The numerical difference between the area and the circumference of a circle is 8 pi π. Find the radius of the circle.​

Answer 1

Let R equal the radius

The area of the circle is given by:

area = π R²

_{and the circumference is given by:}

circumference = 2 π R

The difference is given by:

area — circumference = 8 π

Therefore:

π R² — 2 π R = 8 π

Divide by sides of the equation by π yields:

R² — 2 R = 8 or R² — 2 R — 8 = 0

Factoring this quadratic equation yields:

(R — 4) (R + 2) = 0

There are two solutions:

R = 4 and R = -2

The radius must be positive, therefore the only solution is:

R = 4

As a check:

area = π 4² = 16 π

circumference = 2 π 4 = 8 π



difference = 16 π — 8 π = 8 π

Answer 2

Circumference (C) of a circle = 2*pi*r.

Area (A) of the circle = pi*r².

A - C = 8pi

pi*r² - 2*pi*r = 8pi

Subtract 8pi from both sides and factor out pi

pi*(r² - 2r - 8) = 0

Divide both sides by pi to get

r² - 2r - 8 = 0

This is a quadratic equation which can be factored as.

(r-4)*(r+2) = 0

The solutions are r are +4 and -2

If either value is used in A - C the answer is 8pi.