Question

The area of a circle is directly proportional to the square of its radius. If the radius is halved, by what factor is the area multiplied? a) 1/8 b) 2 c) 1/2 d) 4 e) 1/4

Can you deeply and clearly explain to me, please? Thanks.

Answer 1

If the area of a circle is directly proportional to the square of the radius, then:
\[A=k\times r^2\]
i.e. A = some constant times \(r^2\).

Imagine then, that instead of r, you have \(\frac{r}{2}\). How does that equality change?

Answer 2

Give that the area of the circle, A is directly proportional to the square of its radius, R.

\[A \alpha R ^{2}\]
\[A= k R ^{2}\] , where k is some constant.

WHen your radius is halved, R/2.

SUbstitute R/2 into the original R,

\[ k \left( \frac{R}{2} \right) ^{2}\]
\[= k \left( \frac{R^{2}}{4} \right) \]
\[= \frac{1}{4}k R ^{2}\]

Since A = k R^2,
Substituting into the above equation, we will get 1/4 A.

Area of the circle decreases by a factor of 1/4