Surface Area of Three-Dimensional Shapes

Three-Dimensional Figures

Tutorial

Objective

In this lesson, you will represent three-dimensional
figures using nets made up of rectangles and triangles
and find the surface area of these figures.

Introduction
In his science class, Boris conducted an experiment in which he used a glass prism to make a rainbow. He now wants to find the surface area of the prism. His teacher gets him a paper model of the prism and spreads it into a flat shape called a net. Boris can see that the net has exactly the same number of faces as the prism. The teacher tells Boris that he can use this net to find the surface area of the prism.
In this lesson, you'll learn how to find the surface area of three-dimensional figures using two-dimensional nets.

Exploring Three-Dimensional Figures
Three-dimensional objects are part of your everyday life. Your computer monitor, the box that contains your favorite cereal, and the books you read are all three-dimensional objects, each with a specific length, width, and height.
Three-dimensional figures are geometrical figures whose sizes we can specify using three quantities: length, width, and height. Prisms, pyramids, and cubes are three-dimensional objects that you're likely familiar with.
Two-dimensional figures are those figures whose sizes we can specify using only two quantities: length and width. All polygons are two-dimensional figures.
Recall that the area of a two-dimensional figure is the number of square units that can fit inside it. When we talk about three-dimensional figures, we use the term "surface area" instead of area. The surface area of a three-dimensional figure is the number of square units that fit on all of the figure's faces.

Let's review some of the parts of three-dimensional figures:
•face—a flat surface on a three-dimensional
figure. The top and bottom faces are called
bases. Other faces are called lateral faces. The
rectangular prism shown here has two bases
and four lateral faces.
•edge—a line segment where two faces
meet. The rectangular prism shown here has
12 edges.
•vertex—a point where three edges meet. The
rectangular prism shown here has eight
vertices. (The plural of vertex is vertices.)

Prisms
A prism, such as the one Boris used in his science class, is a three-dimensional figure whose lateral faces are rectangles and whose bases are parallel, congruent polygons. (Congruent means the polygons are the same size and shape.)
The shape of the base of a prism determines the kind of prism it is. A prism with triangular bases is a triangular prism, and a prism with rectangular bases is a rectangular prism. A cube is a prism with square bases and square lateral faces.
A triangular prism has five faces, six vertices, and nine edges, and a rectangular prism has six faces, eight vertices, and twelve edges.

Pyramids
The pyramids in Egypt are the oldest of the seven wonders of the ancient world. A pyramid is a three-dimensional figure whose base is a polygon and whose lateral faces are all triangles. Some pyramids have bases that are regular polygons; that is, polygons whose angles and sides have equal measures. The lateral faces of such pyramids are congruent. If the base of a pyramid is not a regular polygon, its lateral faces are not congruent.
As with a prism, the shape of a pyramid's base determines the kind of pyramid it is. A pyramid with a triangular base is called a triangular pyramid, and a pyramid with a rectangular base is called a rectangular pyramid. A triangular pyramid has four faces, four vertices, and six edges. A rectangular pyramid has five faces, five vertices, and eight edges.

Surface Area of Three-Dimensional Figures
Each face of a prism or a pyramid is a polygon, a two-dimensional figure. For example, a triangular prism has triangles as its bases and three triangles as its lateral faces. A rectangular pyramid has a rectangle as its base and four triangles as its lateral faces.
The surface area of a three-dimensional figure is the total area covered by all of its faces together. In other words, the surface area of a three-dimensional figure is equal to the sum of the areas of its bases and lateral faces. If we spread the surface of a three-dimensional figure into a flat surface, we get a two-dimensional shape. This flat surface is called the net of the three-dimensional figure.

