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Geometry

Unit 9

Equations of Circles,

Circle Formulas, and

Volume

1

Warm-up

1. Use the Pythagorean Theorem to find the length of a right triangle’s hypotenuse if the two legs are length 8 and 14. Leave your answer in simplest radical form.

2. Solving the quadratic by completing the square:

a. 2 6 32 5x x

b. 2 10 29x x

2

Learning Task: Deriving the Equation of a Circle

Part 1 – The Return of Pythagoras Below is a circle of radius 3, drawn on a coordinate plane centered at the origin. Four points are marked on the circle. Each member of your group should choose one of the four points and draw a right triangle that connects the chosen point to the origin. Use the given coordinates to label the legs of your right triangle. Then apply the Pythagorean Theorem to determine the hypotenuse of the triangle.

Calculation for your right triangle

222 hypotenuselegleg

1. Share your answers with your group. What do you notice?

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

x

y

5,2 5,2

8,1 5.4,5.4

3

2. Now that you have seen how the Pythagorean Theorem relates to the radius of a circle, you will develop that relationship in a more general sense. An arbitrary point has been placed on the circle of radius 3. A right triangle has been drawn in for you as well. Label the triangle’s legs and hypotenuse, and then write the Pythagorean Theorem that models your triangle.

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

x

y

(x, y)

3. Finally, try to move to a more general circle. This time, not only is the point arbitrary, but the radius is as well. Call the radius r. Similar to problem #2, label the right triangle’s legs and hypotenuse and then write the Pythagorean Theorem that models your triangle.

4

4

2

2

4

6

8

10

15 10 5 5 10 15

8

6

4

2

2

4

6

8

15 10 5 5 10 15

8

6

4

2

2

4

6

8

15 10 5 5 10 15

Part 2 – Circles, Transformed In Part 1, you developed the general equation for a circle centered at the origin. Can we still use this equation if the circle changes location? Using our knowledge of transformations, we can adjust this equation to move a circle to any location in the plane. 1. Look at the circles below; their centers have translated horizontally (from dotted circle into solid circle). Use what you know about translations to rewrite these circles equations.

Dotted Circle: 922 yx Dotted Circle: 2522 yx

Transformed Circle: __________________ Transformed Circle: ____________________

2. Now try to do the same with a circle that have been translated vertically.

Dotted Circle: 1622 yx

Transformed Circle: ___________________

Standard Form of a Circle

where (h,k) is ______________ and r is the ______________

5

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8-7-6-5-4-3-2-1

1234567

x

y

Part 3 – Using a Circle’s Equation Using what you have developed in Parts 1 and 2, you should now be able to write and interpret the equations of circles on a coordinate plane. 1. Write the equation of each circle below.

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8

x

y

Equation: _________________

Equation: ________________

Equation: __________________

Equation: _________________

-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3

-4-3-2-1

123456789

x

y

-8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

x

y

6

2. Given the equation of the circle, identify the radius and center for each circle. Leave answers in simplest radical form.

100)8( 22 yx

Radius:

Center: ),(

1)4()1( 22 yx

Radius:

Center: ),(

50)3()10( 22 yx

Radius:

Center: ),(

45)2()2( 22 yx

Radius:

Center: ),(

3. Use the equation to graph each circle.

16)3( 22 yx

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8

x

y

1)5()1( 22 yx

36)4( 22 yx

-10 -8 -6 -4 -2 2 4 6 8 10

-8

-6

-4

-2

2

4

6

8

x

y

25)5()5( 22 yx

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-7

-6

-5

-4

-3

-2

-1

1

2

3

x

y

-9 -8-7-6-5-4-3 -2-1 1 2 3 4 5 6 7 8 9 1011

-6-5-4-3-2-1

123456789

101112

x

y

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Part 4 – Converting General Form to Standard Form Now complete the square to rewrite the following equations in standard form. An example is shown below: Example:

2 2

2 2

2 2

2 2 2 2

2 2

8 6 56 0

8 6 56

( 8 ) ( 6y) 56

8 6 8 68 6 56

2 2 2

56 56

2

,

x y x y

Move the

x y x y

Group like terms together x x y

Take half of

co

the middle x x y y

coefficient s

ns

q

tant

uare

2 2 2 2

2 2

2 2

as a binomial squared, 4 3 56 4 3

and simplify the squared terms

4 3 56 16 9

4 3 81

it and

add it to both sides

Factor x y

x y

Simplify x y

Guided Practice: Change the following equations to standard form. Then, identify the radius and center.

