surface area all shape

Math 11 – Apprenticeship Marsh

Surface Area, Volume, and Capacity

3.1 Surface Area of Prisms

3.2 Surface Area of Pyramids, Cylinders, Spheres, and Cones

3.3 Volume and Capacity of Prisms and Cylinders

3.4 Volume and Capacity Spheres, Cones, and Pyramids

Name: _____________________________


3.1 Surface Area of Prisms


Review of Surface Area of 2D Figures

Examples


Calculate the area of each figure: (a) (b) (c)

Worksheet: Surface Area of 2D Figures

Surface Area of 2D Figures Worksheet


1. For each picture, name the shape and calculate the area.

(a) (b) (c) (d)

2. For each picture, name the shape and calculate the area.


(a)

(b) (c) (d)

(e) (f) What do you notice about the area calculations? Answers: 1. a) 6082.1 mm2 b) 26.5 m2 c) 78.4 cm2 d) 72 in2 2. a) 44.2 m2 b) 44.2 m2 c) 22.1 m2 d) 22.1 m2 e) 22.1 m2

Composite Shapes


A composite shape is a __________________________________________________. First we need to _______________ the shape up into smaller more ________________

shapes that we can find the area of. We then determine the area of each

____________and then _______________.

Examples

Find that area of the following figure.

Method 1:

Method 2:

Worksheet: Surface Area of Composite Shapes

Surface Area of Composite Shapes


1. Find the areas of the following figures. (a) (b) (c) (d)


(e) (f) (g)


2. Determine the area of the shaded area. (a)

(b)

Answers: 1. a) 128.5 m2 b) 36.6 m2 c) 46 m2 d) 41.7 m2 e) 15.23 m2 f) 26 m2 g) 2.3 m2 2. a) 32.14 m2 b) 44.76 m2 Surface Area of Prisms


A prism is a __________________________________ with:

________________________________________________________________

_______________________________________________________(the same).

________________________________________________________________

_______________________________________________________________.

Naming Prisms:

A prisms name is ____________________________________________________and

whether the sides _______________________________________________________.

A Prism is a “____________________________” prism if the base and the sides are

____________________________to each other (make 90o).

An____________________________ prism is when the base and the sides are not

perpendicular to each other.

Example

Name the following prisms:

(a) (b)

(c) (d)

Practice


Name the following prisms:


Working with Nets

A net is a _____________ pattern that can be folded to form a _____________ shape.

Think of a pizza box: it is made up of one piece of cardboard, folded into the shape of a

right rectangular prism.

The surface area of a prism is the area that it would take up if it were laid out flat, as in

its net.

Example

Draw nets for the following prisms:

Worksheet: Working with Nets Activity: Hexomino


Working with Nets Worksheet

1. Draw nets for the following prisms and label the dimensions. (a) (b)

2. Ralph says that diagram A is the net of a right octagonal prism. Mandy disagrees.

She says that diagram B is the correct net for a right octagonal prism. Who is correct? Justify your answer.


Surface Area of 2D Nets

Remember that when a 2D net is folded together it turns into a 3D shape. We are going to get some more practice determining the surface area of 3D objects by using their “nets”. Examples Find the surface area. (a)


(b)

Alternative Solution


Carl has been hired to paint the walls and ceiling of a living room in a house. The room is 22.5 feet long, 13.5 feet wide, and 8.5 feet high. There is one window that is 10.5 feet by 6 feet, two windows that are 3.5 feet by 2.5 feet, and two doors that are 2.5 feet by 8 feet.

a) What surface area must he paint?

b) One gallon of paint covers approximately 255 sq. ft. How many gallons will he have to buy?

c) If paint costs $55.40 per gallon and he wants to make a profit of about $225.00, how much should he charge to paint the room?

Worksheet: Surface Area of 3D Figures


Surface Area of 3D Figures Worksheet

1. For each diagram, draw a net and use it to calculate the surface area. (a) (b)

2. Calculate the surface area. (a)


(b)

3. Aaron is building a storage chest in the shape of a rectangular prism. The chest will be 90 cm long, 70 cm deep, and 60 cm high.

a) What will be the outer surface area of the box?

b) Aaron needs to buy 20% more wood than the surface area, to account for wastage during cutting. What area of wood will he need to buy?


