1. Module 4

1.21. Page 4

Mathematics 10-3 Module 4 Lesson 4

Module 4: Area

 

Bringing Ideas Together

 

In Get Started and Explore you reviewed the nets of cubes, rectangular prisms, and pyramids. You can use nets to develop formulas for surface area.

 

m10_3_selfcheck.jpgSelf-Check

 

Take a look at the net for a cube with s units on a side.

 

This illustration shows a cube with side lengths “s.” The illustration also shows a net for the cube with the area of one of the faces given as s squared.

 

Answer the following questions. When you are finished, check your answers.

 

SC 4. What do you notice about the areas of the squares in the net?

 

SC 5. How many squares make up the net for a cube?

 

SC 6. What would you suggest for the formula for the surface area of a cube?

 

Compare your answers.

 

Next, consider the net for a rectangular prism with length l script.eps, width w, and height h.

 

This illustration shows a rectangular prism with length l, width w, and height h. The illustration also shows a net for the prism with the dimensions of the faces given.

 

There are two faces with area eqn213.eps

 

There are two faces with area eqn214.eps

 

There are two faces with area hw.

 

So, the surface area of a rectangular prism is eqn215a.eps

 

If you worked on a rectangular prism in the Share activity, how does the formula you came up with compare with this formula?

 

Surface Area by Example

 

In the next few examples, you will sometimes apply these formulas and you will sometimes need to sketch the faces of the object to determine the surface area.

 

This is a photo of sugar cubes.

© Feng Yu/24837881/Fotolia

Example 1

 

What is the surface area of a sugar cube measuring 0.5 in on each side?

 

Solution

 

Each side, s, of the cube is 0.5 in.

 

 

eqn087.eps

 

The surface area of this sugar cube is 1.5 in2.

 

This answer is reasonable because, if you rearranged the six faces of the cube, the faces would form one square with sides of 1 in and a rectangle half the area of the square.

 

 

This illustration shows two groups of squares. One group is four squares arranged into a larger square. The other group is two squares beside each other.

 

Example 2

 

What is the surface area of a cardboard container 2 ft long, 18 in wide, and 1 ft high?

 

Solution

 

All dimensions must be in the same units.

 

 

eqn091.eps

 

So, eqn211.eps

 

 

eqn092.eps

 

The surface area of the box is 13 ft2.

 

You could also solve this question using square inches. Since 1 ft is 12 in, 2 ft is 2 × 12 = 24 in. Then, l script.eps= 24 in, w = 18 in, and h = 12 in.

 

 

eqn103.eps

 

The surface area of the box is 1872 in2.

 

Example 3

 

What is the surface area of a pyramid with a square base measuring 10 cm on a side and 4 identical triangular faces with heights of 8 cm measured along the face?

 

Solution

 

In this solution you will see how to find the surface area of a pyramid. As you watch, pause the recording and try to predict the next step in solving the question.

 

View “Example 3 Solution.”

 

Example 4

 

Find the surface area of the triangular prism. Each triangular end is 4 cm along the base, 3 cm high, and 5 cm along the longest side. The prism is 10 cm long.

 

 

This illustration shows a triangular prism with a height of 3 centimetres, a width of 4 centimetres, and a length of 10 centimetres. The slant height of 5 centimetres is also shown.

 

Solution

 

Sketch the faces of the prism and find their areas.

 

The base of the prism is a rectangle 10 cm by 4 cm.

 

 

This illustration shows a rectangle 4 centimetres wide and 10 centimetres long.

 

 

eqn110.eps

 

The back of the prism is a rectangle 10 cm by 3 cm.

 

 

This illustration shows a rectangle with a width of 3 centimetres and a length of 10 centimetres.

 

 

eqn112.eps

 

The slant face of the prism is a rectangle 10 cm by 5 cm.

 

 

This illustration shows a rectangle of width 5 centimetres and length of 10 centimetres.

 

 

eqn113.eps

 

Each triangular end is a triangle with a base of 4 cm and a height of 3 cm.

 

 

This illustration shows two triangles. Each has a height of 3 centimetres and a base of 4 centimetres.

 

 

eqn114.eps

 

surface area of prism = area of base + area of back + area of slant face + (2 × area of triangle)

 

 

eqn118.eps

 

The area of the prism is 132 cm2.

 

Now check your mastery of surface area.

 

m10_3_selfcheck.jpg Self-Check

 

Answer the following questions. When you are finished, check your answers.

 

SC 7. A cubical shipping carton is 30 in on a side. What is the carton’s surface area in cubic feet?

 

SC 8. Over 90% of non-bulk goods shipped by ship, rail, and truck are in rectangular containers. A common container is 48 ft long, 8 ft wide, and 8 ft high. What is the surface area of this container?

 

SC 9. Jim is going to paint the walls and ceiling of his room. His room is 9 ft wide, 12 ft long, and 8 ft high. He wishes to estimate the area he will cover in order to buy enough paint. What is Jim’s estimate? (Notice that Jim won’t be painting the floor. In doing your estimate, you can ignore the door and window in Jim’s room.)

 

SC 10. A tetrahedron has 4 identical triangular faces. What is the surface area of the tetrahedron shown below?

 

This illustration shows a tetrahedron with a slant height of 5 centimetres and a side length of 5.8 centimetres.

 

Compare your answers.

 

m10_3_mastcon.jpg Mastering Concepts

 

Try this question. When you are finished, check your answer.

 

Star plans to sew a slip cover for her couch. She needs to estimate the fabric required. How can she minimize her calculations?

 

This illustration shows a couch consisting of several rectangular prisms. The seat is one rectangular prism, the two arms are rectangular prisms, and the back is a rectangular prism.

 

Compare your answers.