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. 3 inches
C. 1 inch
D. NONE
so you want to set the area equal to the circumference ..
πr^2 = 2πr
r^2 - 2r = 0
r(r-2) = 0
r = 0 , the trivial case
or
r = 2
so the diameter is 4 inches
A = pi * r^2
C = pi * d
Which value for the radius fits both formulas?
Let's solve this step-by-step:
1. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
2. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
3. 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.
4. Therefore, we can set up the equation A = C: πr^2 = 2πr.
5. Simplifying this equation, we can divide both sides by π: r^2 = 2r.
6. Moving all terms to one side, we have the quadratic equation r^2 - 2r = 0.
7. Factoring out r, we have r(r - 2) = 0.
8. So, either r = 0 or r - 2 = 0. The radius cannot be 0, so we have r = 2.
9. The diameter of a circle is twice the radius, so the diameter of the circle is 2r = 2(2) = 4 inches.
Therefore, the diameter of the circle is 4 inches.
The correct answer is A. 4 inches.
To find the diameter of the circle, we can use the formulas for the area and circumference of a circle.
The formula for the area of a circle is given by:
Area = πr^2
And the formula for the circumference of a circle is given by:
Circumference = 2πr
In this case, it is 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 have:
πr^2 = 2πr
We can simplify this equation by canceling out the common factor of π:
r^2 = 2r
To solve for the diameter, we know that the diameter is twice the radius. So let's substitute 2r for the diameter:
(2r)^2 = 2(2r)
4r^2 = 4r
Now, we can solve for r by subtracting 4r from both sides of the equation:
4r^2 - 4r = 0
We can factor out the common factor of 4r:
4r(r - 1) = 0
Now, we have two possible solutions:
1) 4r = 0, which means r = 0 (not possible since a circle cannot have a radius of 0)
2) r - 1 = 0, which means r = 1
Since r = 1, the diameter of the circle is twice the radius, which is equal to 2(1) = 2 inches.
Therefore, the correct answer is not listed in the options provided. The correct answer should be:
D. NONE