CHAPTER 7: GEOMETRY

College Prep Essential Math Chapter 7: Geometry

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CHAPTER 7: GEOMETRY

Chapter Objectives

By the end of this chapter, students should be able to:

Identify basic vocabulary in geometry such as point, line, segment, ray, etc. Recognize common plane geometric shapes. Find the perimeter and area for common shape polygons. Calculate the circumference and area of a circle. Find the surface area of a common prism. Solve area, perimeter, and volume application word problems. Apply the Pythagorean Theorem to find a missing side of a right triangle or

solve application problems.

Contents CHAPTER 7: GEOMETRY .............................................................................................. 1

SECTION 7.1 Definitions ............................................................................................. 2

A. Geometric Definitions ...................................................................................... 2

B. Angles ............................................................................................................. 3

C. Special Quadrilaterals ..................................................................................... 5

Exercises ................................................................................................................. 7

Section 7.2 Perimeter and Area ................................................................................... 9

A. Perimeter ........................................................................................................ 9

B. Rectangles and Parallelograms .................................................................... 10

C. Trapezoids .................................................................................................... 13

D. Triangles ....................................................................................................... 15

Exercises ............................................................................................................... 18

Section 7.3 Circles ..................................................................................................... 21

A. Circumference............................................................................................... 21

B. Area .............................................................................................................. 22

Exercises ............................................................................................................... 24

Section 7.4 Pythagorean Theorem ............................................................................ 26

A. Square Roots ................................................................................................ 26

B. Pythagorean Theorem .................................................................................. 27

Exercises ............................................................................................................... 30

Section 7.5 Volume and Surface Area ....................................................................... 31

A. Volume and Surface Area of Rectangular Solid ............................................ 31

B. Cylinders ....................................................................................................... 35

Exercises ............................................................................................................... 37

CHAPTER REVIEW .................................................... Error! Bookmark not defined.


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SECTION 7.1 Definitions

A. Geometric Definitions

This chapter will have us explore geometry. We will begin by reviewing some key

elements of geometry. Some of these terms may be familiar to you.

Geometric Definitions

Name Definition/Properties Picture

Point A location in space. A point has no width, length, or height.

Line A collection of points that extend along a straight path in two directions without end.

Line Segment A part of a line that has two endpoints.

Ray A part of a line that has one endpoint.

Media Lesson Lines, Line Segments, and Rays (Duration 3:37)

View the video lesson and follow along.

YOU TRY:

Draw the following. a) Point

b) Line Segment

c) Line d) Ray

https://www.youtube.com/watch?time_continue=55&v=JcqCf762y9w


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B. Angles

An angle is formed by two rays that share a common endpoint. Each ray is called a side

of the angle and the common endpoint is called the vertex.

Angles are measured in degrees. We use the symbol ° to represent degrees. Below are

some angles with their angle measure.

We measure an angle by the amount of rotation from one ray to the other ray.

There is one very important angle and that is the right angle. A right angle measures

90°. It is the type of angle you find at the corner of a room and the corner of a piece of

paper. The angle below is a right angle. Notice how this angle has a square at its

vertex. When you see this on an angle it is being identified as a right angle.

We can classify angles by their measurements.


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Right Angles Right angles measure exactly 90°.

Acute Angles Acute angles measure

less than 90°.

Obtuse Angles Obtuse angles measure

more than 90° and less than 180°.

Straight Angle Straight angles measure

exactly 180°.

Media Lesson Recognizing Angles (Duration 2:14)


YOU TRY:

Determine whether the given angle is an acute, right, or obtuse angle. e)

f)

g)

https://www.youtube.com/watch?time_continue=1&v=B0R3MJOrST0


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C. Special Quadrilaterals

A quadrilateral is a 4-sided shape. The following are all examples of quadrilaterals.

There are quadrilaterals you may be more familiar with like the square, rectangle,

parallelogram, and the rhombus.

