1. Surface Area
&
Volume
Geometry/Trig 2
Fall 2007

2. Rectangular Prism – A three dimensional object with two rectangular bases and four rectangular lateral faces.
Base (B)
Base (B)
Right
Left
Front
Back

3. Surface Area – the sum of the areas of all of the faces (bases and lateral faces) of a solid.
Lateral Area – the sum of the areas of the lateral faces of a prism

4. How do we find Surface Area?
Example 1
Find the area of each face:
Front: ____________
Back: ____________
Top: _____________
Bottom: __________
Left Side: ________
Right Side: _______
Surface: ___________
6cm
80cm2
8cm
80cm2
10cm
60cm2
60cm2
48cm2
48cm2
376cm2

5. How do we find Lateral Area?
Example 1
Find the area of each face:
Front: ____________
Back: ____________
Top: _____________
Bottom: __________
Left Side: ________
Right Side: _______
Lateral Area: ___________
6cm
80cm2
8cm
80cm2
10cm
60cm2
60cm2
48cm2
48cm2
256cm2

6. Formula for Lateral Area:
Formula for Surface Area:
Lateral Area = Perimeter of Base x Height
L.A. = ph
Surface Area = Lateral Area + 2(Area of the Base)
T.A. = L.A. + 2B

7. Example 2 – Find the Lateral Area and the Surface Area
L.A. = Perimeter of Base x Height L.A. = ph
p = = 52m
h = 6m
L.A. = 52 x 6 = 312m2
T.A. = L.A. + 2(Area of the Base) T.A. = L.A. + 2B
L.A. = 312m2
B = 20 x 6 = 120m2
T.A. = (120) = 552m2

8. Example 3 – Find the Lateral Area and the Surface Area
L.A. = Perimeter of Base x Height L.A. = ph
p = = 36in
h = 9in
L.A. = 36 x 9 = 324in2
T.A. = L.A. + 2(Area of the Base) T.A. = L.A. + 2B
L.A. = 324in2
B = 9 x 9 = 81in2
T.A. = (81) = 486in2

9. Volume - the amount of space that an object occupies
The units for Volume are always cubed. Examples: in3, m3, cm3.
Formula for Volume of a Rectangular Prism:
V = Area of the base x height
V = Bh

10. V = Area of the base x height V = Bh
Example 1
6cm
B = 6 x 10 = 60cm2
8cm
h = 8cm
10cm
V = 60 x 8 = 480cm3

11. V = Area of the base x height V = Bh
Example 2 6m 6m
B = 6 x 20 = 120m2
20m
h = 6m
V = 120 x 6 = 720m3

12. V = Area of the base x height V = Bh
Example 3
9in
B = 9 x 9 = 81in2
9in
h = 9in
9in
V = 81 x 9 = 729in3

13. L.A. = ph T.A. = L.A. + 2B V = Bh L.A. = 40 x 5 T.A. = 200 + 2(91)
Find the Lateral Area, Surface Area, and
Volume of the rectangular prism.
Example 4
p = = 40in
7in
h = 5in
5in
B = 7 x 13 = 91in2
13in
L.A. = ph
T.A. = L.A. + 2B
V = Bh
L.A. = 40 x 5
T.A. = (91)
V = 91 x 5
L.A. = 200in2
T.A. = 382in2
V = 455in3

14. We use the same formulas for lateral area, surface area and volume when dealing with other right prisms.
L.A. = ph
T.A. = L.A. + 2B
V = Bh
Triangular Prism
Trapezoidal Prism
Hexagonal Prism

15. Area of the Base (B) = ½(12)(4) = 24m2
Example 1
12m
L.A. = ph = (27)(12) 4m
L.A. = ph = 324m2 8m 7m
12m
T.A. = L.A. + 2B = 324m2 + 2(24)
T.A. = 372m2
V = Bh = (24)(12)
V = 288m3
Area of the Base (B) = ½(12)(4) = 24m2
Perimeter of Base (p) = = 27m
Height (h) = 12m

