Lesson 12-5: Solve Problems Involving Area of Circles

Property

From the relationship $A = \frac{1}{2}Cr$ and $A = \pi r^2$, we can derive the formula for the circumference of a circle in terms of its radius $r$:

Examples

• A bicycle wheel has a radius of $12$ inches. The distance it travels in one full rotation is its circumference: $C = 2\pi(12) = 24\pi$ inches.
• The circumference of a circular pool is $40\pi$ feet. To find its diameter, we can use the formula $C = \pi d$. So, $40\pi = \pi d$, which means the diameter is $40$ feet.
• A hula hoop has a radius of $50$ cm. Its circumference is calculated as $C = 2\pi(50) = 100\pi$ cm.

Explanation

The circumference is the distance around a circle. It's always a little more than three times the distance across the circle (the diameter). This 'a little more than three' is exactly $2\pi$ times the radius, making $\pi$ the key conversion factor.

Property

The area enclosed by a circle is given by

where $r$ is the radius of the circle. An exponent tells us how many times the base occurs as a factor in a product. So $r^2$ means $r \times r$. We compute powers before we compute the rest of a product.

Examples

• A pizza has a radius of 7 inches. Its area is $A = \pi \times 7^2 = 49\pi \approx 153.94$ square inches.
• A circular garden has a diameter of 20 feet. Its radius is 10 feet, so its area is $A = \pi \times 10^2 = 100\pi \approx 314.16$ square feet.
• A circular rug has an area of 50 square feet. Its radius is found by solving $r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{50}{\pi}} \approx \sqrt{15.91} \approx 3.99$ feet.

Explanation

The area of a circle tells you the amount of space inside it. To find it, you first square the radius (multiply it by itself), and then multiply that result by pi ($\pi$). This gives the total surface coverage.