lesson 7.1.3

1

Lesson 7.1.3Lesson 7.1.3

Surface Areas of Cylinders

and Prisms

Surface Areas of Cylinders

and Prisms

2

Lesson

7.1.3Surface Areas of Cylinders and PrismsSurface Areas of Cylinders and Prisms

California Standards:Measurement and Geometry 2.1Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

Measurement and Geometry 3.5Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones.

Mathematical Reasoning 1.3Determine when and how to break a problem into simpler parts.

What it means for you:You’ll see how to work out the surface area of 3-D shapes like cylinders and prisms.

Key words:• net• surface area• cylinder• prism

3

Lesson


Nets are very useful for finding the surface area of 3-D shapes. They change a 3-D problem into a 2-D problem.

4

Draw a Net to Work Out the Surface Area

Lesson


The net of a three-dimensional solid can be folded to make a hollow shape that looks exactly like the solid.

The surface area of a three-dimensional solid is the total area of all its faces — it’s the area you’d paint if you were painting the shape.

So one way to work out the surface area of the solid is to work out the surface area of the net.

5

Example 1

Solution follows…

Lesson


What is the surface area of this cube?

Solution

The area of each square is 8 × 8 = 64 in2.

The net of the cube is six squares. So the surface area of the cube is equal to the area of six squares.

So the surface area of the entire cube is 6 × 64 = 384 in2.

8 in

8 in

6

Example 2

Solution follows…

Lesson


What is the surface area of this prism?

Solution

The net of this prism has three identical rectangles.

The area of each rectangle is 10 × 20 = 200 cm2.

So the total surface area of the three rectangles is 3 × 200 = 600 cm2. 20 cm

8.7 cm10 cm

10 cm

10 cm

10 cm

10 c

m

20 cm 10 cm

8.7 cm

600 cm2

Solution continues…

7

Example 2

Lesson


What is the surface area of this prism?

Solution (continued)

20 cm

8.7 cm10 cm

10 cm

10 cm

10 cm

10 c

m

20 cm 10 cm

8.7 cmThere are also two identical triangles.

Each has a base of 10 cm and a height of 8.7 cm. The area of each triangle is 0.5 × 10 × 8.7 = 43.5 cm2.

So the surface area of both the triangles together is 2 × 43.5 = 87 cm2.

87 cm2

So the total surface area of the prism is 600 + 87 = 687 cm2.

8

Guided Practice

Solution follows…

Lesson


Work out the surface area of the shapes shown in Exercises 1–3.

1. 2.

3.

96 + 60 + 80 = 236 in2

6 in8 in

5 in 9 cm

10 cm3 cm

9 cm

2 m

7 m

30 m

Height = 8 cm

54 + 30 + 80 = 164 cm2

28 + 120 + 420 = 568 m2

9

Finding the Surface Area of Cylinders

Lesson


The net of a circular cylinder has a rectangle and two circles.

So you need to use the formula for the area of a circle to find its surface area.

10

Example 3

Solution follows…

Lesson


What is the surface area of this cylinder? Use = 3.14.

Solution

The net of the cylinder has one rectangle

and two identical circles.

3 ft

5 ft

3 ft

5 ft

To work out the area of the rectangle, you need to know its length.

It’s the same as the circumference of the circles, so it is 3 × = 9.42 ft.

9.42 ft


11

Example 3

Lesson




3 ft

5 ft

3 ft

5 ft

9.42 ft

So the area of the rectangle

is 9.42 × 5 = 47.1 ft2.The circles have a diameter of 3 feet. So they have a radius of 1.5 feet.

The area of each circle is × 1.52 = × 2.25 = 7.065 ft2.

47.1 ft2

7.065 ft2

7.065 ft2


12

Example 3

Lesson




3 ft

5 ft

3 ft

5 ft

9.42 ft

Together the two circles have a surface area of 2 × 7.065 = 14.13 ft2.

So the total surface area of the cylinder is 47.1 + 14.13 = 61.23 ft2.

47.1 ft2

7.065 ft2

7.065 ft2

13

Guided Practice

Solution follows…

Lesson


Find the surface areas of the cylinders in Exercises 4–6. Use = 3.14.

4. 5. 6. 2 in

10 in 3 ft

3 ft

1 yd

9 yd

2 × 3.14 = 6.286.28 × 10 = 62.83.14 × 12 = 3.143.14 × 2 = 6.2862.8 + 6.28 = 69.08 in2

3 × 3.14 = 9.429.42 × 3 = 28.263.14 × 1.52 = 7.0657.065 × 2 = 14.1328.26 + 14.13 = 42.39 ft2

2 × 3.14 = 6.286.28 × 9 = 56.523.14 × 12 = 3.143.142 × 2 = 6.2856.52 + 6.28 = 62.8 yd2

14

Use Formulas For Prism and Cylinder Surface Areas

Lesson


The part between the bases is sometimes called the lateral area.

The way you work out the surface area of a cylinder, and the way you work out the surface area of a prism are similar. The surface area of either is twice the area of the base plus the area of the part between the bases of the net.

Area = (2 × base) + lateral area

Lateral

areaLateral

area

Two basesTwo bases

15

Independent Practice

Solution follows…

Lesson


Work out the surface areas of the shapes shown in Exercises 1–4. Use = 3.14.

1. 2.

3. 4.

6 cm

5 cm

1 cm

30 in

20 in

10 ft

8 ft

10 ft

30 ft

Vertical height = 6.8 feet

7 in

7 in

7 in

82 cm2 2512 in2

894.4 ft2 294 in2

16


Solution follows…

Lesson


5. A statue is to be placed on a marble stand, in the shape of a regular-hexagonal prism.

Find the area of the stand’s base, given that the stand has a surface area of 201.5 square feet and dimensions as shown.

3 ft

6 ft

46.75 ft2

17


Solution follows…

Lesson


The inside of a large tunnel in a children’s play area is to be painted. The tunnel is 6 meters long and 1 meter tall. It is open at each end.

6. What is the area to be painted?

7. Cans of paint each cover 5 m2. How many cans do they need to buy?

1 m

6 m

18.84 m2

4

18

Round UpRound Up

Lesson


Working out the surface area of a 3-D shape means adding together the area of every part of the outside.

One way to do that is to add together the areas of different parts of the net.

Just make sure you can remember the triangle, rectangle, and circle area formulas.

lesson 7.1.3

Documents