1. 2 2 2 10 22x y x y 2. 2 2 14 24 0x y y

Standard Form: ____________________ Standard Form: ____________________ Radius: _____ Center: _________ Radius: _____ Center: _________

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On your own: Rewrite in Standard form for a circle: 2 2 2( ) ( ) .x h y k r Then, state the radius and

center.

1. 84822 xyx

2. 2 2 18 65 0x y y

3. 2 2 20 26 268 0x y x y

4. 2 2 14 22 134 0x y x y

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Skills Practice: Converting General Form to Standard Form.

Directions: Write each circle in Standard Form by completing the square. Then state the center and radius.

1.

2.

3.

4.

10

5.

6.

7.

8.

Answers to Converting General Form to Standard Form Skills Practice:

1. 2.

3. 4.

5. 6.

7. 8.

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Learning Task: Algebraic Proof

1. Proof #1. Prove or disprove that the point (1, 3) lies on the circle centered at the origin and

passing through the point (0, 2).

a. What do we need to show in order to prove or disprove this statement?

b. Write an equation for the circle described in the problem.

c. Substitute the point in for the equation and comment on the results. Did you prove the statement or disprove it?

Guided Practice:

2. a. Write the equation of a circle centered at (5,-2)

b. The equation of the circle passes through the point (6,5). Substitute the values into x and y to find the radius.

c. Prove or disprove that the point A(10, 3) lies on a circle centered at C(5, -2) and passing

through the point B(6, 5).

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r = 10cm

50

A

C

B

Learning Task: Arc Length and Circumference Define circumference: Define Arc Length:

FORMULAS OF A CIRCLE: “Perimeter” (Circumference) of a Circle: 2 r or d

Arc Length: 2

360

r , where is the central angle (or intercepted arc measure)

Use the formulas to answer the questions below. Be sure to leave all answers in terms of pi. EXAMPLE 1: Find the circumference of the circle.

Example 2: Use the diagram of the circle to find the arc length of BC. Example 3: Use the diagram of the circle to find the arc length of BC with a radius of 4 inches.

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A

B

C

13

12

120C

B

A

Example 4: If a central angle measures 80° and the diameter of the circle measures 24 feet, find the arc length. Sketch a picture to help you solve the problem. Example 5: Use the formula that you have developed for arc length and find the circumference of the circle.

Skills Practice: Calculating Arc Length and Circumference

Use the diagram to find the indicated measure. Leave answers in term of pi. 1. Find the circumference. 2. Find the circumference.

3. Find the radius. Find the indicated measure. a. The exact radius of a circle with circumference 36 meters

b. The exact diameter of a circle with circumference 29 feet

c. The exact circumference of a circle with diameter 26 inches

d. The exact circumference of a circle with radius 15 centimeters

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4.Find the length of . a. b. c.

In D shown below, ADC BDC. Find the indicated measure

5. mCB

6. mACB

7. Length ofCB

8. Length of ABC

9. mBAC

10. Length of ACB

11. Find the indicated measure. a. The radius of circle Q b. Circumference of Q and Radius of Q

AB

15

36

D

E

C

80

120

D

E

Find the perimeter of the region. Round to the nearest hundredth. 12.

13. Birthday Cake A birthday cake is sliced into 8 equal pieces. The arc length of one piece of cake

is 6.28 inches as shown. Find the diameter of the cake.

14. Radius = 5 in

Length of Arc CE = _______

15. Find the radius of the circle. r = ___________

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For #16-18, solve for the requested variable. C is the center of each circle. 16. r = _______ 17. x° = ________ 18. d = ______

19. Circumference = 10 m; Find the arc length of JT = ______

20. The arc length of OP = inches; 21. The arc length ofQT = cm.;

r = _______ d = _______ (to the tenth)

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Area and Area of a Sector

Formulas:

Area of a Circle: 2r

Area of a Sector: 2

360

r , where is the ________________

Example 1: Find the exact area of the circle. Example 2:

a. The area of a circle is 58 square inches. Find the radius.

b. The area of a circle is 37 square meters. Find the radius.

c. The area of a circle is 106 square centimeters. Find the diameter.

d. The area of a circle is 249 square feet. Find the diameter. Example 3: Given the diagram to the right to find the area of the shaded sector. 150m CBA

150 o

6 in B

C

A

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Example 4: Find the area of the shaded region.

a. r = 8 cm b. r = 16 cm Example 5: Find the area of the shaded region. a. b.

