If the number of square inches in the area of a circle is equal to the number of inches in its circumference,the diameter of the circle is

a. 4 inches
b. 2 inches
c. 1 inches
d. pi inches
e. 2 pi inches

please answer and explain

A = C

pi*r^2 = pi*D
pi*r^2 = pi*2r
Divide both sides by pi*r
r = 2 Inches.
D = 2r = 2 * 2 = 4 Inches.

Let's start by examining the formulas for the area and circumference of a circle.

The area of a circle is given by the formula A = πr^2, where A represents the area and r represents the radius.

The circumference of a circle is given by the formula C = 2πr, where C represents the circumference and r represents the radius.

In this case, we are given that the number of square inches in the area of the circle is equal to the number of inches in its circumference.

So, we can set up the equation A = C:

πr^2 = 2πr

We can simplify this equation by dividing both sides by π:

r^2 = 2r

Now, we have a quadratic equation. Let's rearrange it to be in standard form:

r^2 - 2r = 0

Factor out r:

r(r - 2) = 0

Now, we have two possibilities:

1) r = 0

2) r - 2 = 0

The first case, r = 0, doesn't make sense in this context, as it would mean the circle has no size.

So, we consider the second case, r - 2 = 0:

r = 2

Since the diameter of a circle is twice the radius, the diameter of the circle is 2 * 2 = 4 inches.

Therefore, the correct answer is option a. 4 inches.

To solve this problem, we need to set up an equation using the formulas for the area and circumference of a circle.

The formula for the area of a circle is:
Area = π * r^2

The formula for the circumference of a circle is:
Circumference = 2 * π * r

Let's assume the diameter of the circle is D inches, which means the radius is D/2 inches.

We are told that the number of square inches in the area of the circle is equal to the number of inches in its circumference. Therefore, we can set up the equation:

π * (D/2)^2 = 2 * π * (D/2)

To simplify the equation, we can cancel out the π factors on both sides of the equation:

(D/2)^2 = 2 * (D/2)

Squaring both sides of the equation gives us:

(D/2)*(D/2) = 2 * (D/2)

Simplifying further:

D^2 / 4 = D

Multiplying both sides of the equation by 4 gives us:

D^2 = 4 * D

Rearranging the equation:

D^2 - 4 * D = 0

Now, we can solve this quadratic equation by factoring:

D * (D - 4) = 0

This equation will be true if either D = 0 (which is not possible for the diameter of a circle) or (D - 4) = 0.

Therefore, the diameter of the circle is 4 inches.

So, the correct option is a. 4 inches.