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Lesson 12-1
Three-Dimensional Figures
Lesson 12-2
Nets and Surface Area
Lesson 12-3
Surface Areas of Prisms
Lesson 12-4
Surface Areas of Cylinders
Lesson 12-5
Surface Areas of Pyramids
Lesson 12-6
Surface Areas of Cones
Lesson 12-7
Surface Area of Spheres
Example 1 Use Orthogonal Drawings
Example 2 Identify Solids
Example 3 Slicing Three-Dimensional Figures
Draw the back view of a figure given its orthogonal
drawing.
Use blocks to make a model. Then use your model to draw
the back view.
The top view indicates one row of different heights
and one column in the front right.
The front view indicates that there are four standing columns.
The first column to the left is 2 blocks high, the second
column is 3 blocks high, the third column is 2 blocks high,
and the fourth column to the far right is 1 block high. The
dark segments indicate breaks in the surface.
The right view indicates that the front right column is only
1 block high. The dark segments indicate breaks in the
surface.
Check the left side of your model. The back left column is
2 blocks high. The dark segments indicate breaks in the
surface.
Check to see that all views correspond to the model.
Now that your model is accurate, turn it around to the
back and draw what you see. The blocks are flush, so
no heavy segments are needed.
Answer:
Draw the corner view of the figure.
Turn your model so you are looking at the corners of the
blocks. The lowest columns should be in the front so the
differences in height between the columns are visible.
Connect the dots on isometric dot paper to
represent the edges of the solid. Shade the
tops of each column.
Answer:
a. Draw the back view of the figure given its orthogonal
drawing.
Answer:
b. Draw the corner view of the figure.
Answer:
Identify the solid. Name the bases, faces, edges,
and vertices.
The bases are rectangles, and the four remaining
faces are parallelograms.
Answer: This solid is a rectangular prism.
Bases:
Faces:
Edges:
Vertices: A, B, C, D, E, F, G, H
Identify the solid. Name the bases, faces, edges,
and vertices.
The bases are circular
and congruent.
Answer: This solid is a cylinder.
Bases:
Identify the solid. Name the bases, faces, edges,
and vertices.
The base is a triangle, and the
remaining three faces meet at
a point.
Answer: This solid is a triangular pyramid.
Base:
Faces:
Edges:
Vertices: A, B, C, D
a. Identify the solid. Name the bases, faces, edges, and
vertices.
Answer: triangular prism
Bases:
Faces:
Edges:
Vertices: A, B, C, E, F, G
b. Identify the solid. Name the bases, faces, edges,
and vertices.
Answer: cylinder
Bases:
c. Identify the solid. Name the bases,
faces, edges, and vertices.
Answer: pentagonal pyramid
Bases: ABCDE
Faces:
Edges:
Vertices: A, B, C, D, E, F
BAKERY A customer ordered a two-layer sheet cake.
Describe the possible cross sections of the cake.
Answer:
If the cake is cut horizontally, the cross section will be
a rectangle.
If the cake is cut vertically, the cross section will also be
a rectangle.
A solid cone is going to be sliced so that the resulting
flat portion can be dipped in paint and used to make
prints of different shapes. How should the cone be
sliced to make prints of a circle, triangle, and an oval?
Answer:
If the cone were to be cut
parallel to the base, the
cross-section would be
a circle.
Answer:
If the cone were to be
cut perpendicular to the
base, the slice would be
a triangle.
If the cone were to be
cut on an angle to the
base, the slice would be
an oval.
Example 1 Draw a Solid
Example 2 Nets for a Solid
Example 3 Nets and Surface Area
Sketch a rectangular prism 4 units long, 3 units
wide, and 2 units high using isometric dot paper.
Step 1 Draw the corner of the solid: 2 units down,
4 units to the left, and 3 units to the right.
Sketch a rectangular prism 4 units long, 3 units
wide, and 2 units high using isometric dot paper.
Step 2 Draw a parallelogram for the top of the solid.
Sketch a rectangular prism 4 units long, 3 units
wide, and 2 units high using isometric dot paper.
Step 3 Draw segments 2 units down from each vertex
for the vertical edges.
Sketch a rectangular prism 4 units long, 3 units
wide, and 2 units high using isometric dot paper.
Answer:
Step 4 Connect the corresponding vertices. Use
dashed lines for the hidden edges. Shade
the top of the solid.
Sketch a rectangular prism 1 unit high, 5 units long,
and 2 units wide.
Answer:
MULTIPLE- CHOICE TEST ITEM Which net could be
folded into a triangular prism if folds are made only
along dotted lines?
Read the Test Item You are given four nets, only one of
which can be folded into a triangular prism.
Solve the Test Item A triangular prism has two triangular
bases and three rectangular faces. Analyze each answer
choice carefully and locate the one that has each of these
elements.