16. V = Bh = (48)(14) V = 672m3 L.A. = ph = (36)(14) L.A. = 504m2
Example 2
10m
10m 6m
16m
14m
V = Bh = (48)(14)
V = 672m3
L.A. = ph = (36)(14)
L.A. = 504m2
T.A. = L.A. + 2B = (48)
T.A. = 600m2

17. B = ½h(b1 + b2) B = ½10(8 + 30) B = 190m2 V = Bh = (190)(40)
Example 3
10m
B = ½h(b1 + b2)
B = ½10(8 + 30)
40m
B = 190m2
18m
14m 8m
V = Bh = (190)(40)
V = 7600m3
L.A. = ph = (70)(40)
L.A. = 2800m2
T.A. = L.A. + 2B = (190)
T.A. = 3180m2

18. Area of Base (B) = ½ (36) = 54 Area of Base (B) = ½ap 6cm Example 430
Area of Base (B) = ½ap
3cm

19. Cylinders – Cylinders are very similar to the right prisms that we have been examining. The only difference is that instead of polygons (rectangle, triangle, trapezoid, hexagon) as bases, a cylinder has circular bases. The formulas to calculate lateral area, Surface area, and volume will be nearly the same as prisms.

20. The formula for Volume remains the same. (V = Bh)
The formula for Volume remains the same. (V = Bh). Because in this case the base is a circle, we must use the formula for finding area of a circle.
Recall that area of a circle is calculated by using
A = pr2
The Lateral Area and Surface Area are calculated in a similar manner. However we must replace “perimeter of base” with
circumference of base.
C = 2pr

21. Therefore, to calculate Surface Area and Volume of a cylinder you must find three key pieces of information:
1. Area of the Base – pr2
2. Circumference of the Base – 2pr
3. Height of the object - given

22. Radius – Area of Base = Circumference = Height = Volume = = L.A. = =
Example 1 – Find the Lateral Area, Surface Area, and Volume of the Cylinder.
Radius –
Area of Base =
Circumference =
Height =
Volume = =
L.A. = =
T.A. = =
4in
10in
16pin2
8pin
4in
10in
(16p)(10)
160pin3
(8p)(10)
80pin2
80p + 2(16p)
112pin2

23. Radius – Area of Base = Circumference = Height = Volume = = L.A. = =
Example 2 7m
Radius –
Area of Base =
Circumference =
Height =
Volume = =
L.A. = =
T.A. = = 7m
14m
49pm2
14pm
14m
(49p)(14)
686pm3
(14p)(14)
196pm2
196p + 2(49p)
294pm2

24. Radius – Area of Base = Circumference = Height = Volume = = L.A. = =
Example 3
75cm
Radius –
Area of Base =
Circumference =
Height =
Volume = =
L.A. = =
T.A. = =
50cm
2500pcm2
100cm
100pcm
75cm
(2500p)(75)
187500pcm3
(100p)(75)
7500pcm2
7500p + 2(2500p)
12500pcm2

25. Radius – Area of Base = 182.25pin2 Circumference = Height = Volume = =
Example 4
You can have decimals in your answers, if there are decimals in the problems.
Radius –
Area of Base = pin2
Circumference =
Height =
Volume = =
L.A. = =
T.A. = =
22.8in
13.5in
27in
27pin
22.8in
(182.25p)(22.8)
4155.3pin3
(27p)(22.8)
615.6pin2
615.6p + 2(182.25p)
980.1pin2

26. Homework Answers:
p. 6
1. V = 9072m3 L.A. = 3276m T.A. = 3708m2
2. V = 60000m3 L.A. = 11200m2 T.A. = 12400m2
3. V = m3 L.A. = 25500m2 T.A. = 39500m2
p. 7
4. V = 19200m L.A. = 5520m2 T.A. = 6160m2
5. V = L.A. = 720cm2 T.A. =
6. V = 68600m L.A. = 8120m2 T.A. = 10080m2

27. Homework Answers:
p. 11
1. V = 7943pft3 L.A. = 1222pft2 T.A. = 1560pft2
2. V = 864pft3 L.A. = 288pft2 T.A. = 360pft2
3. V = 180pin L.A. = 60pin T.A. = 132pin2

