11.3 Circles

Links and useful resources

Concept summary and lesson

Circle: All points that are a fixed distance (the radius) from a single point (the center)
$π$ (PI) - The ratio of the circumference of a circle to its radius (about 3.14159)
Area of a circle: $π r^{2}$

What happens if you double the area of a circle? The radius is multiplied by $\sqrt{2}$ .

\begin{aligned} area & = π r^{2} \\ \sqrt{\frac{area}{π}} & = r_{1} \\ 2 \cdot area & = π r_{2}^{2} \\ \sqrt{2 \frac{area}{π}} & = r_{2} \\ \sqrt{2} \sqrt{\frac{area}{π}} & = r_{2} \\ \sqrt{2} r_{1} & = r_{2} \end{aligned}

What happens when you double the radius? The area is multiplied by 4.

How about doubling the circumference? The area is multiplied by 4! Doubling the circumference is the same thing as doubling the radius!
$c = 2 π r$ , so $2 c = 2 π \cdot 2 r$

Media resources

Guided practice

Find radius, arc length, and area given the others.
What is the radius of a circle that has circumference $54 π$ centimeters?
If 1 gallon of paint is enough to paint the interior of a circle with diameter 10 meters, then what is the diameter of the largest circle whose interior I can paint with 4 gallons of paint?
My rectangular house is 30 feet by 40 feet.
- My goat Cassidy is outside on a 20-foot long leash that is connected to a corner of my house. What is the area of the region in which Cassidy can roam?
- If I lengthen Cassidy’s leash to 50 feet, then what is the area of the region in which Cassidy can roam?
Two quarter-circles are drawn with their centers at opposite vertices of a square, as shown at the right. If the side length of the square is 6, then what is the area of the shaded region between the quarter-circles?

Homework

#hw (math) (gr7) 11.3 Circles (p. 432) 2025-03-12