# Use 2 parallel cuts to divide a pizza into 3 segments with equal area

1. Oct 4, 2011

### JamesGold

Problem: Where would you have to make two parallel cuts on a pizza so that all 3 segments have the same amount of pizza?

Given: the pizza's diameter is 14 in.

Attempt: With the leftmost point of the pizza as the origin of my coordinate system, the equation for the top half of the pizza is y = √7^2 - (x - 7)^2. If I set the integral of that equation from 0 to A equal to one sixth the area of the pizza then A should be how far from the edge of the pizza you'd have to make the first cut, right? I'm just having trouble evaluating this integral. I used a trig substitution x - 7 = 7sinσ and simplified until I had:

(49/2)∫1 + cos(2σ)dσ from (3pi/2) to inversesin(A/7 - 1) which simplifies to

(49/2) * [σ + 0.5sin(2σ)] with the same limits of integration. Plugging the limits into the antiderivative made things real ugly and I can't handle the algebra. Did I make a mistake up to this point? If not, could someone please guide me through the algebra?

2. Oct 4, 2011

### phinds

You seem to be going about it the right way, but I'm puzzled as to why you would want to create the ugliness of the equations that you get when you put the origin somewhere other than the center of the pizza?

3. Oct 4, 2011

### SammyS

Staff Emeritus
I would use non-Euclidean geometry. (LOL)

4. Oct 4, 2011

### JamesGold

Hm, I hadn't thought about that. I guess you could use the center of the pizza as your origin, set the integral from 0 to A equal to one sixth the area, and A would give you half the width of the middle segment.

Give me some time to rework!

5. Oct 4, 2011

### JamesGold

Okay, with the origin at the center I have

(49/2)[inversesin(A/7) + 0.5sin{2inversesin(A/7)}] = 49pi/6

Now what?

Last edited: Oct 4, 2011
6. Oct 5, 2011

### Ray Vickson

Now you solve the equation numerically.

RGV

7. Oct 15, 2011

### JamesGold

The correct equation is actually

(49/2)[inversesin(A/7) + 0.5sin{2inversesin(A/7)}] = 49pi/12

and just plugging in numbers I got A ≈ 1.854, so you should make two parallel cuts 1.854 inches from either side of the pizza's center.

Thanks, gentlemen.