4. Sharon is installing an L-shaped heating duct in a house. If the duct has the

measurements shown in the diagram below, what will be the total surface area of the duct? The ends of the duct are open.

5. Alex is going to construct a fish tank that is 1.2 m long, 0.6 m wide, and 0.4m high. How much glass will he need to make it? (Note: There will be no glass on the top.)

Answers:

1. a) 376 cm2 b) 108 in2 2. a) 14.5 m2 b) 2250 cm2 3. a) 31800 cm2 b) 38160 cm2 4. 48.5 ft2 5. 2.2 m2

Activity: Composite Shapes


3.2 Surface Area of Pyramids, Cylinders, Spheres, and Cones

Last section we looked at the surface area of mostly rectangular and triangular prisms. In this section we are looking into: Cylinders:

A 3D shape with _________________________ that are parallel and congruent. The side is a rectangle that is “_________________________” the circular face at the ends. Pyramids:

A 3D shape with a base of a polygon and the lateral sides that are triangles that meets at the vertex. Cones:

A 3D shape with a _________________________________________________. Spheres:

A 3D shape, whose surface consists of points that are all _______________________

_________________________ of the sphere.

To determine the surface areas of these shapes we add up all the different sides, there are just different processes for each.


Working with Circles: Review In this section, you will need to be able to calculate the area and circumference of a circle.

rC

rAcircle

2

2

Example Find the area of the following diagrams.


Working with Cylinders

To find the surface area, you have to find the area of the two circles and the area

between them. If you draw a net of a cylinder, you will find that it is made up of a

_________________ and two _____________________. The length of the rectangle

will be the ____________________________ of the circle, and the width will be the

______________ of the cylinder.

Example

Find the area of the following cylinder.

Worksheet: Surface Area of Cylinders


Surface Area of Cylinders Worksheet

1. Find the surface area of a cylindrical tank that has a radius of 1.5 m and a height

of 5 m.

2. Find the surface area of a pipe that has a diameter of 4.5 cm and is 18.8 cm long.

3. Find the surface area of the figure below. The upper cylinder is centered on the lower one.

Answers: 1. 61.2 m2 2. 265.8 cm2 3. 872.6 mm2


Working with Spheres A sphere is like a ball. All points on the sphere are equidistant (of equal distance) from

the centre. It is _________________________ to draw a net of a sphere. The formula

for a sphere’s surface area depends only on the ___________ and ___________.

The formula for the surface area of a sphere is:

24 rSA

Example

(a) The radius of the following sphere is 28 cm. What is its surface area?

(b) A ball has a surface area of approximately 9900 cm2. What is its radius?


(c) The earth’s radius is 6370km.

Lets determine:

(a) the circumference

(b) the surface area of Earth

(c) the surface area of each hemisphere

Worksheet: Surface Area of Spheres


Surface Area of Spheres Worksheet

1. Find the surface area of a sphere with a radius of 1.3 m.

2. Find the surface area of a sphere with a diameter of 24.8 mm.

3. Find the surface area of a hemisphere with a radius of 18.5 cm.

4. A tennis ball has a diameter of 6.7 cm. What is its surface area?

Answers: 1. 21.2 m2 2. 1932.2 mm2 3. 3225.8 cm2 4. 140.95 cm2


Working with Pyramids To determine the surface area of a pyramid, we need to find out the SA of the base and then the 4 sides. Example Find the surface area of the square-based pyramid.

(a)

Slant Height ©

(b)

Worksheet: Surface Area of Pyramids


Surface Area of Pyramids Worksheet

1. Find the surface area of the square-based pyramid below.

2. Find the total surface area of a square pyramid with a base of 12 cm by 12 cm and a height of

8 cm.

3. A triangular pyramid has faces that are all equilateral triangles. Each side length is 16 cm.

What is the surface area of the pyramid? Hint: you need to find the height using Pythagoras.

4. If the surface area of the sides of a square-based pyramid is 680 cm2 and the side lengths of

the square are 16 cm, what is the height of the pyramid?

Answers:

1. 1960 cm2 2. 336 cm2 3. 444.8 cm2 4. 13.25 cm


Working with Cones A cone is like a __________________, except it has a circular base. The surface area of the lateral area of the cone (the area not including the base) is calculated with the formula:

rsA

Example

(a) Find the surface area of a cone that has a radius of 12 feet and slant height of 15 feet.