Media Lesson Introduction to Types of Quadrilaterals (Duration 8:06)


The table lists special quadrilaterals and their properties.

Rectangle A quadrilateral with four right angles. Opposite sides of rectangles have the same length.

Square A square is a rectangle with all sides

of the same length.

Parallelogram A quadrilateral whose opposite sides

are parallel. Opposite sides of parallelograms have the same length.

Rhombus A parallelogram with all sides of the

same length.

https://www.youtube.com/watch?v=H-ykHosJW9c


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YOU TRY:

Identify what best describes the following shapes: parallelogram, rhombus, rectangle, or square. h)

i)


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Exercises

In the following exercises identify whether you are given a point, line, line segment, ray,

or angle.

1)

2)

3)

4)

5)

6)

In the following exercises draw the indicated shape.

7) Point

8) Line Segment

9) Line

10) Ray

In the following exercises identify whether the given angle is acute, right, obtuse, or

straight.

11)

12)

13)

14)

15)

16)


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In the following exercises draw the indicated angle.

17) Right 18) Obtuse

19) Straight 20) Acute

In the following exercises identify what best describes the quadrilateral: parallelogram,

rhombus, rectangle, or square.

21)

22)

23)


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Section 7.2 Perimeter and Area

A. Perimeter

Perimeter is the distance around a figure. The perimeter of a figure is the sum of the

lengths of its sides.

Example: The perimeter of the following figure is 24.

If we add the length of all the sides we get:

𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 7 + 7 + 6 + 4

= 24

Media Lesson Determine the perimeter of a Rectangle Involving Whole Numbers (Duration 2:41)

View the video lesson, take notes and complete the problems below.

Determine the perimeter of the rectangle.

Media Lesson Ex: Find the Perimeter of an L-Shaped Polygon Involving Whole Numbers (Duration 2:48)


https://www.youtube.com/watch?v=UuyW51OVPJ0

https://www.youtube.com/watch?v=0HfnFXm7zHI

https://www.youtube.com/watch?v=0HfnFXm7zHI


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YOU TRY:

j) What is the perimeter of the figure?

k) Find the perimeter of a square with side length 2.17 feet.

B. Rectangles and Parallelograms

If we want to know the size of a wall that needs to be painted or a floor that needs to be

carpeted, we will need to find its area. The area is a measure of the amount of surface

that is covered by the shape. Area is measured in squared units.

We will discover formulas to find the area of several shapes. We will explore the area of

rectangles and parallelograms first.

For a rectangle, the formula for area is

𝐴𝑟𝑒𝑎 = 𝑙𝑒𝑛𝑔𝑡ℎ × 𝑤𝑖𝑑𝑡ℎ or 𝐴 = 𝑙𝑤

Example: Find the area of the following rectangle.

The length is 5 feet and the width is 4 feet.

𝐴 = 𝑙𝑤

= 5 × 4 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑒 𝑡ℎ𝑒 𝑣𝑎𝑢𝑒𝑠 𝑓𝑜𝑟 𝑙𝑒𝑛𝑔ℎ𝑡 𝑎𝑛𝑑 𝑤𝑖𝑑𝑡ℎ

= 20 𝑓𝑡2 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦. 𝐼𝑛𝑐𝑙𝑢𝑑𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

The rectangle has an area of 20 square feet.


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For a parallelogram, the formula for the area is

𝐴𝑟𝑒𝑎 = 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 or 𝐴 = 𝑏ℎ

Notice that the base is a side of the parallelogram and that the height is not. The height

will always form a right angle with the base.

Example: Find the area of the following parallelogram.

The base is 11 inches and the height is 3 inches.

𝐴 = 𝑏ℎ

= 11 × 3 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑒 𝑡ℎ𝑒 𝑣𝑎𝑢𝑒𝑠 𝑓𝑜𝑟 𝑏𝑎𝑠𝑒 𝑎𝑛𝑑 ℎ𝑒𝑖𝑔ℎ𝑡.