This net has two triangles
for the bases, but only
one rectangle.
This net has four nonoverlapping triangles and
a square for the base. It
folds into a rectangular
pyramid.
This net has four
squares and one
triangle.
This net has three adjacent
rectangles and two triangles. The
triangles are adjacent to each of the
shorter sides of the middle rectangle.
Each base and face of the prism is
represented. This choice is correct.
Answer: D
Which net could be folded into a triangular pyramid
if folds are made only along the dotted lines?
Answer: B
Draw a net for the triangular prism.
We need to know the length of the legs of the isosceles
triangular bases in order for us to know the width of two
of the rectangular faces of the prism. Use the
Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Take the square root of each side.
Use rectangular dot paper to draw a net. Let one unit on
the dot paper represent 1 centimeter.
Answer:
Use the net to find
the surface area
of the prism.
To find the surface area
of the prism, add the
areas of the three
rectangles and the two
triangles.
Write an equation to solve for the surface area.
Answer: The surface area of the triangular prism
is 60 square centimeters.
a. Draw a net for the rectangular pyramid.
Answer:
b. Use the net to find the surface area of the prism.
Answer: 84 cm2
Example 1 Lateral Area of a Pentagonal Prism
Example 2 Surface Area of a Triangular Prism
Example 3 Use Surface Area to Solve a Problem
Find the lateral area of the regular hexagonal prism.
The bases are regular
hexagons. So the perimeter
of one base is 6(5) or 30
centimeters.
Lateral area of a prism
P = 30, h = 12
Multiply.
Answer: The lateral area is 360 square
centimeters.
Find the lateral area of the regular octagonal prism.
Answer: 216 cm2
Find the surface area of the square prism.
Surface area of a prism
L = Ph
Substitution
Simplify.
Answer: The surface area is 360 square centimeters.
Find the surface area of the triangular prism.
Answer: 416 units2
SCULPTURE A solid block of marble will be used
for a sculpture. If the block is 3 feet wide, 4 feet long,
and
feet high, find the surface area of the block.
The block is
shaped like a
rectangular prism.
The perimeter of a base is 2(3) + 2(4) or 14 feet.
The area of a base is 3(4) or 12 square feet.
Formula for surface area
Simplify.
Answer: The surface area of the block of marble is 157
square feet.
A television is packaged in a box that is
feet
long, 2 feet wide, and 2 feet high. Find the surface
area of the box.
Answer: 28 ft2
Example 1 Lateral Area of a Cylinder
Example 2 Surface Area of a Cylinder
Example 3 Find Missing Dimensions
A fruit juice can is cylindrical with aluminum sides
and bases. The can is 12 centimeters tall, and the
diameter of the can is 6.3 centimeters. How many
square centimeters of aluminum are used to make
the sides of the can?
The aluminum sides of the can represent the lateral
area of the cylinder. If the diameter of the can is
6.3 centimeters, then the radius is 3.15 centimeters.
The height is 12 centimeters. Use the formula to find
the lateral area.
Lateral area of a cylinder
Use a calculator.
Answer: About 237.5 square centimeters of aluminum
are used to make the sides of the can.
A set of toy blocks are sold in a cylindrical shape
container. A product label wraps around all sides
of the container without any overlaps or gaps. How
much paper is used to make the label the appropriate
size if the diameter of the container is 12 inches and
the height is 18 inches?
Answer: about 678.6 in2
Find the surface area of the cylinder.
The radius of the base and the height of the cylinder are
given. Substitute these values in the formula to find the
surface area.
Surface area of a cylinder
Use a calculator.
Answer: The surface area is approximately 2814.9
square feet.
Find the surface area of the cylinder.
Answer: about 1156.1 ft2
Find the radius of the base of a right cylinder if the
surface area is
square feet and the height is 10
feet.
Use the formula for surface area to write and solve an
equation for the radius.
Surface area of a cylinder
Substitution
Simplify.
Divide each side by
Subtract 264 from each side.
Factor.
Solve.
Since a radius of a circle cannot have a negative value,
–22 is eliminated.
Answer: The radius of the base is 12 feet.
Find the radius of the base of a right cylinder if the
surface area is
square inches and the height
is 22 inches.
Answer: 14 in.
Example 1 Use Lateral Area to Solve a Problem
Example 2 Surface Area of a Square Pyramid
Example 3 Surface Area of Pentagonal Pyramid
CANDLES A candle store offers a pyramidal candle
that burns for 20 hours. The square base is 6
centimeters on a side and the slant height of the
candle is 22 centimeters. Find the lateral area of the
candle.
We need to find the lateral area of the square pyramid.
The sides of the base measure 6 centimeters, so the
perimeter is
Lateral area of a regular pyramid
Multiply.
Answer: The lateral area of the candle is 264 square
centimeters.
