Open Middle®
Directions: Which circle is bigger: one with an area of 30 square units or one with a circumference of 30 units? How do you know?
Hint
What information to we need to compare these two circles?
Answer
The circle with a circumference of 30 feet is bigger. A procedural justification involves separately setting the formulas for area and circumference both equal to 30 and solving for the radius. That gives the circle with an area of 30 square units a radius of ~3.09 units. The circle with a circumference of 30 units has a radius of ~4.77 units. One possible conceptual justification would be that as a circle’s radius (and therefore size) increases, its area increase at a faster rate than its circumference. Therefore, a growing circle would reach an area of 30 square units before reaching a circumference of 30 units, so the circle with a circumference of 30 units would be bigger.
Source: Robert Kaplinsky
The circle that is bigger is the one with a 30-unit circumference because it has a bigger radius than the one with an area of 30 units.