Slicing pizza over and over til you get the right amount

Geometry Level 3

It's easy that if you want to slice a circular pizza into 2 halves (of the same exact size).

However, suppose you want to slice a segment of a circular pizza such that the segment is only 1/3 of the area of the pizza. Roughly how big is the crust, with respect to the circumference to the pizza?

Choose the closest answer.

40% 35% 25% 30%

Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Parth Sankhe
Nov 20, 2018

Area of a segment of a circle = r 2 ( θ − sin ⁡ θ ) 2 \frac {r^2(\theta - \sin \theta )}{2} , where θ \theta is the angle subtended by it at the centre.

Therefore, r 2 ( θ − sin ⁡ θ ) 2 = π r 2 3 \frac {r^2(\theta - \sin \theta )}{2}=\frac {πr^2}{3}

θ − sin ⁡ θ = 2 π 3 \theta - \sin \theta = \frac {2π}{3}

The solution of the above equation is at θ ≈ 2.605 \theta ≈ 2.605

Hence, percentage of crust in the piece = θ 2 π × 100 ≈ 40 \frac {\theta }{2π}×100 ≈ 40

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David Vreken
Nov 20, 2018

Let r r be the radius of the pizza and θ \theta be the the central angle that subtends the segment.

Then the area of 1 3 \frac{1}{3} of the pizza is A = 1 3 π r 2 A = \frac{1}{3} \pi r^2 and the area of the circular segment is A = 1 2 r 2 ( θ − sin ⁡ θ ) A = \frac{1}{2}r^2(\theta - \sin \theta) .

Setting these equal to each other and solving numerically gives θ ≈ 2.605 \theta \approx 2.605 (radians), which is approximately 2.605 2 π ≈ 40 % \frac{2.605}{2 \pi} \approx \boxed{40\%} of the circle (the same ratio as the length of the crust to the circumference of the pizza).

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