From: Lori

I have a pizza. The radius is 10 inches long.The pizza was cut into 16 equal slices. When one slice was left, my sister and I both wanted it, so we agreed to cut it in half, but I like the crust more than she does, so we decided to cut it the "other way." In other words, the two pieces would not be symmetrical. The inside piece would contain all topping, and the outer piece would contain some topping and some crust.

Finally, the question: How far up the radius from the center of the circle will I need to cut so we will both have an equal area of pizza? Note: The area of the crust and topping make no difference. Just think of it as a regular circle.

Hi Lori.

Here's a sketch of the pizza slice:

I've drawn a couple of orange lines and labelled some points as well.

We want to find the distance CD that will make the area of ABC the same as the area of the odd shape above it. This sound hard, but actually there is a fairly straightforward way to solve the problem. First we find the area of the whole pizza slice.

The area of a circle is πr², but we are only interested in this 16^th of it. So if the radius is 10 inches, the total amount of pizza to be divided in half is ¹/₁₆π10² = 19.63 inches². Each person gets half of this, so that means your sister's triangular piece ABC is 9.82 inches².

Notice that ABC is composed of the two congruent right triangles ADC and BDC.
So the area of ADC = ^9.82/₂ = 4.91 inches².

The area of a triangle is half the base length times the height. In the right triangle ADC, the base is AD and the height is DC (which is the length that the question is asking for). That means 4.91 = (AD)(DC) / 2. We don't know either of these, but we can work out ACD and use that.

ACD is half the angle of a slice, which is one 16^th of a circle. A circle is 360°, so ACD = (360° / 16) / 2 = 11.25°. Now we can use this to figure out the ratio of the sides AD to DC.

The Tangent function gives the ratio of the opposite side over the adjacent side of a right triangle. That means that in our case, tan(11.25°) = AD/DC. That means AD = (DC)tan(11.25°). Now we can use this expression in the calculation of the area of ADC.

Let's set x = DC (just for simplicity):

From here, all that's left is to solve for x.

Hope this helps!
Stephen La Rocque.