Shaded area = _____________ Shaded area = ____________ c. d. radius = 10 cm. Shaded area = _____________ Shaded area =____________

10cm

240 o

A

B

C A

B

D

C

60

12 in L

M

N

4ft

300

80 o 10 cm

A

B

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Skills Practice: Area and Area of a Sector 1. Find the exact area of the circle.

a. b.

2. Find the area of each sector below. Leave all answers in terms of pi.

Find the areas of the sectors formed by angle ACB. 3. 5.

4.

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Use the diagram to find the indicated measure. 6. Find the area of S 7. Find the area S 8. Find the area S

9. The area of Z is 124.44 square centimeters. The area of sector XZY is 28 square centimeters. Find the indicated measure.

a. Radius of Z

b. Circumference of Z

c. m

d. Length of

e. Perimeter of shaded region

f. Perimeter of unshaded region

XY

XY

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10. Find the area of the shaded region. a. b.

11. Pizza A pizza is cut into 8 congruent pieces as shown. The diameter of the pizza is 16 inches. Find the area of one piece of pizza.

12. Clock A wall clock has an area of 452.39 inches. Find the diameter of the clock. Then find the area

of the sector formed when the time is 3:00 as shown.

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Warm-Up For problems 1-2, find the area of the given figure. 1. 2.

Solve for x. When necessary, round to hundredths place and leave pi in answers that contain it.

3. 1

(2)(12)(5)3

x 4. 21(5) (22)

3x

Cross Sections Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section. For example, the diagram shows that an intersection of a plane and a triangular pyramid is a triangle. Also, an intersection of a plane in a rectangular prism is a rectangle.

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Describe the cross section formed by the intersection of the plane and the pentagonal prism. 1. 2. 3.

Describe the cross section formed by the intersection of the plane and the solid. 4. 5. 6.

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Learning Task: Volumes of Cylinders, Cones, Pyramids, and Spheres

Volume Formulas Graphic Organizer

Shape Formula Example 1 Example 2

Sphere

A beach ball has a diameter of 8 inches. Find its volume.

Find the volume of the hemisphere.

Find the volume of prisms and cylinders.

V=Bh

(where B is the area of the base)

ARectangle= bh

ACircle= πr2

Find the volume. 4 m 2 m 10 m

Find the volume.

Find the volume of pyramids and cones.

V = 1

3Bh

(where B is the area of the base)

Find the volume. 15 yd

5 yd

Find the volume. 15.8 in 44 in 30 in 28 in

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Learning Task: Cavaleri’s Principle Materials: 10 quarters

In order to explore Cavalieri’s Principle construct a right cylinder using the 10 quarters.

Recall that the volume of a cylinder is V Bh . The height of the stack of quarters is 2 cm.

1. The diameter of a quarter is 2.4 cm. Calculate the area of the circular base.

2. Find the volume of the cylinder.

Now we are going to create an oblique cylinder. Slightly move the quarters to form an oblique cylinder like the image below.

3. Does the height of your right cylinder change? Does the area of the base change? Does the

volume change?

4. Look at the bases of all of your quarters. What two things do you notice?

**The bases of the quarters are also referred to as cross sections

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5. What can we conclude about the area of each cross section?

6. Create a conjecture about the volume of a right and oblique cylinder with congruent bases and heights.

Guided Practice: Use volume formulas to calculate the volume of the figure described in the following problems. Leave answers in terms of pi or the nearest hundredth. 1. 2.

Cavalieri’s Principle States: The volumes of two solids are equal if the areas of corresponding sections drawn parallel to some given plane are equal.

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3.

4. The volume of a cylindrical watering can is 100cm3. If the radius is doubled, then how much

water can the new can hold?

