Lesson 8.7 – Surface Area HW: 8.7/1-10

Let’s start in the beginning…

Before you can do surface area or volume, you have to know the following formulas.

Circle A = π r²

C = πd

Rectangle A = lw

Triangle A = ½ bh

Surface Area• What does it mean to you?• Does it have anything to do with what is in

the inside of the prism.?• Surface area is found by finding the area

of all the sides and then adding those answers up.

• How will the answer be labeled?• Units2 because it is area!

SA of Prisms

Find the SA of any prism by using the basic formula for SA:

SA = 2B + LSA– LSA= Lateral Surface Area– LSA= perimeter of the base ● height of the prism– B = area of the base of the prism.

Rectangular PrismHow many faces are on here? 6

Find the area of each of the faces.

A

B

C

4 in

5 in

6 inDo any of the faces have the same area?

A = 5 x 4 = 20 x 2 =40

B = 6 x 5 = 30 x 2 = 60

C = 4 x 6 = 24 x 2 = 48

If so, which ones?

148 in2

Opposite faces are the same.

Find the SA

SA of a Rectangular Prism

SA = 2bh+2bw+2hw

bw

h

SumSum of the areasareas of all the

faces.

CubeAre all the faces the same? YES

4m How many faces are there? 6

Find the Surface area of one of the faces.

A

4 x 4 = 16

Times the number of faces* 6

96 m2 SA for a Cube = 6BSA = 6 ● area of the base

Area of one base/face

TOP

BOTTOM

BACK

FRONT

SIDE SIDE

Triangular PrismHow many faces are there? 5

How many of each shape does it take to make this prism?

2 triangles and 3 rectangles = SA of a triangular prism

4

3

5

10 m

½ (4 ∙ 3) = 6How many triangles were there? 2 ∙ 2= 12

Find the area of the 3 rectangles.

5 ∙ 10 = 50 = front

4 ∙ 10 = 40 = back

3 ∙ 10 = 30 = bottom

SA = 132 m2What is the final SA?

Example:

8mm

9mm

6 mm 6mm

Find the AREA of each SURFACE

1. Top or bottom triangle:

A = ½ bh

A = ½ (6)(6)

A = 18

2. The two dark sides are the same.

A = lw

A = 6(9)

A = 54

3. The back rectangle is different

A = lw

A = 8(9)

A = 72

ADD THEM ALL UP!

18 + 18 + 54 + 54 + 72

SA = 216 mm²

Cylinders

6 10m

What does it take to make this?2 circles and 1 rectangle= a cylinder

2 B B = π x 9 = 9π * 2 = 18π

+ LSA(C x H)

SA = 18π + 60 π

Formula SA = 2B + LSALSA is a rectangle with b = circumference of Base

H = height of cylinder

SURFACE AREA of a CYLINDER.

You can see that the surface is made up of two circles and a rectangle.

The length of the rectangle is the same as the circumference of the circle!

Imagine that you can open up a cylinder like so:

EXAMPLE: Round to the nearest hundredth.

Top or bottom circle

A = πr²

A = π(3.1)²

A = (9.61) π

A ≈ 30.1754

Rectangle

Length = Circumference

C = π d

C = (6.2) π

C ≈ 19.468

Now add:

SA = 30. 1754 + 30. 1754 + 233.62Now the area

A = lw

A ≈ 19.468(12)

A ≈ 233.62 SA ≈ 293.97 in²

Find the surface area of the triangular prism.

Area of each triangle is 12m² (there are two of them)

Area of one of the rectangles is 63m² (7 9)∙

9 m

Area of another one of the rectangles is 56m² (7 8)∙

Area of the final rectangles is 21m² (7 3)∙

Find the surface area of the triangular prism.

12m² + 12m² + 63m² + 56m² + 21m² 9 m

S.A. = 164m²

Definition• Surface Area – is the total number of unit

squares used to cover a 3-D surface.

Find the SA of a Rectangular Solid

Front

Top

RightSide

A rectangular solid has 6 faces.

They are: Top Bottom Front Back Right Side Left Side

Which of the 6 sides are the same? Top and Bottom Front and Back Right Side and Left Side

We can only see 3 faces at any one time.

Surface Area of a Rectangular Solid

Front

Top

RightSide

We know thatEach face is a rectangle.

and theFormula for finding the area of a

rectangle is:A = lw

Steps:Find:

Area of Top Area of Front Area of Right Side

Find the sum of the areas Multiply the sum by 2.

The answer you get is the surface area of the rectangular solid.

Find the Surface Area of the following:

12 m

8 m

5 m

Top

Front

RightSide

Front

Top

RightSide

Find the Area of each face:12 m

5 m

12 m

8 m

5 m

8 m

A = 12 m x 5 m = 60 m2

A = 12 m x 8 m = 96 m2

A = 8 m x 5 m = 40 m2

Sum = 60 m2 + 96 m

2 + 40 m

2 = 196 m

2

Multiply sum by 2 = 196 m2 x 2 = 392 m

2

The surface area = 392 m2

A. 22 in2

B. 36 in2

C. 76 in2

D. 80 in2

Find the surface area of the rectangular prism.