= 33 𝑖𝑛2 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦. 𝐼𝑛𝑐𝑙𝑢𝑑𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

The parallelogram has an area of 33 square inches.

Media Lesson Area – Rectangle and Parallelogram (Duration 3:53)


Area of a Rectangle:

Area of a Parallelogram:

Important: The height must meet the base at a perfect ________ angle.

https://www.youtube.com/watch?v=IFRGGD07Cgw


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Find the area.

Media Lesson Ex: Determine the Area of a Rectangle Involving Whole Numbers (Duration 2:06)


Media Lesson Ex: Area of a Parallelogram (Duration 1:57)


https://www.youtube.com/watch?v=TN4tm_rONNc

https://www.youtube.com/watch?v=npY54EKjWaA


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YOU TRY:

l) Find the area of the parallelogram.

m) Find the area of the square.

C. Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. We call the parallel

sides the bases. The following figures are trapezoids.

Most of the times you will see the formula for the area of a trapezoid as

𝐴 =1

2× ℎ𝑒𝑖𝑔ℎ𝑡 × (𝑏𝑎𝑠𝑒1 + 𝑏𝑎𝑠𝑒2) or 𝐴 =

1

2ℎ(𝑏1 + 𝑏2)

The height needs to be perpendicular to both bases.


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Media Lesson Area - Trapezoid (Duration 4:49)


Area of a trapezoid:

The bases (a and b) must be _________________.

The _____________ connects the _____________.

Find the area:

YOU TRY:

Find the area of the trapezoids. n)

o)

https://www.youtube.com/watch?v=S3p4ET5mcNQ


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D. Triangles

A triangle is a three sided polygon. If we think of a triangle as half of a parallelogram, it

is easy to see the area of a triangle will be half the area of a parallelogram with the

same base and height. The formula for the area of a triangle is

𝐴 =1

2𝑏ℎ

, where 𝐴 is area, 𝑏 is base, and ℎ is height.

Media Lesson Area – Triangle (Duration 2:58)


Area of a triangle:

Find the area:

https://www.youtube.com/watch?v=7wskaOGeIEA


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YOU TRY:

p)


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Reference Page

Print out this page to use with the chapter.

Area of Rectangle (and Square) Area of Trapezoid

𝐴𝑟𝑒𝑎 = 𝑙𝑒𝑛𝑔ℎ𝑡 × 𝑤𝑖𝑑𝑡ℎ 𝐴𝑟𝑒𝑎 = 𝑙 × 𝑤

𝐴𝑟𝑒𝑎 =(𝑏𝑎𝑠𝑒1 + 𝑏𝑎𝑠𝑒2) × ℎ𝑒𝑖𝑔ℎ𝑡

2

𝐴𝑟𝑒𝑎 =(𝑏1 + 𝑏2) × ℎ

2

Area of Parallelogram (and Rhombus) Area of a Triangle

𝐴𝑟𝑒𝑎 = 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝐴𝑟𝑒𝑎 = 𝑏 × ℎ

𝐴𝑟𝑒𝑎 =1

2× 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡

𝐴𝑟𝑒𝑎 =1

2× 𝑏 × ℎ


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Exercises

In the following exercises find the perimeter of the figures.

1)

2)

3)

4)

5)

6)

In the following exercises find the perimeter and area of each rectangle.

7) The length of a rectangle is 85 feet and the width is 45 feet.

8)


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9) A rectangle that is 20 feet wide and 35 feet long.

10)

In the following exercises find the area of the figures.

11) A triangle with a base of 12 inches and height of 5 inches.

12)

13) Find the area of a triangle with a base of 24.2 feet and height of 20.5 feet.

14)

15)

16)


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17)

18) A rhombus with sides of 6 feet and a height of 3 feet.

19)

20)

21) A trapezoid with height 12 feet and bases of 9 and 15 feet.