Nets
A net is the two-dimensional shape that results if you unfold the faces of a three-dimensional figure. There can be more than one net for the same three-dimensional figure. For example, a cube has 11 different nets.
A three-dimensional figure and its net have the same number of faces, and their faces have the same shapes. The faces of the net of a geometric figure are connected by their edges. Visualizing how a three-dimensional figure opens up into its net can help you understand how the net matches the original shape.
Let's see how the cube in the upper image is opened up to its net. A cube has six faces in all, of which two are the bases and four are the lateral faces. You can fold the net shown in the center image into the cube in the upper image. But if you fold the net in the lower image along the edges, you will not get a cube because you cannot form the top and bottom faces.

Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we can make use of its net. A triangular prism has two congruent triangles as its bases and three rectangles as its lateral faces. So if you open up a triangular prism, the resulting net will have two triangles and three rectangles.
The image shows you a triangular prism and its net. The edge lengths of the prism are labeled p, q, and r. You can see how the labels in the polygons in the net match those in the prism. In this net, the base triangle is an isosceles triangle. If the base triangle were an equilateral triangle, then r and p would be equal lengths.

12

Let's find the surface area of a triangular prism. The surface area is the sum of
the areas of all the faces. A triangular prism has two triangles as its bases and
three rectangular lateral faces. So, we can write this equation for the surface area of the triangular prism: surface area of triangular prism = area of two triangles + area of three rectangles.
We'll apply the formula for the area of a triangle and multiply the area by 2 to get the total area of the two triangular faces:

Multiple Choice

Which three-dimensional figure does the net in the image represent?

• a square prism

• a triangular prism

• a cube

• a rectangular prism

Surface Area of a Rectangular Prism
You know that a rectangular prism has two rectangles as bases and four rectangles as lateral faces, as you can see in the three-dimensional figure and the net shown here. The length, width, and height of the prism are marked l, w, and h, respectively. Opposite pairs of faces are congruent and are shaded the same color in the image.
To find the surface area of the prism, let's list the area of each face and add all the areas. Because opposite pairs of faces are congruent and there are six sides, there are three pairs of sides with identical areas:
rectangle 1: area = l × h
rectangle 3: area = l × h
rectangle 2: area = l × w
rectangle 4: area = l × w
rectangle 5: area = h × w
rectangle 6: area = h × w
So, the surface area of a rectangular prism with length l, width w, and height h is given by this equation:
surface area of a rectangular prism = 2(l × h) + 2(l × w) + 2(h × w)
= 2[(l × h) + (l × w) + (h × w)].

A cube has 11 nets.

A cube is a rectangular prism whose faces are all congruent squares. Its length, width, and height are all equal: length = width = height.
For a cube with side length a, we can derive this formula for the surface area of a cube:
area of one face = area of a square = a2
surface area of a cube = 6 × area of one face
= 6 × a2= 6a2.
For example, the surface area of a cube whose side is 3 cm long is
6 × 3 cm × 3 cm
= 6 × 9 cm2= 54 cm2.

16

What is the surface area of a cube if each side length is 5 inches?

150 square inches

125 square inches

180 square inches

100 square inches

Correct. The surface area of a cube is the sum of the
areas of its six faces:
area of one face = 5 inches × 5 inches = 25 square
inches
surface area of the cube = 6 × 25 square inches = 150
square inches.

Multiple Choice

What is the surface area of a cube if each side length is 5 inches?

• 100 square inches
• 150 square inches
• 50 square inches
• 25 square inches

Let's calculate the surface area of a rectangular prism from its net. The rectangular prism in the image is 8 centimeters long, 12 centimeters wide, and 5 centimeters high. There are three different edge lengths, so the net of this three-dimensional figure contains three pairs of congruent rectangles.
The area of each rectangle is the product of its length and its width. The pairs of rectangles have three different areas:
•8 cm × 5 cm
•8 cm × 12 cm
•5 cm × 12 cm
We find the surface area of the prism by adding the areas of all the rectangles:
surface area of the rectangular prism = 2(l × h) + 2(l × w) + 2(h × w)
= (2 × 8 cm × 5 cm) + (2 × 8 cm × 12 cm) + (2 × 5 cm × 12 cm)
= 80 cm²+ 192 cm² + 120 cm²
= 392 cm².