28. Slant Height (l) Height (h) Radius (r) A cone has one circular base.
Day 2 – Pyramids & Cones
A cone has one circular base.
Slant Height (l)
Height (h)
Radius (r)

29. Cone Formula for Volume Volume = Area of the Base x Height
Formula for Lateral Area
L.A. = ½Circumference of Base x Slant Height
Cone
Formula for Surface Area (Surface Area)
Surface Area = Lateral Area + Area of Base

30. Therefore, to calculate Surface Area and Volume of a Cone you must find four key pieces of information:
1. Area of the Base – pr2
2. Circumference of the Base – 2pr
3. Height of the object – h
4. Slant Height - l

31. 6m 36pm2 12pm 8m 10m 60pm2 60p + 36p = 96pm2 96pm3 Radius (r) –
Example 1
Radius (r) –
Area of the Base (B) –
Circumference of Base (C) –
Height (h) –
Slant Height (l) –
Lateral Area (L.A.) –
Surface Area (T.A) –
Volume (V) - 6m
36pm2
10m 8m
12pm 6m 8m
10m
60pm2
60p + 36p = 96pm2
96pm3

32. 9m 81pm2 18pm 12m 15m L.A. = 135pm2 135p+ 81p = 216pm2 324pm3
Example 2
Radius (r) –
Area of the Base (B) –
Circumference of Base (C) –
Height (h) –
Slant Height (l) –
Lateral Area (L.A.) –
Surface Area (T.A) –
Volume (V) - 9m
81pm2
15m
12m
18pm 9m
12m
15m
L.A. = 135pm2
135p+ 81p = 216pm2
324pm3

33. Example 3 –
Lateral Area – 260pcm2
Surface Area – 360pcm2
Volume – 800pcm3
Example 4 –
Lateral Area – 15pin2
Surface Area – 24pin2
Volume – 12pin3

34. A regular square pyramid has a square base.
Slant Height (l)
Height (h)
Base Edge (e)

35. Pyramid Formula for Volume Volume = Area of the Base x Height
Formula for Lateral Area
L.A. = ½ Perimeter of Base x Slant Height
Pyramid
Formula for Surface Area
Surface Area = Lateral Area + Area of Base

36. Therefore, to calculate Surface Area and Volume of a Pyramid you must find four key pieces of information:
1. Area of the Base – e2
2. Perimeter of the Base – 4e
3. Height of the object – h
4. Slant Height - l

37. Example 1
Base Edge (e) –
Height (h) –
Slant Height (l) –
Area of Base (B) –
Perimeter of Base (p) –
Lateral Area (L.A.) –
Surface Area (T.A) –
Volume (V) -
12in
10in = l
8in
8in = h
10in
144in2
12in = e
48in
240in2
= 384in2
384in3

38. Example 2
Base Edge (e) –
Height (h) –
Slant Height (l) –
Area of Base (B) –
Perimeter of Base (p) –
Lateral Area (L.A.) –
Surface Area (T.A) –
Volume (V) -
20in
26in
24in
24in = h
26in = l
24in
26in
10in
400in2
20in = e
80in
1040in2
= 1440in2
3200in3

39. Example 3 – height = 12m
Lateral Area – 260m2
Surface Area – 360cm2
Volume – 400cm3
Example 4 –
Lateral Area – 544ft2
Surface Area – 800ft2
Volume – 1280ft3

40. Spheres

41. Sphere – the set of all points a given distance away from a center point.
Volume -
Surface Area -

42. Example 1 – Find the Surface Area and Volume of the Sphere
Radius – 6in
Volume -
Surface Area -
6in
T.A = 4pr2
T.A = 4p62
T.A = 144pin2

43. Example 2
r = 15mm
V = 4500pmm3
T.A. = 900pmm2
Example 3
r = 1cm
V =
T.A. = 4pcm2
Example 4
r = 13.1ft
V = pft3
T.A. = 686.4pft2

44. Similar Solids Theorem 12-1
If the scale factor of two similar solids is a:b, then
1. The ratio of their perimeters is a:b.
2. The ratio of their base areas, lateral areas, and Surface areas is a2:b2.
3. The ratio of their volumes is a3:b3.

