How Much of a 6-Inch Pizza Equals One Slice of a 12-Inch Pizza?

When comparing different sizes of pizza, it's important to understand how much of a smaller pizza equals the size of one slice from a larger pizza. This question can be answered through straightforward geometric calculations involving the area of circles. In this article, we'll explore the process of calculating the area of both a 12-inch and a 6-inch pizza, and then determine how much of the smaller pizza corresponds to one slice of the larger pizza.

Calculating the Area of the 12-Inch Pizza

The first step is to calculate the area of the 12-inch diameter pizza. The formula for the area ( A ) of a circle is given by:

[ A pi r^2 ]

Here, the radius ( r ) is half of the diameter. For a 12-inch pizza, the radius is:

[ r frac{12}{2} 6 text{ inches} ]

Substituting this into the formula gives us the total area of the 12-inch pizza:

[ A_{12} pi (6^2) 36pi text{ square inches} ]

Calculating the Area of One Slice of the 12-Inch Pizza

The 12-inch pizza is cut into 12 equal slices. Therefore, the area of one slice is one-twelfth of the total area:

[ A_{text{slice}} frac{A_{12}}{12} frac{36pi}{12} 3pi text{ square inches} ]

Calculating the Area of the 6-Inch Pizza

Next, we calculate the area of a 6-inch diameter pizza. The radius of this pizza is:

[ r frac{6}{2} 3 text{ inches} ]

Using the same area formula, the total area of the 6-inch pizza is:

[ A_6 pi (3^2) 9pi text{ square inches} ]

Determining the Fraction of the 6-Inch Pizza Corresponding to One Slice

To find out how much of the 6-inch pizza is equivalent to one slice of the 12-inch pizza, we calculate the fraction of the 6-inch pizza area that corresponds to the area of one slice:

[ text{Fraction} frac{A_{text{slice}}}{A_6} frac{3pi}{9pi} frac{3}{9} frac{1}{3} ]

This means that one slice of the 12-inch pizza is equivalent to one-third of the 6-inch pizza.

Conclusion

In summary, when dealing with different sized pizzas, using the area formula ( pi r^2 ) and the concept of fractions, we can determine that one slice of a 12-inch pizza is equivalent to (frac{1}{3}) of a 6-inch pizza. This understanding can be helpful when making decisions about which size of pizza to order or when sharing pizza with friends and family.

Additional Tips for Calculating Pizza Area

Here are a few additional tips for calculating the area of pizzas and understanding pizza size comparisons:

Radius vs. Diameter: Always remember that the radius is half the diameter, and use the appropriate formula for the area of a circle. Equal Slices: The area of each slice is the total area of the pizza divided by the number of slices. Proportional Thinking: Understanding how to set up and solve proportions can help with other similar size comparison questions.

By following these steps and understanding the geometric principles involved, you can easily determine how much of smaller pizza sizes correspond to the slices of larger pizzas.