5. Approximate the volume of the backpack that is 17 in. x 12 in. x 4 in. The top of the backpack is half a cylinder and the bottom of the backpack is a rectangular prism.

6. Find the volume of the Grain Silo shown below that has a diameter of 20 ft. and a height of 50 ft. The top of the Grain Silo is a hemisphere and the bottom of the silo is a cylinder.

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7. The diameter of a baseball is about 1.4 in. How much rubber is needed to fill it?

Calculate the volume of the cylinder pictured in problems 8-9. 8. 9. 10. A sphere of ice cream is placed onto your ice cream cone. Given that the cone has a diameter of 8 centimeters, find the volume of the ice cream.

11. Tennis balls with a diameter of 2.5 in are sold in cans of three. The can is a cylinder. What is the volume of the space not occupied by tennis balls? Assume the balls touch the can on the sides, top, and bottom.

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Skills Practice: Volume and Cavalieri’s Principle Directions: Find the volume for each of the solids pictured or described in the following problems. Leave answers in terms of pi or round to the nearest hundredth. 1. Find the volume of a sphere when the diameter is 24 cm.

2. 3. Find the volume of the hemisphere.

4. 5. 6. 7.

4

9

16

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8. Find the volume of a right cylinder with a radius of 4 ft. and a height of 13 ft. 9. Determine the volume of a right cylinder with a diameter of 16 in. and a height of 6 in. 10. A right circular cone has a diameter of 6.2 in. and a height of 7 in. Calculate the volume of the cone. 11. Find the volume of the figure to the right. 12. Calculate the volume of cylinder 13. Calculate the volume. with a hemisphere taken out of the top.

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14. Find the volume of the cone below. 15. Calculate the volume, given that the height is three times the length of the radius.

Learning Task: Density

____________ is the amount of matter that an object has in a given unit of volume. The density of an object is calculated by dividing its mass by its volume.

massdensity

volume

Different materials have different densities, so density can be used to distinguish between materials that look similar. For example, table salt and sugar look alike. However, table salt has a density of 2.16 grams per cubic centimeter, while sugar has a density of 1.58 grams per cubic centimeter.

Example 1: A piece of copper with a volume of 2.85 cubic centimeters has a mass of 73.92 grams. A piece of iron with a volume of 5 cubic centimeters has a mass of 39.35 grams. Which metal has the greater density?

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Another use of the word density occurs in the term population density. The population density of a city, country, or state is a measure of how many people live within a given area.

number of peoplepopulation density

area of land

Population density is usually given in terms of square miles, but can be expressed using other units such as city blocks. The area of a trapezoid can be calculated by :

Use the area of a trapezoid formula to help answer the next question.

Example 2: The population of Vermont in 2009 was 621,760. The state can be modeled by a trapezoid with vertices at (0,0) , (0, 160), (80, 160), and (40,0), with each unit on the coordinate plane being 1 mile.

a. Calulcate the area of Vermont.

b. Find the population density of Vermont.

Skills Practice

Answer all the questions below. When necessary round all answers to the nearest hundredth.

1. A piece of tin has a mass of 16.52 g and a volume of 2.26 3cm . What is the density of tin?

2. A man has a 50.0 3cm bottle completely filled with 163 g of slimy green liquid. What is the density of

the liquid?

3. Different kinds of woods have different densities. The density of oak wood is generally .74 3/g cm .

If a 35 3cm piece of wood has a mass of 21g, is the wood likely to be oak?

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4. The density of pine is generally about 0.5. What is the mass of 800 3cm piece of pine?

5. What is the volume of 325g of metal with a density of 9.0 3/g cm ?

6. Diamonds have a density of 3.5 3/g cm . How bug is a diamond that has a mass of 0.10g?

7. Which has more mass: a solid cylinder of gold with a height of 5 cm and a diameter of 6 cm, or a

solid cone of platinum with a height of 21 cm and a diameter of 8cm? Use the following table to

help you answer the question.

a. Find the volume of the cylinder of gold. Then use the density formula and volume to calculate the mass.

b. Find the volume of the cone of platinum. Then use the density formula and volume to

calculate the mass. c. Which has more mass, the cylinder of gold or the cone of platinum?

Metal Density

Gold 19.30 3/g cm

Platinum 21.40 3/g cm