1. Circles
Lesson 3.1.3

2. Circles 3.1.3 California Standard: What it means for you: Key words:
Lesson
3.1.3
Circles
California Standard:
Measurement and Geometry 2.1
Use formulas routinely for finding the perimeter and area of basic two-dimensional figures, and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
What it means for you:
You’ll find the circumference and area of circles using formulas.
Key words:
pi (p)
radius
diameter
irrational number
circumference

3. Lesson
3.1.3
Circles
You’ve already met the special irrational number p or “pi”. Now you’re going to use it to find the circumference and area of circles.
Area
Circumference

4. Circles 3.1.3 Circles Have a Radius and a Diameter
Lesson
3.1.3
Circles
Circles Have a Radius and a Diameter
The distance of any point on a circle from the center is called the radius.
diameter
The distance from one side of the circle to the other, through the center point, is called the diameter.
radius
Notice the diameter is always twice the radius.
diameter = 2 • radius
d = 2r

5. Circles 3.1.3 If a circle has a diameter of 4 in., what is its radius?
Lesson
3.1.3
Circles
Example 1
If a circle has a diameter of 4 in., what is its radius?
Solution
Use the formula: d = 2r. r
4 in
Rearrange to give r in terms of d,
so r = d ÷ 2
Substitute d from question:
r = 4 ÷ 2 = 2 in.
Solution follows…

6. Circles 3.1.3 Guided Practice
Lesson
3.1.3
Circles
Guided Practice
1. If a circle has a radius of 2 in., what is its diameter?
2. A circle has a diameter of 12 m. What is its radius?
d = 2r = 2 • 2 = 4 in
r = d ÷ 2 = 12 ÷ 2 = 6 m
Solution follows…

7. Circles 3.1.3 Circumference is the Perimeter of a Circle
Lesson
3.1.3
Circles
Circumference is the Perimeter of a Circle
The circumference is the distance around the edge of a circle. This is similar to the perimeter of a polygon. d r
There’s a formula to find the circumference.
Circumference = p • diameter
C = pd
Because the diameter = 2 × radius, Circumference = 2 × p × radius
C = 2pr

8. Lesson
3.1.3
Circles
Example 2
12 cm
Find the circumference of the circle shown. Use the approximation, p » 3.14.
Solution
Use the formula that has diameter in it: C = pd.
Substitute in the values and evaluate with your calculator.
C = pd » 3.14 × 12 cm = cm » 37.7 cm.
Solution follows…

9. Circles 3.1.3 Guided Practice
Lesson
3.1.3
Circles
Guided Practice
Find the circumference of the circles in Exercises 3–6.
2 m
13 in
5 ft
10 cm
2 • p × 2 ≈ 6.28 m
2 • p × 10 ≈ 62.8 cm
2 • p × 5 ≈ 31.4 ft
2 • p × (13 ÷ 2) ≈ 40.8 in
Solution follows…

10. Circles 3.1.3 Guided Practice
Lesson
3.1.3
Circles
Guided Practice
7. Find the radius of a circle that has a circumference of 56 ft.
8. Find the diameter of a circle that has a circumference of 7 m.
C = 2pr r = C ÷ (2p) = 56 ÷ (2 • 3.14) = 8.92 ft
C = pd d = C ÷ p ≈ 7 ÷ 3.14 ≈ 2.23 m
Solution follows…

11. Circles 3.1.3 The Area of a Circle Involves p Too
Lesson
3.1.3
Circles
The Area of a Circle Involves p Too
The area of a circle is the amount of surface it covers. The area of a circle is related to p — just like the circumference.
Area
There’s a formula for it:
A = pr2
Area = p • (radius)2

12. Circles 3.1.3 Find the area of the circle shown, using p » 3.14. 12 ft
Lesson
3.1.3
Circles
Example 3
Find the area of the circle shown, using p » 3.14.
12 ft
Solution
Use the formula: A = pr2
Substitute in the values and evaluate the area.
A » 3.14 × (12 ft)2 = 3.14 × 144 ft2 = ft2 » 452 ft2
Solution follows…

13. Lesson
3.1.3
Circles
Example 4
If you know the area of a circle you can calculate its radius:
The area of a circle is 200 cm2. What is the radius of this circle? Use p » 3.14.
Solution
The question gives the area, and you need to find the radius.
This means rearranging the formula for the area of a circle to get r by itself.
Solution continues…
Solution follows…

14. Lesson
3.1.3
Circles
Example 4
If you know the area of a circle you can calculate its radius:
The area of a circle is 200 cm2. What is the radius of this circle? Use p » 3.14.
Solution (continued)
A = pr2
A ÷ p = r2
Divide both sides by p
r = (A ÷ p)
Take the square root of both sides
Substitute in the values and find the radius
r = (200 ÷ 3.14) » » 8 cm

15. Circles 3.1.3 Guided Practice
Lesson
3.1.3
Circles
Guided Practice
9. Find the area of a circle that has a diameter of 12 in.
10. Find the area of a circle that has a radius of 5 m.
11. If a circle has an area of 45 in2, what is its radius?
A = pr2 = p(d ÷ 2)2
≈ 3.14(12 ÷ 2)2 ≈ 113 in2
A = pr2 ≈ 3.14 • 52 = 78.5 m2
A = pr2 r2 = A ÷ p ≈ 45 ÷ 3.14 ≈ r ≈ 3.78 in
Solution follows…

16. Circles 3.1.3 Independent Practice
Lesson
3.1.3
Circles
Independent Practice
In Exercises 1–3, find the area of the circles shown. 3.
8 m
area ≈ 201 m2
2 in
area ≈ 3.14 in2
C = 45 cm
area ≈ 161 cm2
Solution follows…

17. Circles 3.1.3 Independent Practice
Lesson
3.1.3
Circles
Independent Practice
4. Find the circumference and area of a circle with a diameter of 6 m.
5. Lakesha has measured the diameter of her new whirlpool bath as 20 ft. Find its surface area.
circumference » m, area » 28.3 m2
area ≈ 314 ft2
20 ft
Solution follows…

18. Circles 3.1.3 Independent Practice
Lesson
3.1.3
Circles
Independent Practice
6. Find the circumference of the base of a glass with a 1.5 inch radius.
7. Find the area of the base of the glass in Exercise 6.
8. A circle has an area of 36 cm2. Find its radius and circumference.
circumference ≈ 9.42 in
area ≈ 7.07 in2
r ≈ 3.39 cm and C ≈ 21.3 cm
Solution follows…

19. Lesson
3.1.3
Circles
Round Up
This Lesson is all about circles, and how to find their circumferences and areas.
There are a few formulas that you need to master — make sure you practice rearranging them.