Area of Circles

At a local pizza place, Tetromino’s Pizza, 1 large pizza costs the same as 2 medium pizzas. Which option do you think you should buy if you want as much pizza as possible? 1 large 2 medium. To decide which is the better purchase, we need to know the area circumference of each circular pizza.

As we’ve done in previous chapters, let’s use what we know about polygons to help us understand the area of circles. Estimate the area of each pizza by cutting them into 8 equal triangles.

Recall that the area of a triangle can be found by using the formula 12xbasexheight, as long as the base forms a right acute obtuse angle with the base.

The larger triangle has square cm of pizza and the smaller triangle has square cm of pizza. In total, 1 large pizza has about square cm of pizza and the 2 medium pizzas have about square cm of pizza. So, it seems the 2 medium pizzas give more pizza.

This process is helpful for making good estimates, but our answers are less than more than the actual areas. Maybe the 1 large pizza actually gives more pizza? Let’s keep working to see if we can develop a process for calculating the exact area of a circle.

In the Circumference chapter, we thought about the distance around a square to help us think about the distance around a circle. Let’s take the same approach with area. Below is a square pizza with one dimension labeled as “R.” While it’s not typical to think about the radius of a square, this line is the shortest distance from the center of the square to the outside side length.

Cut out a square piece of this pizza that is a square with side lengths “R.”

The area of one of those pieces is RxR R+R 2xR. Recall that this can also be written as R2.

If each person receives one of the R2-sized pieces, this pizza can serve people.

Therefore the area of this square is . Each side length of the square is , we can also find the area of the square by doing . This also equals 4R2. Either way, we see that it takes 4 “R by R” squares to fill in a square when R represents the radius of the square. Let’s see if we can use this idea to discover a way to find the area of a circle.

Below is a circular pizza with the radius labeled.

Coming Soon!

Draw in a square with side length “R.”

How many of these “R by R” squares do you think you can make from this pizza if rearranging the pizza is allowed TODO.

4 R2’s is bigger smaller than the area of the pizza. Click on one piece to remove it from the picture.

Let’s see how the extra part of the R2 compares to the 14 of the pizza left.

3R2 is just not quite enough to cover up the whole pizza. We need a little bit more than less than 3R2 to cover the pizza. Do you recall another number we’ve learned about in thinking about circles that is a little more than 3? Do you think this could be the same number? Let’s keep exploring this idea and find out!

Before answering this question, let’s come back to our initial pizza question using the idea of 3R2

Coming Soon!

The large pizza has a diameter of 44 cm, so the radius is cm. The area of one R2 with this radius is square cm, so 3R2 for the large pizza is square cm. The radius of the medium pizza is cm so 3R2 for 1 one of the medium pizzas is square cm. The area of two of these medium pizzas is square cm.

So, using 3R2 as an estimate for finding the area of a circle gives us a better estimate than splitting the circle into triangles. Unfortunately, 3R2 is still underestimating the area of a circle.

Delivered pizzas can sometimes sit around for too long and get cold. People often reheat pizzas before eating it.

Unfortunately, most pizzas are too big to fit in standard ovens. Rearrange the 8 slices onto this baking sheet that can fit in the oven.

8 slices is pretty common in pizzas, but it could be cut into any number of equal slices. If we increase the number of slices to ${n1}, the shape begins to look more and more like a parallelogram triangle square.

We used all the pizza and the slices aren’t overlapping, so the area of the parallelogram is equal to different than the area of the circle. The formula for the area of a parallelogram is base x height when the height makes a right acute obtuse angle with the base. Click on the side of the parallelogram you want to use as the base. Select the height that corresponds with this base.

Move the slider back and forth to see the base and height of the parallelogram in the circle.

The area of the parallelogram is the same as the area of the circle, so we can find the area of a circle by doing 12xCxR!

People like to eat pizza in all sorts of unusual ways.

For example, some people cut their pizza into rings rather than slices.

This pizza becomes a little more difficult to arrange on the baking sheet but it is possible.

Move the slider to see the rings being arranged on the baking sheet. If we increase the number of rings up to ${n2}, the shape starts to look more and more like a triangle parallelogram circle.

We used all the pizza and the rings aren’t overlapping, so the area of the triangle is equal to different than the area of the circle. The formula for the area of a triangle is xbasexheight when the height makes a right angle with the base. Click on the side of the triangle you want to use as the base. Draw in the height that corresponds with this base.

Move the slider back and forth to see the base and height of the triangle in the circle.

So, whether we cut the pizza in the triangular slices or rings, we end up with a formula for the area of the circle as 12xCxR.

Can you think of another way to rearrange parts of a circle into other shapes we know how to find the area of? This could be a good time to step away from your device and try this on your own. Perhaps you’ll come up with a new way to rearrange a circle to show that the area formula is 12xCxR.

One way to find the area of a circle would be to multiply the length of the radius by the length of the circumference and then multiply it by . However, to determine the length of the circumference, we would have to use another formula. So, while 12xCxR is a correct approach to finding the area of a circle, mathematicians prefer to have formulas that are as simple as possible. Let’s work with the formula 12xCxR and see if we can simplify it.

The formula for the area of a circle is πR2. Let’s not lose sight of how exciting this is! Remember that earlier in the chapter, we were trying to determine how many R2 are needed to cover a circle:

Coming Soon!

We saw that 4R2 was too big too small and that 3R2 was a little too small too big. We have now proved the exact number of R2 we need - π “R2” are needed to fill in the circle. Recall from the Circumference chapter, that we often approximate π with 3.14.

π is a powerful number! not only does it tell us how many diameters radii are needed to go around the circumference, it also tells us how many R2 are needed to completely fill in a circle.

When using the formula πR2, remember to first multiply the radius by itself and then multiply that product by π. Let’s add this formula to our toolkit.

We are now ready to answer our original question. Which option has more pizza?

Coming Soon!

We can now finally conclude that the 2 medium pizzas 1 large pizza have more pizza!

Some people don’t eat the crust on the pizza. If we compare the 1 large pizza to the two medium pizzas without the crust, which option do you think will be the better purchase? 2 medium pizzas 1 large pizza.

The crust on these pizzas is 4 cm thick.

Coming Soon!

If we are just interested in finding the area of the pizza without the crust, the diameter of the large pizza-only circle is cm and the diameter of the medium pizza only-circle is cm.

Compare this with our earlier result:

If you eat the crust, the better purchase is 2 medium pizzas 1 large pizza. If you don’t eat the crust, the better purchase is 1 large pizza 2 medium pizzas.

Below are three different square pizzas with different sized pepperonis on each pizza.

Coming Soon!

Let’s find the amount of each pizza not covered by pepperoni. First, rearrange the pizzas into the order that you think goes from least to greatest in terms of amount of pizza NOT covered by pepperoni.

To find the area of each pizza not covered by pepperoni, we can find the area of the whole pizza and then subtract add divide the area of the pepperoni.

It turns out all the pizzas have the same different areas of pizza not covered by pepperoni!