45. Ratio of Surface Areas:
Cubes 6 10
Scale Factor:
Ratio of Surface Areas:
Ratio of Volumes:
6 to 10 = 3 to 5
9 to 25
All Cubes are Similar.
27 to 125

46. Similar Cylinders 2 3
Scale Factor:
Ratio of Surface Areas:
Ratio of Volumes:
3 to 2
9 to 4
27 to 8

47. All Spheres are Similar.8 2
Scale Factor:
Ratio of Surface Areas:
Ratio of Volumes:
8 to 2 = 4 to 1
16 to 1
64 to 1
All Spheres are Similar.

48. Warm Up
The circumference of the earth is approximately 40820km. If you assume that the Earth is a perfect sphere, find its volume and Surface area. (Use p = 3.14).
Answer for Volume will need to be in scientific notation.

50. p. 30 1. Sphere T.A. = 64pm2 2. Cone V = 320pm3
Homework Answers:
p. 30
1. Sphere
T.A. = 64pm2
2. Cone V = 320pm3
L.A. = 136pm2 T.A. = 200pm2
3. Cylinder V = 17.2pcm3
L.A. = 17.2pcm T.A. = 25.2pcm2

51. p. 31 2. Triangular Prism V = 330m3 L.A. = 330m2 T.A. = 390m2
Homework Answers:
p. 31
2. Triangular Prism V = 330m3
L.A. = 330m2 T.A. = 390m2
3. Square Pyramid V = 400m3
L.A. = 260m T.A. = 360m2

52. 3. Trapezoidal Prism V = 40.5in3 L.A. = 78in2 T.A. = 91.5in2
Homework Answers:
p. 32
2. Sphere V = pft3
T.A. = pft2
3. Trapezoidal Prism V = 40.5in3
L.A. = 78in T.A. = 91.5in2

53. Unit 9, Worksheet 1
Sorry! Labels wouldn’t fit. Make sure all of the areas have units2 and all of the volumes have units3. #
L.A
T.A V 1. 28 52 24 2. 48 78 45 3. 54 94 60 4. 72
100 56 5. 36 6. 27 7. 80 64 8. 88 9.
108 #
L.A
T.A V
10. 80
110 75
11.
320
446
630
12.
304
424
480
13.
240
528
720
14.
810
1162
2640
15.
1200
1416
2700
16.
132
356
336
17. 50
700
300
18. 35 52
340

54. 1. Rectangular Prism V = 2400in3 L.A. = 480in2 T.A. = 1280in2
Homework Answers:
p. 17
1. Rectangular Prism V = 2400in3
L.A. = 480in2 T.A. = 1280in2
2. Triangular Prism V = ft3
L.A. = 1334ft2 T.A. = 1659ft2
3. Cylinder V = pm3
L.A. = 17600pm T.A. = 18568pm2

55. 3. Rectangular Prism (Cube) V = 3375in3 L.A. = 900in2 T.A. = 1350in2
Homework Answers:
p. 18
1. Trapezoidal Prism V = 12000m3
L.A. = 4200m2 T.A. = 4600m2
2. Cylinder V = 896pm3
L.A. = 224pm2 T.A. = 352pm2
3. Rectangular Prism (Cube)
V = 3375in3
L.A. = 900in T.A. = 1350in2

56. Homework Answers:
p. 21
1. V = 320pin3 L.A. = 136pin2
T.A. = 200pin2
2. V = 768pft3 L.A. = 240pft2
T.A. = 384pft2
3. V = 100pcm3 L.A. = 65pcm2
T.A. = 90pcm2

57. Homework Answers:
p. 24
1. V = 9680m3 L.A. = 2684m2
T.A. = 3168m2
2. V = 540ft3 L.A. = 369ft2
T.A. = 450ft2
3. V = 1568in3 L.A. = 700in2
T.A. = 896in2