CAMPING A pyramidal shaped tent is put up by two
campers. The square base is 7 feet on a side and the
slant height of the tent is 7.4 feet. Find the lateral area
of the tent.
Answer: 103.6 ft2
Find the surface area of the regular pyramid to the
nearest tenth.
To find the surface area,
first find the slant height
of the pyramid. The slant
height is the hypotenuse
of a right triangle with
legs that are the altitude
and a segment with a
length that is one-half the
side measure of the
base.
Pythagorean Theorem
Use a calculator.
Now find the surface area of a regular pyramid. The
perimeter of the base is
and the area
of the base is
Surface area of a regular
pyramid
Use a calculator.
Answer: The surface area is 179.4 square meters to
the nearest tenth.
Find the surface area of the regular pyramid to the
nearest tenth.
Answer: 89.8 m2
Find the surface area of the regular pyramid. Round to
the nearest tenth.
The altitude, slant height, and
apothem form a right triangle.
Use the Pythagorean Theorem
to find the apothem. Let x
represent the length of the
apothem.
Pythagorean Theorem
Simplify.
Now find the length of the sides of the base. The central
angle of the hexagon measures
Let a
represent the measure of the angle formed by a radius
and the apothem.
Use trigonometry to find the length
of the sides.
Multiply each side by 9.
Multiply each side by 2.
Use a calculator.
Next, find the perimeter and area of the base.
Finally, find the surface area.
Surface area of a regular pyramid
Simplify.
Answer: The surface area is approximately 748.2
square centimeters.
Find the surface area of the regular pyramid.
Answer: about 298.2 cm2
Example 1 Lateral Area of a Cone
Example 2 Surface Area of a Cone
ICE CREAM A sugar cone has an altitude of 8 inches
and a diameter of
inches. Find the lateral area of
the sugar cone.
Explore We are given the altitude
and the diameter of the
base. We need to find
the slant height of the
cone.
Plan
The radius of the base, height, and slant height
form a right triangle. Use the Pythagorean
Theorem to solve for the slant height. Then use
the formula for the lateral area of a right circular
cone.
Solve
Write an equation and solve for .
Pythagorean Theorem
Simplify.
Take the square root of
each side.
Next, use the formula for the lateral area of
a right circular cone.
Lateral area of a cone
Use a calculator.
Examine Use estimation to check the reasonableness
of this result. The lateral area is approximately
Compared to
the estimate, the answer is reasonable.
Answer: The lateral area is approximately 31.8 square
inches.
A hat for a child’s birthday party has a conical shape
with an altitude of 9 inches and a diameter of 5
inches. Find the lateral area of the birthday hat.
Answer: 73.4 in2
Find the surface area of the cone. Round to the
nearest tenth.
Surface area of a cone
Use a calculator.
Answer: The surface area is approximately
20.2 square centimeters.
Find the surface area of the cone. Round to the
nearest tenth.
Answer: about 63.6 cm2
Example 1 Spheres and Circles
Example 2 Surface Area
Example 3 Surface Area
In the figure, O is the center of the sphere, and
plane P intersects the sphere in circle R. If OR 6
centimeters and OS 14 centimeters, find RS.
The radius of circle R is segment
and S is a point on
circle R and on sphere O. Use the Pythagorean Theorem
for right triangle ORS to solve for RS.
Pythagorean Theorem
Simplify.
Subtract 36 from each side.
Use a calculator.
Answer: RS is approximately 12.6 centimeters.
In the figure, O is the center of the sphere, and plane
U intersects the sphere in circle L. If OL 3 inches and
LM 8 inches, find OM.
Answer: about 8.5 in.
Find the surface area of the sphere, given a great
circle with an area of approximately 907.9 square
centimeters.
The surface area of a sphere is four times the area of the
great circle.
Surface area of a sphere
Multiply.
Answer: The surface area is approximately 3631.6
square centimeters.
Find the surface area of a hemisphere with a radius
of 3.8 inches.
A hemisphere is half of a sphere. To find the surface area,
find half of the surface area of the sphere and add the
area of the great circle.
Surface area of a
hemisphere
Substitution
Use a calculator.
Answer: The surface area is approximately
136.1 square inches.
a. Find the surface area of the sphere, given a great circle
with an area of approximately 91.6 square centimeters.
Answer: about 366.4 cm2
b. Find the surface area of a hemisphere with a radius of
6.4 inches.
Answer: about 386.0 in2
Find the surface area of a ball with a circumference of
24 inches to determine how much leather is needed to
make the ball.
First, find the radius of the sphere.
Circumference of a circle
Use a calculator.
Next, find the surface area of the sphere.
Surface area of a sphere
Use a calculator.
Answer: The surface area is approximately 183.3
square inches.
Find the surface area of a ball with a circumference of
18 inches to determine how much leather is needed to
make the ball.
Answer: about 103.1 in2
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