22)

23) A coach tells her athletes to run one lap around a soccer field. The length of the

soccer field is 100 yards, while the width of the field is 60 yards. Find the total

distance that each athlete will have run after completing one lap around the

perimeter of the field.

24) A fence company is measuring a rectangular area to install a fence around its

perimeter. If the length of the rectangular area is 130 feet and the width is 75 feet,

what is the total length of the distance to be fenced?

25) On a regulation soccer field, the penalty box is 44 yards by 18 yards. What is the

area of a penalty box?


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Section 7.3 Circles

A. Circumference

The distance around a circle has a special name called the circumference. It is the

perimeter of a circle. Before we find the formula for the circumference of a circle, we will

first need to define a few attributes of a circle.

Mathematically, a circle is defined as the set of all points equidistant to tits center. The

radius is the distance from the center of a circle to its edge. The diameter is the

distance across the circle passing through the center. The diameter is twice as long as

the radius.

There is a special relationship between the radius and circumference of a circle. The

ratio between the radius and circumference is a factor we call pi. The symbol 𝜋 is used

for this factor and we pronounce it “pie.”

To find the circumference we use the following formula:

𝐶 = 2𝜋𝑟 or 𝐶 = 𝜋𝑑

, where C is the circumference, r is the radius, and d is the diameter.

Note sometimes your answers can be left in terms of 𝜋. Alternatively, you might be

asked to use 𝜋 ≈ 3.14 or 22

7.

Example: Find the circumference of the following circle using 𝜋 = 3.14.

The radius is 4 cm and 𝜋 is 3.14.

𝑟 = 4 𝑐𝑚 𝑎𝑛𝑑 𝜋 = 3.14

Substitute into the equation and then multiply. 𝐶 = 2 × 3.14 × 4 𝑐𝑚

= 6.28 × 4 𝑐𝑚

= 25.12 𝑐𝑚 The circumference is 25.12 centimeters.


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Media Lesson Examples: Determine the Circumference of a Circle (Duration 2:59)


Determine the circumference.

YOU TRY:

q) A circle has a radius of 10 centimeters. Find its circumference using 𝜋 = 3.14.

B. Area

The area of a circle is found with the following formula

𝐴 = 𝜋𝑟2

Where A is the area and r is the radius.

Example: Find the area of the following circle. Leave your answers in terms of 𝜋.

The radius is 8 feet.

𝐴 = 𝜋𝑟2

= 𝜋 × 82 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠.

= 64𝜋 𝑓𝑡2 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦. 𝐼𝑛𝑐𝑙𝑢𝑑𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

The circle has an area of 64𝜋 square feet.

https://www.youtube.com/watch?v=sHtsnC2Mgnk


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Media Lesson Example: Determine the Area of a Circle (Duration 2:57)


Determine the area.

YOU TRY:

r) Find the area of the following circle. Leave answers in terms of 𝜋.

https://www.youtube.com/watch?v=SIKkWLqt2mQ


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Exercises

Determine if you are given the radius or diameter in the following circles.

1)

2)

Find the circumference and the area of the following circles. Leave your answers in

terms of 𝜋.

3)

4)

5)

6)

In the following exercises find the circumference and area. Use 𝜋 = 3.14. Round

answers to the hundredths place.

7)

8)


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9) A circle with diameter 9.5 inches.

10) A circle with radius 1.1 centimeters.

11) A farm sprinkler spreads water in a circle with radius of 8.5 feet. What is the area of the watered circle? (Use 𝜋 = 3.14 and round to the hundredths place.)

12) A reflecting pool is in the shape of a circle with diameter of 20 feet. What is the

circumference of the pool? (Use 𝜋 = 3.14.)

13) A circular rug has radius of 3.5 feet. Find the circumference and the area of the

rug. (Use 𝜋 = 3.14.)

