What is the percentage increase in area when ordering a 14 inch rather than a 12 inch pizza?
The Correct Answer and Explanation is:
To calculate the percentage increase in area when ordering a 14-inch pizza instead of a 12-inch pizza, we use the formula for the area of a circle: Area=πr2\text{Area} = \pi r^2Area=πr2
For a 12-inch pizza, the radius is 6 inches: Area12=π×62=π×36≈113.1 square inches\text{Area}_{12} = \pi \times 6^2 = \pi \times 36 \approx 113.1 \text{ square inches}Area12=π×62=π×36≈113.1 square inches
For a 14-inch pizza, the radius is 7 inches: Area14=π×72=π×49≈153.9 square inches\text{Area}_{14} = \pi \times 7^2 = \pi \times 49 \approx 153.9 \text{ square inches}Area14=π×72=π×49≈153.9 square inches
Now calculate the percentage increase: Percentage Increase=(153.9−113.1113.1)×100≈36.1%\text{Percentage Increase} = \left( \frac{153.9 – 113.1}{113.1} \right) \times 100 \approx 36.1\%Percentage Increase=(113.1153.9−113.1)×100≈36.1%
Explanation (
When comparing the value of different pizza sizes, it is important to focus on area rather than diameter. A common mistake is to assume that a 14-inch pizza is only slightly bigger than a 12-inch pizza because the diameter increases by just 2 inches. However, the area of a circle increases with the square of the radius. This means even small increases in diameter result in much larger increases in area.
The formula for the area of a circle is pi times the radius squared. For a 12-inch pizza, the radius is 6 inches, giving an area of about 113.1 square inches. A 14-inch pizza has a radius of 7 inches and an area of about 153.9 square inches. The difference between these two areas is 40.8 square inches.
To understand what this means in practical terms, consider that a 14-inch pizza offers more than one-third extra area compared to a 12-inch pizza. Specifically, the increase is approximately 36.1 percent. This is significant because it shows that buying a larger pizza usually provides more food for less money per square inch.
Many pizza chains price pizzas based on size, but the larger pizzas often give better value when measured by area. If the cost difference between the 12-inch and 14-inch pizzas is less than 36 percent, then the 14-inch pizza is the better deal. This calculation can be a useful tool when trying to get the most food for your money.