46 in2

Find the surface area of the rectangular prism.

S.A. of a Triangular Prism

There is NO FORMULA!There is NO FORMULA!

Simply find the area of each face Simply find the area of each face and add them all togetherand add them all together.

Find the Surface Area

6 cm

4 cm

14 cm

Area of Top = 6 cm x 4 cm = 24 cm2

Area of Front = 14 cm x 6 cm = 84 cm2

Area of Right Side = 14 cm x 4 cm = 56 cm2

Find the sum of the areas:

24 cm2 + 84 cm

2 + 56 cm

2 = 164 cm

2

Multiply the sum by 2:

164 cm2 x 2 = 328 cm

2

The surface area of this rectangular solid is 328 cm

2.

24 m2

84 m2 56 m

2

NetsNets

• A net is all the surfaces of a rectangular solid laid out flat.

Top

RightSide

Front

10 cm

8 cm

5 cm

Top

Bottom

Front

Back

Right SideLeft Side

10 cm

5 cm

8 cm

8 cm

8 cm

5 cm

Top

RightSide

Front

10 cm

8 cm

5 cm

Top

Bottom

Front

Back

Right SideLeft Side

10 cm

5 cm

8 cm

8 cm

8 cm

5 cm

Find the Surface Area using nets.

Each surface is a rectangle.A = lw

Find the area of each surface.Which surfaces are the same?Find the Total Surface Area.

80

80

50 50

40

40

What is the Surface Area of the Rectangular solid?

340 cm2

Why should you learn about surface area?Is it something that you will ever use in

everyday life?If so, who do you know that uses it?Have you ever had to use it outside of

math?

You can tell the base and height of a triangle by finding the right angle:

TRIANGLES

CIRCLESYou must know the difference between RADIUS and DIAMETER.

r

d

Let’s start with a rectangular prism.

Surface area can be done using the formula

SA = 2 lw + 2 wl + 2 lw OR

Either method will gve you the same answer.

you can find the area for each surface and add them up.

Volume of a rectangular prism is V = lwh

Example:

7 cm

4 cm

8 cm Front/back 2(8)(4) = 64

Left/right 2(4)(7) = 56

Top/bottom 2(8)(7) = 112

Add them up!

SA = 232 cm²

V = lwh

V = 8(4)(7)

V = 224 cm³

To find the surface area of a triangular prism you need to be able to imagine that you can take the prism apart like so:

Notice there are TWO congruent triangles and THREE rectangles. The rectangles may or may not all be the same.

Find each area, then add.

There is also a formula to find surface area of a cylinder.

Some people find this way easier:SA = 2πrh + 2πr²

SA = 2π(3.1)(12) + 2π(3.1)²

SA = 2π (37.2) + 2π(9.61)

SA = π(74.4) + π(19.2)

SA = 233.7 + 60.4

SA = 294.1 in²

The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

Find the radius and height of the cylinder.

Then “Plug and Chug”…

Just plug in the numbers then do the math.

Remember the order of operations and you’re ready to go.

The formula tells you what to do!!!!

2πrh + 2πr² means multiply 2(π)(r)(h) + 2(π)(r)(r)

Volume of Prisms or Cylinders

You already know how to find the volume of a rectangular prism: V = lwh

The new formulas you need are:

Triangular Prism V = (½ bh)(H)

h = the height of the triangle and

H = the height of the cylinder

Cylinder V = (πr²)(H)

Volume of a Triangular Prism

We used this drawing for our surface area example. Now we will find the volume.

V = (½ bh)(H)

V = ½(6)(6)(9)

V = 162 mm³

This is a right triangle, so the sides are also the base and height.

Height of the prism

Try one:

Can you see the triangular bases?

V = (½ bh)(H)

V = (½)(12)(8)(18)

V = 864 cm³

Notice the prism is on its side. 18 cm is the HEIGHT of the prism. Picture if you turned it upward and you can see why it’s called “height”.

V = (πr²)(H)

V = (π)(3.1²)(12)

V = (π)(3.1)(3.1)(12)

V = 396.3 in³

Volume of a Cylinder

We used this drawing for our surface area example. Now we will find the volume.

optional step!

Try one:

10 m

d = 8 m

V = (πr²)(H)

V = (π)(4²)(10)

V = (π)(16)(10)

V = 502.7 m³Since d = 8,

then r = 4

r² = 4² = 4(4) = 16

Here are the formulas you will need to know:

A = lw SA = 2πrh + 2πr²

A = ½ bh V = (½ bh)(H)

A = π r² V = (πr²)(H)

C = πd

and how to find the surface area of a prism by adding up the areas of all the surfaces