14) The Earth’s equator is the circle around the Earth that is equidistant to the North and South Poles, splitting the Earth into what we call the Northern and Southern Hemispheres. The radius of the Earth is approximately 3,958.75 miles. What is

the circumference of the equator? Use 𝜋 = 3.14 and round your answer to the hundredths place.

15) Anderson rollerbladed around a circular lake with a radius of 3 kilometers. How

far did Anderson rollerblade? Use 𝜋 = 3.14 and round your answer to the hundredths place.


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Section 7.4 Pythagorean Theorem

A. Square Roots

To start this section, we need to review some vocabulary and notation.

Remember when a number 𝑛 is multiplied by itself, we can write this as 𝑛2. We read it

as “𝑛 squared.” For example, 82 is read as “8 squared.” We say 64 is the square of 8.

The square root of a number is that number which, when multiplied times itself, gives

the original number. For example,

4 × 4 = 42 = 16

So we say “the square root of 16 is 4”. We denote square roots with the following

notation.

√16 = 4

A perfect square is a number whose square root is a whole number. The list below

shows the square roots of the first 15 perfect square numbers.

√1 √4 √9 √16 √25 √36 √49 √64 √81 √100 √121 √144 √169 √196 √225

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The square root of a non-perfect square is a decimal value. For example, 19 is not a

perfect square because √19 ≈ 4.36 is not a whole number.

Media Lesson Pythagorean Theorem – Square Roots (Duration 2:49)


Square Roots: _____________________________

√25 means: _______________________________

√49 √225

YOU TRY:

s) √144

t) √36

https://www.youtube.com/watch?v=xSfjMTNXsq4


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B. Pythagorean Theorem

There is a special kind of triangle called a right triangle. A right triangle has a 90° angle.

The triangle below is a right triangle.

The side of the triangle opposite the 90° angle is called the hypotenuse. The sides next

to the 90° angle are called legs.

The Pythagorean Theorem is a special property of right triangles. It states that in any

right triangle, the sum of the squares of the two legs equals the sum of the hypotenuse:

𝑙𝑒𝑔2 + 𝑙𝑒𝑔2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒2

Often the sides will be labeled with a, b, and c. The letters a and b will be used for the

legs and the letter c will be the hypotenuse. So, you might see Pythagorean Theorem

written as

𝑎2 + 𝑏2 = 𝑐2

We often use the Pythagorean Theorem to solve missing lengths of sides.

Example: Solve for the missing side of this right triangle.

We are given the length of the legs and need to find the hypotenuse.

𝑎2 + 𝑏2 = 𝑐2

82 + 62 = 𝑐2 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑔𝑠 𝑓𝑜𝑟 𝑎 𝑎𝑛𝑑 𝑏. 64 + 36 = 𝑐2 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 82 𝑎𝑛𝑑 62.

100 = 𝑐2 𝐴𝑑𝑑.

√100 = √𝑐2 𝑇𝑎𝑘𝑒 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠. 10 = 𝑐

The hypotenuse measures 10 centimeters.


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Media Lesson Pythagorean Theorem – Find Hypotenuse (Duration 5:00)


Name the sides of the right triangle:

Pythagorean Theorem: _______________

c is always the _____________________

Find the missing side.

The base of a ladder is four feet from a building. The top of the ladder is eight feet up

the building. How long is the ladder?

Media Lesson Pythagorean Theorem – Find Leg (Duration 5:00)


When finding a leg with the Pythagorean Theorem, first _______________ ,

then _______________ .

Recall: The opposite of squaring is to _______________ .

Find the missing leg.

https://www.youtube.com/watch?v=z55FGJlAaVU

https://www.youtube.com/watch?v=BNspHwYGgPc


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A young boy is flying a kite. He let our 21 meters of string until the kite was flying over

the head of his sister who was 9 meters away. How high is the kite?

YOU TRY:

Use the Pythagorean Theorem to find the length of the missing side. u)

v)


30

Exercises

In the following exercises find the square root.