(b) Find the surface area of the cone.

Worksheet: Surface Area of Cones


Surface Area of Cones Worksheet

1. Find the surface area of a cone that has a slant height of 82 cm and a radius of 28 cm.

2. Find the surface area of a cone with a diameter of 13.6 cm and a slant height of 9.8 cm.

3. Find the total surface area of a cone with a radius of 16 inches and a height of 20 inches.

4. Find the surface area of a cone with a radius of 45.7 mm and a height of 39.7 mm.

5. Find the lateral surface area of a cone whose height is 32 cm and whose diameter is 28 cm.

Answers: 1. 9676.1 cm2 2. 354.6 cm2 3. 2091 in2 4. 15247 mm2 5. 1535 cm2


Surface Area of Composite Figures: Cylinders, Pyramids, Cones, Spheres

Let’s put everything together to calculate the surface area of some composite figures.

Example

Find the surface area of the composite figure. (a)

(b)

Worksheet: Surface Area of Composite Figures


Surface Area of Composite Figures Worksheet

1. Three cylinders with radii of 4 feet, 3 feet, and 2 feet are stacked one on top of the other. Each has a height of 2 feet. What is the total exposed surface area? Do not include the bottom.

2. Find the surface area of the cone topped by the hemisphere shown in the following diagram.

Answers: 1. 163.4 ft2 2. 933 cm2


3.3 Volume and Capacity of Prisms and Cylinders

What do you think is the difference between volume and capacity?

Volume is the ____________________________________________________

a solid 3D object occupies; we usually measure this in ________.

o Can we think of some examples of when we need to determine the

volume? Example:

o A single piece of paper doesn’t have a very large volume but when we stick 500 together their volume becomes very noticeable.

Capacity is the _________________________________________a 3D object,

we usually measure this in ________________ or ________________.

What are some things that just have volume?

What are some things that have both volume and capacity? The volume of a prism and cylinder can be determined by multiplying the area of the base by the objects height.

hAVbase

Volume and capacity are closely related. A volume of ________________ is equivalent

to a capacity of ______________.

Example

1. A rectangular prism has a base that is 15 cm by 12 cm and a height of 20 cm.

a) Calculate the volume of the prism.

b) Calculate the capacity of the prism.


2. Determine the capacity of the box if the base has a width and length of 18 in and 24 in and a height of 16 in. 3. Calculate the volume and capacity of the composite prism.

4. A rectangular prism has a square base with side lengths of 7 cm. Its volume is 392 cm3. Calculate the height of the prism.

Worksheet: Volume and Capacity of Prisms


Volume and Capacity Worksheet

1. Find the volume and capacity of the following rectangular prisms. a) The base is 15.7 cm by 18.8 cm and the height is 12.5 cm.

b) The base is a square with sides of 2.75 m, and the height is 4.5 m.

c) The base is 1 ½ inches by 3 ¾ inches, and the height is 2 ¼ inches.

2. A rectangular prism has a base of 5.2 m by 7.8 m. Its volume is 142 m3. What is the height of the prism?

3. One rectangular prism has dimensions of 18 cm by 12 cm by 32 cm. A second prism has a base that is 14 cm by 20 cm. Approximately what must its height be if it has the same volume as the first prism?

4. Patrice is in charge of the excavation for the foundation of a building. If a hole must be dug that is 35 m by 25 m by 12 m, how many trips will be required to remove the dirt if a trailer can carry only 15 cubic meters of dirt?


5. Find the volume of this figure.

6. The capacity of the composite prism is 1.8 liters. Determine the depth, w.

Answers: 1. a) 3960 cm3 ;3.69 L b) 34 m3 or 34000000 cm3 ;34000 L c) 208 cm3 ;0.208 L or 208 mL 2. 3.5 m 3. 24.7 cm 4. 700 trips 5. 4464 m3 6. 6.2 cm


Working with Cylinders The volume of a prism is calculated using the following formula:

hAVbase

The area of the base of a cylinder can be calculated as:

2rA

If you combine these two formulas, the formula for the volume of a cylinder is as follows:

hrV 2

Examples 1. A can of tomato sauce has a radius of 3.8 cm and a height of 10.2 cm.

a) What is the volume of the can? b) How much tomato sauce (in liters) does the can hold?

