Two circles have circumferences of pi and 3 pi. What is the ratio of the area of the circles? The diameters? The radii?
the ratios of the diameters and radii are the same as the ratio of the circumferences
the ratio of the areas is the square of the other ratios
To find the ratio of the area of the circles, we can use the formula A = πr², where A is the area and r is the radius.
Let's denote the radius of the first circle as r1 and the radius of the second circle as r2.
Since the circumference of the first circle is π, we can use the formula C = 2πr to find r1. Substituting in the given circumference, we get:
π = 2πr1
Dividing both sides by 2π, we get:
r1 = 1/2
Similarly, for the second circle with a circumference of 3π:
3π = 2πr2
Dividing both sides by 2π, we get:
r2 = 3/2
Now, let's calculate the areas of the circles:
Area of the first circle (A1) = πr1² = π(1/2)² = π/4
Area of the second circle (A2) = πr2² = π(3/2)² = 9π/4
The ratio of the areas is:
A1:A2 = (π/4): (9π/4) = 1:9
Therefore, the ratio of the areas of the circles is 1:9.
Next, let's calculate the diameters of the circles. The diameter (d) is twice the radius (r):
Diameter of the first circle (d1) = 2r1 = 2(1/2) = 1
Diameter of the second circle (d2) = 2r2 = 2(3/2) = 3
The ratio of the diameters is:
d1:d2 = 1:3
Finally, let's calculate the ratio of the radii:
Radius of the first circle (r1) = 1/2
Radius of the second circle (r2) = 3/2
The ratio of the radii is:
r1:r2 = (1/2):(3/2) = 1:3
So, the ratio of the areas is 1:9, the ratio of the diameters is 1:3, and the ratio of the radii is 1:3.
To find the ratio of the area of the circles, we need to compare the areas of the circles.
The formula for the area of a circle is A = πr², where A is the area and r is the radius.
Since we are given the circumferences of the circles, we can find the radii using the formulas:
C = 2πr, where C is the circumference and r is the radius.
For the first circle, with a circumference of π, its radius can be found by rearranging the formula as:
π = 2πr
r = 1/2
For the second circle, with a circumference of 3π, its radius can be found similarly:
3π = 2πr
r = 3/2
Now that we have the radii, we can calculate the areas of the circles:
For the first circle:
A₁ = π(1/2)²
A₁ = π/4
For the second circle:
A₂ = π(3/2)²
A₂ = (9π)/4
To find the ratio of the area, we can divide the area of the second circle by the area of the first circle:
Ratio of area = A₂ / A₁ = ((9π)/4) / (π/4) = 9/1 = 9
Therefore, the ratio of the areas of the circles is 9:1.
To find the ratio of the diameters, we can double the radius:
For the first circle:
D₁ = 2r₁
D₁ = 2(1/2)
D₁ = 1
For the second circle:
D₂ = 2r₂
D₂ = 2(3/2)
D₂ = 3
The ratio of the diameters is 3:1.
Finally, to find the ratio of the radii, we can compare the radii directly:
Ratio of radii = r₂ / r₁ = (3/2) / (1/2) = 3/1 = 3
Therefore, the ratio of the radii of the circles is 3:1.