1) √25 2) √144

3) √256 4) √100

5) √625 6) √400

Use the Pythagorean Theorem to solve for the missing side of the triangle.

7)

8)

9)

10)

11)

12)

13)

14)

15) A wheelchair ramp raises a wheel chair a vertical distance of 3 feet, in a

horizontal distance of 4 feet. How long is the ramp the when chair travels on?

16) Two trains left a station at the same time. One train traveled south, and one train

traveled west. When the southbound train had gone 75 miles, the westbound

train had gone 125 miles. How far apart were the trains at this time? Label the

picture below to help you solve the problem.


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Section 7.5 Volume and Surface Area

A. Volume and Surface Area of Rectangular Solid

Volume

Volume is the amount of space that an object takes up. We measure volume in cubic

units.

The rectangular solid below has length 3 units, width 2 units, and height 2 units. If we

were to look at it layer by layer we could count the number of cubic units it is made of.

It is made up of 12 cubic units. Notice that this is the product of the length, width, and

height. The formula for the volume of a rectangular solid is

𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙𝑒𝑛𝑔𝑡ℎ × 𝑤𝑖𝑑𝑡ℎ × ℎ𝑒𝑖𝑔ℎ𝑡or 𝑉 = 𝑙𝑤ℎ.

Example: Find the volume of the following rectangular solid.

The length is 6 meters, the width is 2 meters, and the height is 4 meters.

𝑉 = 𝑙𝑤ℎ

= 6 × 2 × 4 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑙, 𝑤, 𝑎𝑛𝑑 ℎ.

= 48 𝑚3 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦. 𝑈𝑠𝑒 𝑐𝑢𝑏𝑖𝑐 𝑢𝑛𝑖𝑡𝑠.

The volume of the rectangular solid is 48 cubic meters.

A cube is a rectangular solid where the length, width, and height have the same

measure. The figure below is a cube with a length, width, and height of s.

If we know the measure of one side of a cube we can solve for the volume. The volume

of a cube is

𝑉 = 𝑠3

, where 𝑠 is the length of the side of a cube.


32

Media Lesson Introduction to Volume (Duration 3:52)


Find the volume of the shape below.

a) The cube below.

b) A box with sides of length 2 ft, 3 ft, and 5 ft.

Introduction to Surface Area

Surface area is the total area of all the sides. If we were to be looking for the surface

area of the rectangular solid below we could think of finding the area of each side.

Example: Find the surface area of the box below.

To find the total area of all the sides we would need to find the area of each side and

add it up.

Front of Box 𝐴 = 6 × 4 = 24 𝑢𝑛𝑖𝑡𝑠2

Top of Box 𝐴 = 6 × 3 = 18 𝑢𝑛𝑖𝑡𝑠2

Right Side of Box 𝐴 = 3 × 4 = 12 𝑢𝑛𝑖𝑡𝑠2

It is easy to see the top and bottom of the box have the same area, the front and back

have the same area, and the left and right sides have the same area.

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝑡𝑜𝑝 + 𝑏𝑜𝑡𝑡𝑜𝑛 + 𝑓𝑟𝑜𝑛𝑡 + 𝑏𝑎𝑐𝑘 + 𝑟𝑖𝑔ℎ𝑡 𝑠𝑖𝑑𝑒 + 𝑙𝑒𝑓𝑡 𝑠𝑖𝑑𝑒

= 18 𝑢𝑛𝑖𝑡𝑠2 + 18 𝑢𝑛𝑖𝑡𝑠2 + 24 𝑢𝑛𝑖𝑡𝑠2 + 24 𝑢𝑛𝑖𝑡𝑠2 + 12 𝑢𝑛𝑖𝑡𝑠2 + 12 𝑢𝑛𝑖𝑡𝑠2

= 108 𝑢𝑛𝑖𝑡𝑠2

The surface area of the box is 108 square units.

https://www.youtube.com/watch?v=AB9zpbc8fUE&feature=youtu.be


33

Media Lesson Surface Area of a Box (Duration 2:24)


YOU TRY:

Find the surface area of the following box. w)

A shortcut to find the surface area is to use the following formula

𝑆𝐴 = 2(𝑙𝑤 + 𝑙ℎ + ℎ𝑤)

, where 𝑆𝐴 is surface area, 𝑙 is the length, 𝑤 is the width, and ℎ is the height.

