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

2. Rectangle A = lw Triangle A = ½ bh Circle A = π r² C = πd
Let’s start in the beginning…
Before you can do surface area or volume, you have to know the following formulas.
Rectangle A = lw
Triangle A = ½ bh
Circle A = π r²
C = πd

3. 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!

4. 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.

5. Rectangular Prism B C A 5 in 6 in 4 in How many faces are on here? 6
Find the area of each of the faces.
Do any of the faces have the same area?
If so, which ones?
Opposite faces are the same.
A = 5 x 4 = 20 x 2 =40
B = 6 x 5 = 30 x 2 = 60
Find the SA
C = 4 x 6 = 24 x 2 = 48
148 in2

6. SA of a Rectangular Prism Sum of the areas of all the faces.
SA = 2bh+2bw+2hw
Sum of the areas of all the faces. h w b

7. Cube SA for a Cube = 6B Are all the faces the same? YES A
How many faces are there? 6 4m
Find the Surface area of one of the faces.
4 x 4 = 16
Area of one base/face
* 6
96 m2
Times the number of faces
SA for a Cube = 6B
SA = 6 ● area of the base

8. BACK
SIDE
BOTTOM
SIDE
FRONT
TOP

9. Triangular Prism ∙ 2= 12 How many faces are there? 5 4 5
How many of each shape does it take to make this prism?
10 m 3
2 triangles and 3 rectangles = SA of a triangular prism
How many triangles were there? 2
½ (4 ∙ 3) = 6
∙ 2= 12
5 ∙ 10 = 50 = front
4 ∙ 10 = 40 = back
3 ∙ 10 = 30 = bottom
Find the area of the 3 rectangles.
What is the final SA?
SA = 132 m2

10. 3. The back rectangle is different
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
Example:
8mm
9mm
6 mm mm
3. The back rectangle is different
A = lw
A = 8(9)
A = 72
ADD THEM ALL UP!
SA = 216 mm²

11. Imagine taking the prism apart.
Find the area of each shape.
Add all of the areas together to find the surface area.

12. The area of each triangular base is 35 ft².
The area of two of the rectangular faces is ft².
The area of the third rectangular face is 120 ft².
Add all the areas together to get the surface area: or ft².

13. Cylinders SA = 18π + 60 π Formula  SA = 2B + LSA
What does it take to make this?
2 circles and 1 rectangle= a cylinder 6
10m
Formula  SA = 2B + LSA
LSA is a rectangle with b = circumference of Base
H = height of cylinder
2 B 
B = π x 9 = 9π
* 2 = 18π
+ LSA(C x H)
SA = 18π + 60 π

14. SURFACE AREA of a CYLINDER.
Imagine that you can open up a cylinder like so:
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!

15. EXAMPLE: Round to the nearest hundredth.
Top or bottom circle
A = πr²
A = π(3.1)²
A = (9.61) π
A ≈
Rectangle
Length = Circumference
C = π d
C = (6.2) π
C ≈
Now add:
SA =
Now the area
A = lw
A ≈ (12)
A ≈
SA ≈ in²

17. 9 m 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)
Area of another one of the rectangles is 56m² (7∙8)
Area of the final rectangles is 21m² (7∙3)

18. 9 m S.A. = 164m² Find the surface area of the triangular prism.
12m² + 12m² + 63m² + 56m² + 21m²
S.A. = 164m²

19. Definition
Surface Area – is the total number of unit squares used to cover a 3-D surface.

20. Find the SA of a Rectangular Solid
A rectangular solid has 6 faces.
They are:
Front
Top
Bottom
Front
Back
Right Side
Left Side
Top
Right
Side
We can only see 3 faces at any one time.
Which of the 6 sides are the same?
Top and Bottom
Front and Back
Right Side and Left Side

21. Surface Area of a Rectangular Solid
We know that
Each face is a rectangle.
and the
Formula for finding the area of a rectangle is:
A = lw
Front
Top
Right
Side
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.

22. Find the Surface Area of the following:
Find the Area of each face:
12 m
Top
5 m
A = 12 m x 5 m = 60 m2
Top
Right
Side
Front
8 m
Front
8 m
A = 12 m x 8 m = 96 m2
5 m
12 m
12 m
Right
Side
Sum = 60 m m m2 = 196 m2
A = 8 m x 5 m = 40 m2
8 m
Multiply sum by 2 = 196 m2 x 2 = 392 m2
5 m
The surface area = 392 m2

23. Find the surface area of the rectangular prism. A. 22 in2 B. 36 in2
A. 22 in2
B. 36 in2
C. 76 in2
D. 80 in2

24. Find the surface area of the rectangular prism.
46 in2

25. S.A. of a Triangular Prism
There is NO FORMULA!
Simply find the area of each face and add them all together.

26. The surface area of this rectangular solid is 328 cm2.
Find the Surface Area
Area of Top = 6 cm x 4 cm = 24 cm2
Area of Front = 14 cm x 6 cm = 84 cm2
24 m2
Area of Right Side = 14 cm x 4 cm = 56 cm2
Find the sum of the areas:
56 m2
14 cm
84 m2
24 cm cm cm2 = 164 cm2
Multiply the sum by 2:
4 cm
164 cm2 x 2 = 328 cm2
6 cm
The surface area of this rectangular solid is 328 cm2.

27. Nets A net is all the surfaces of a rectangular solid laid out flat.
Back
8 cm
Top
Left Side
Top
Right Side
5 cm
Right
Side
8 cm
Front
8 cm
Front
8 cm
5 cm
10 cm
Bottom
5 cm
10 cm

28. Each surface is a rectangle.
Find the Surface Area using nets.
Top
Back
8 cm
Right
Side
Front
8 cm
Left Side
Top
Right Side
5 cm
5 cm
8 cm
10 cm
Front
8 cm
Each surface is a rectangle.
A = lw 80 80
Bottom
5 cm
Find the area of each surface.
Which surfaces are the same?
Find the Total Surface Area. 40
10 cm 50 50 40
What is the Surface Area of the Rectangular solid?
340 cm2

29. SURFACE AREA 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?

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

31. CIRCLES
You must know the difference between RADIUS and DIAMETER. r d

32. Either method will gve you the same answer.
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

33. 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

34. 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.

35. 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 =
SA = in²
The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

36. The formula tells you what to do!!!!
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)

37. Cylinder V = (πr²)(H) 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)

38. V = 162 mm³ V = (½ bh)(H) V = ½(6)(6)(9) 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

39. V = 864 cm³ V = (½ bh)(H) V = (½)(12)(8)(18) 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”.

40. 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.
V = (πr²)(H)
V = (π)(3.1²)(12)
V = (π)(3.1)(3.1)(12)
V = in³
optional step!

41. V = (πr²)(H) V = (π)(4²)(10) V = (π)(16)(10) V = 502.7 m³ Try one:
d = 8 m
V = (πr²)(H)
V = (π)(4²)(10)
V = (π)(16)(10)
V = m³
Since d = 8,
then r = 4
r² = 4² = 4(4) = 16

42. 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