Suppose a circle has a radius of 4.5 inches. If you double the radius of the circle, does the area of the circle double as well? What happens to the circle’s circumference? Explain and justify your reasoning.
C = 2pi * r
A = pi * r^2
double rand you get
C' = 2pi * (2r)
A' = pi * (2r)^2
Note that C' = 2C but A' = 2^2 A
if you multiply the size of dimensions by n,
perimeter is multiplied by n
area is multiplied by n^2
volume is multiplied by n^3
To determine whether doubling the radius of a circle also doubles its area, let's start by understanding the formulas for calculating the area and circumference of a circle.
1. Area of a circle: The formula to find the area of a circle is A = π * r^2, where A represents the area and r represents the radius.
2. Circumference of a circle: The formula to find the circumference of a circle is C = 2π * r, where C represents the circumference and r represents the radius.
Now, let’s analyze the given scenario:
1. Doubling the radius: If we double the radius from 4.5 inches to 9 inches, the new radius (r') is 2 * 4.5 = 9 inches.
Let's calculate the new area and circumference:
1. New Area (A'): A' = π * r'^2 = π * 9^2 = 81π square inches.
2. New Circumference (C'): C' = 2π * r' = 2π * 9 = 18π inches.
Now, let's compare the original area and circumference with the new ones:
1. Original Area (A): A = π * r^2 = π * (4.5)^2 = 20.25π square inches.
2. Original Circumference (C): C = 2π * r = 2π * 4.5 = 9π inches.
Comparing the new and original area:
A' / A = (81π) / (20.25π) = 4.
Comparing the new and original circumference:
C' / C = (18π) / (9π) = 2.
From the calculations, we can conclude the following:
1. The new area (A') is four times greater than the original area (A). This means doubling the radius of a circle results in the area becoming four times larger. Thus, the statement "doubling the radius of a circle doubles its area" is false.
2. The new circumference (C') is exactly twice the original circumference (C). Therefore, doubling the radius of a circle also doubles its circumference. Hence, the statement "doubling the radius of a circle doubles its circumference" is true.
In summary, doubling the radius of a circle does not double the area; instead, it makes it four times larger. However, doubling the radius does result in doubling the circumference of the circle.