2. A large tin can has a capacity of 3.24 L. If the can has a diameter of 15.56 cm, what is the height of the can? Worksheet: Volume and Capacity of Cylinders


Volume and Capacity of Cylinders Worksheet

1. Calculate the volume and capacity of a cylinder with a diameter of 15 cm and a height of 36 cm.

2. Calculate the volume and capacity of the stacked cylinders below. Each cylinder has a height of 20 cm.

3. Calculate the volume of each figure.

(a) (b)

(c)


4. A silo has a diameter of 24 feet and is filled to a height of 70 feet. What is the volume of grain stored in it?

5. A large cylindrical fuel storage tank has a capacity of 20 000 L. If it has a diameter of 2.4 m, what is the height of the tank?

6. Which of these figures has the larger capacity? Show your work.

Answers: 1. 6362 cm3 ;6.4 L 2. 32987 cm3 3. a) 111.4 cm3 b) 34.6 cm3 c) 70.9 m3 4. 31667.3 ft3 5. 4.4 m 6. 6.4L vs. 5.3 L


3.4 Volume and Capacity Spheres, Cones, and Pyramids

Working with Spheres The volume of a sphere can be calculated using the formula:

3

3

4rV

sphere

The capacity of a spherical container is calculated in similar manner to a cylinder or a

prism. Start by calculating the volume then convert to a unit of capacity

(recall 1000 cm3 = 1 L).

Example

A spherical exercise ball has a diameter of 1.2 m. a) What is its volume? b) What is its capacity? We will need to be able to convert from cubic metres to cubic centimetres.

1 m = 100 cm

lwhV

Worksheet: Volume and Capacity of Spheres


Volume and Capacity of Spheres Worksheet

1. Find the volume of each sphere.

a) A sphere with a radius of 8.5 cm.

b) A sphere with a diameter of 78 cm.

c) 2. A sphere with a radius of 46 cm is centered inside a sphere with a radius of 76

cm.

a) What is the volume of the space between the two spheres?

b) What is the capacity?

3. What is the capacity in US gallons of a spherical water tower with a diameter of

31.2 feet? 1 cubic foot equals 7.48 US gallons.

Answers: 1. a) 2572 cm3 b) 248475 cm3 c) 1047.4 mm3 2. a) 1431058 cm3 b) 1431 L 3. 1180947 US gal.


Working with Pyramids The volume of a pyramid is directly related to the volume of a prism with the same base and height. The volume is calculated using the following formula.

hAVbase

3

1

For a rectangular pyramid this can be written as:

lwhV3

1

Example 1. Calculate the volume and capacity of the pyramid. 2. Calculate the volume of this pyramid.


Worksheet: Volume and Capacity of Pyramids

Volume and Capacity of Pyramids Worksheet

1. Find the volume of each pyramid.


2. Calculate the volume of this prism and pyramid. What is the difference in volume?

3. Calculate the volume of each of these pyramids.

a) b) c) d) Answers: 1. a) 1615 in3 b) 40 ft3 c) 154.5 cm3 2. 3584 in3 3. a) 7309 mm3 b) 2663 in3 c) 637 in3 d) 806 mm3


Working with Cones The volume of a cone is equal to 1/3 of the volume of a cylinder with the same base and height. We can use the same formula for pyramids:

hAVbase

3

1

Since the base of a cone is a _____________, the formula can be rewritten:

hrV 2

3

1

Example 1. A paper cup in the shape of a cone has a radius of 3.2 cm and a height of 6 cm.

How much water can the cup hold? 2. Find the volume of the following figure. Worksheet: Volume and Capacity of Cones


Volume and Capacity of Cones Worksheet

1. Find the volume of the following figure.

a) b)

c) 2. A cone has a radius of 12 mm and a volume of 4071.5 mm3. What is its height? 3. A cone has a slant height of 15 cm and a radius of 8 cm. Determine its volume.

Answers: 1. a) 379.6 in3 b) 198 m3 c) 515.4 mm3 2. 27 mm 3. 851 cm3

surface area all shape

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