I can find the area of one piece of pizza. That’s the question we’re starting with. What is the area of each slice? So, first, I’m going to find the area of my circle. Area of a circle is pi times the radius squared. They didn’t give us the radius, though, we were given the diameter.
The diameter is the distance across the circle through the center. It has its endpoints on the circle. The radius is the distance halfway across the circle, or it has its endpoints on the center and the outside of the circle. The radius, then, is half the diameter.
So, if the diameter’s 18 inches, then the radius is 9 inches. So, we substitute 9 inches for radius. Pi times 9 inches squared.
So, the area is, 9 inches squared is 81 inches squared, times pi is 81 pi inches squared.
Now, that’s the area of the whole pizza. We want to find the area of just one slice, or each slice. So, that tells us we’re going to divide. The area of our pizza is cut into 12 equal slices, so we’re going to take our area, 81 pi inches squared, and divide it by 12. Now, we can simplify that, we can divide our numerator and our denominator both by 3, and we get 27/4 pi inches squared.
So, that’s the area of just one slice of pizza. The next questions says: how many linear inches of crust does each one have? So, they’re talking about the crust, that’s part of the circumference of the circle. How much of the circumference? The circumference of a circle is pi times the diameter. So, the circumference of the circle is pi times 18 inches, since that’s our diameter, which means our circumference is 18 pi inches.
That’s the distance all the way around the circle, but our circumference has been split into 12 equal slices by these 12 pieces of pizza, so if we want to find the linear inches of crust one slice has, then we take our linear inches of crust for the whole pizza, and divide it by 12. So, that’s 18 pi inches divided by 12, and that can be simplified, we can divide our numerator and our denominator by 3, 18 divided by 3 is 6, and 12 divided by 3 is 4, so 6/4 pi inches, or we could be simplifying that, or we can say 4 goes into 6 one time, with 2 leftover, 2/4 or 1/2, so it’s 1 1/2 pi inches. 1 1/2 inches of crust for each slice of pizza.
The last question says, what is the total perimeter of each slice? So, perimeter is the distance around, the perimeter of the slice of pizza would be our crust part, so that’s that 1 1/2 pi inches, plus this segment, which is our radius, which we found earlier was half the diameter, so plus 9 inches, plus the radius again, so plus 9 inches again.
And then we would combine like terms, so the perimeter is 9 inches plus 9 inches is 18 inches, plus 1.5 pi inches. That would be the perimeter of one slice of pizza.