Let's consider a triangular pyramid whose faces are all
congruent equilateral triangles. In this case, surface area will be
four times the area of one triangle:

Jason wants to make a cloth model of a pyramid for a school project. He wants the base to be a square that is 30 centimeters wide. The slant height of each lateral face of the pyramid is 26 centimeters. Jason wants to know how much cloth he will need to make the model pyramid. In other words, he needs to know the surface area of the pyramid. Let's help him out.

To find the surface area of a pyramid with a rectangle as the base, we add the areas of the base rectangle and the lateral triangles: surface area of a rectangular pyramid = area of base rectangle + areas of lateral triangles.
We calculate the area of the base:

Open Ended

Answer the question below. Type your response in the space provided.

What is the surface area of the pyramid formed from the net shown here? The triangles are equilateral, and each triangle has a height of 5.2 centimeters. Write your answer rounded to one decimal point and with the correct unit.

Summary
Three-dimensional figures are geometric figures whose sizes we can specify using three quantities: length, width, and height. The surface area of a three-dimensional figure is the sum of the areas of its bases and its lateral faces. To find the surface area of a three-dimensional figure, we can spread it into a flat shape called a net.
A prism is a three-dimensional figure whose bases are parallel, congruent polygons and whose lateral faces are rectangles. A triangular prism has bases that are triangles. A rectangular prism has bases that are rectangles. We use these formulas to find surface areas of prisms:
• surface area of a triangular prism= 2 × area of base triangle + area of lateral rectangles
• surface area of a rectangular prism = 2(l × h) + 2(l × w) + 2(h × w)
• surface area of a cube with side a = 6a2
A pyramid is a three-dimensional figure whose base is a polygon and whose lateral faces are all triangles. A triangular pyramid has a triangular base, and a rectangular pyramid has a rectangular base. We use these formulas to find surface areas of pyramids:
• surface area of a triangular pyramid = area of base triangle +areas of 3 lateral triangles
• surface area of an equilateral triangular pyramid = 2bh
• surface area of a rectangular pyramid = area of base rectangle + areas of 4 lateral triangles

Lesson Activity

Surface Area of a Triangular Prism

Activity

In his science class, Boris conducted an experiment in which he used a glass prism to make a rainbow. He now wants to find the surface area of the prism. His teacher gets him a paper model of the prism and spreads it into a flat shape called a net. Boris can see that the net has exactly the same number of faces as the prism. The teacher tells Boris that he can use this net to find the surface area of the prism. Use this image of the net of the prism that Boris’s teacher drew to answer the questions in this activity.

Open Ended

Part A

What is the area of triangle i? Show your calculation.

Open Ended

Part B

Triangles i and ii are congruent (of the same size and shape). What is the total area of triangles i and ii? Show

your calculation.

Open Ended

Part C

What is the area of rectangle i? Show your calculation.

Open Ended

Part D

What is the area of rectangle ii? Show your calculation.

Open Ended

Part E

Rectangles i and iii have the same size and shape. What is the total area of rectangles i and iii? Show your

calculation.

Open Ended

Part F

What is the total area of all the rectangles? Show your calculation.

Open Ended

Part G

What areas do you need to know to find the surface area of the prism?

Open Ended

Part H

What is the surface area of the prism? Show your calculation.

Open Ended

Part I

Read this statement: “If you multiply the area of one rectangle in the figure by 3, you’ll get the total area of the

rectangles.” Is this statement true or false? Why?

Open Ended

Part J

Read this statement: “If you multiply the area of one triangle in the figure by 2, you’ll get the total area of the

triangles.” Is this statement true or false? Why?

Open Ended

Self-Evaluation

How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of

your work below. Note what you learned and what challenged you.

Congratulations!

You have completed the tutorial

Three-Dimensional Figures.