For a cube, since all sides are the same we can use

𝑆𝐴 = 6(𝑠2)

https://www.youtube.com/watch?time_continue=4&v=1iSBNSYhvIU


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Applications

Media Lesson Volume and Surface Area – Rectangular Solid (Duration 4:58)


Volume: ____________________

Surface Area: ____________________

Rectangular Solid Volume: ____________________

Rectangular Solid Surface Area: ____________________

On Southwest Airlines, the maximum size of a carryon bag is a length of 24 inches, a

width of 10 inches, and a height of 16 inches. How much can be packed in this

maximum sized bag?

A seamstress needs to cover a box that is 8 cm long, 5 cm high, and 4 cm wide with

material on all sides (including top and bottom). How much material does she need?

Media Lesson Find the Surface Area of an Open Top Box (Duration 2:46)


What is the surface area of the aquarium?

https://www.youtube.com/watch?v=pdCOfYb_6DY

https://www.youtube.com/watch?v=ZWvFH3a0kug&feature=youtu.be


35

B. Cylinders

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is

a solid figure with two parallel circles of the same size at the top and bottom. The top

and bottom of the cylinders are called the bases. The height ℎ of a cylinder is the

distance between the two bases. For all the cylinders we will work with, the sides and

the height ℎ, will be perpendicular to the bases.

Rectangular solids and cylinders are somewhat similar because they both have two

bases and a height. The volume formula for rectangular solids,

𝑉 = 𝑙𝑤ℎ

, can be generalized to find the volume of a cylinder. The base of a rectangular solid

has an area of 𝑙 × 𝑤. Let’s call 𝐵 the area of the base. Then,

𝑉 = 𝐵ℎ

The base for a cylinder is a circle. Since the area of a circle is 𝜋𝑟2, then

𝑉 = 𝜋𝑟2 ℎ

Media Lesson Cylinder Volume and Surface Area | Perimeter, Area, and Volume | Geometry | Khan Academy (Stop at 3:28)


https://www.youtube.com/watch?time_continue=11&v=gL3HxBQyeg0

https://www.youtube.com/watch?time_continue=11&v=gL3HxBQyeg0


36

YOU TRY:

Find the volume of the following cylinders. Leave your answer in terms of 𝜋. x)

y)


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Exercises

Find (a) the surface area and (b) the volume of the following shapes.

1)

2)

3)

4)

In the following exercises, find the volume.

5) A rectangular solid with a length of 8 feet, width of 9 feet, and height of 11 feet.

6) A rectangular solid with a length of 3.5 yards, width of 2.1 yards, and height of 2.4 yards.

In the following exercise, find the volume of the cylinder. Leave answers in term of 𝜋. Round answers to the hundredths place.

7)

8)

9)

10) A cylinder with a radius of 3 feet and height of 9 feet.


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11) A rectangular moving van has a 16 feet length, 8 feet width, and 8 feet height.

What is its volume?

12) Each side of the cube at the Discovery Science Center in Santa Ana is 64 feet

long. Find its volume and surface area.

13) A can of coffee has a radius of 5 cm and a height of 13 cm. Find the volume of

the can. Use 𝜋 = 3.14.

14) A cylindrical column has a diameter of 8 feet and a height of 28 feet. Find its

volume. Use 𝜋 = 3.14.

CHAPTER 7: GEOMETRY

Documents

Transcript of CHAPTER 7: GEOMETRY