The ratio of the area of the small circle to the area of the big circle is 1 to 64. What is the ratio of the radius of the small circle to the big circle's?

Question

Advanced Mathematics By Saxon

Answer 1

Since we know the areas, we can find the radius of both circles and simply divide the radii.

 

Small circle:

 

1 = πr²

 

1 / π = r²

 

1 / √π = r

 

 

Large circle:

 

64 = πr²

 

64 / π = r²

 

8 / √π = r

 

 

Now divide the small radius by the larger radius

 

(1 / √π) / (8 / √π)

 

 

When dividing by fractions, we multiply by the reciprocal of the second fraction.

 

(1 / √π) * (√(π) / 8) =

 

1 / 8

Answer 2

Recall that area of a circle is A=πr^2.

 

So, if the small circle has radius r₁, then its area is A₁=πr₁^2.

 

If the large circle has radius r₂, then its area is A₂=πr₂^2.

 

Since A₁/A₂=1/64, then πr₁^2/πr₂^2=1/64.

 

It follows that r₁/r₂=√(1/64)=1/8.

Answer 3

(Area of smaller circle)/(Area of larger circle) = (scale factor)²

 

1/64 = (scale factor)²

 

√(1/64) = scale factor

 

1/8 = scale factor = ratio of the radii of the two circles

Answer 4

The area of a circle is  ¶ R^2. Let's say we have circle A and circle B

 

Area A= ¶ R_A².

 

Area B= ¶ R_B².

 

Now, let's say that A y bigger than B. That means Area A = 64 times Area B

 

¶ R_A² = 64 x ¶ R_B²

 

R_A² = 64 x R_B²

 

√(R_A² ) = √(64 x ¶ R_B²)

 

R_A = 8 R_B

 

1:8 is the answer

 

 

I hope this can help you

 

 

Fernando