1. page 1

2. Lateral Area, Total Area, & Volume
Geometry/Trig Name: _________________________
Unit 10 – Lateral Area, Total Area and Volume
Lateral Area,
Total Area,
&
Volume
page 2

3. Day 1: Rectangular Prism Notes
Day 1: Rectangular Prism Notes Date: ____________________________________
Rectangular Prism: _____________________________________________________________ ___________________________________________________________________________
Total Area: ___________________________________________________________________ ____________________________________________________________________________
Lateral Area: _________________________________________________________________
How do we find Total Area?
Example 1
Find the area of each face:
Front: ___________ Back: ____________
Top: ____________ Bottom: __________
Left Side: ________ Right Side: _______
Total: ___________
How do you find the Lateral Area? _________________________________________
Formula for the Lateral Area: ________________________________
Formula for the Total Area of a Rectangular Prism: ________________
Example 2
Find the lateral area: ________________
Find the total area: _______________
6cm
8cm
10cm
Lateral Area: ___________
Total Area: __________
Example 3
Find the lateral area: ________________
Find the total area: _______________
9in 6m
20m
Volume: _____________________________________________________________________ Units: ______________________________________________________________________
Formula for Volume of a Rectangular Prism: ___________________________________________
Revisit Example 1: Dimensions: 6cm, 8cm, 10cm Find the volume: __________________
Revisit Example 2: Dimensions: 6m, 6m, 20m Find the volume: __________________
Revisit Example 3: Dimensions: 9in, 9in, 9in Find the volume: __________________
page 3

4. Base is a Regular Hexagon
Other Right Prisms - Notes Date: ____________________________________
page 4
Volume (V) = Area of the Base x height of the prism (V = Bh) Perimeter of Base (p)
Lateral Area (L.A) = Perimeter of Base x height (L.A = ph) height (h)
Total Area (T.A) = Lateral Area + 2(Area of the Base) (T.A. = L.A. + 2B) Area of Base (B)
Example 1 – Triangular Right Prism
Example 2 – Triangular Right Prism
12m 7m 8m
12m
10m
10m 4m 6m 8m 7m
16m
14m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Example 3 – Trapezoidal Right Prism
Example 4 – Hexagonal Right Prism 8m
18m
14m
30m
40m
12cm
6cm
Base is a Regular Hexagon
10m
6cm
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________

5. Unit 10, Worksheet 1 Date: ____________________________________
Find the Volume, Lateral Area, and Total Area of each figure. 1. 2.
100m
42m
42m
24m
20m
28m
36m
18m
12m
18m
24m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________ 3. 4.
100m
60m
75m
16m
40m
70m
24m
28m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________
page 5

6. Base is a Regular Hexagon
Unit 10, Worksheet 2 Date: ____________________________________
Find the Volume, Lateral Area, and Total Area of each figure.
page 6 1.
8cm
Base is a Regular Hexagon 2.
10m 6m 6m 5m
15cm
7.5m
12.5m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________ 3. 4.
4in
70m
16in
20m
24m
22m
28m
50m
Base is a square.
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________

7. Day 2: Cylinder Notes Date: ____________________________________
page 7
A cylinder is like the right prisms with which we have been working this week, except that the bases of a cylinder are circles. The volume and total area can be calculated in a very similar manner.
In a cylinder, the formula for Volume is exactly the same. Multiply the Area of the Base (B) by the height (h). In this case the base is a circle. Recall that the area of a circle is calculated by using A = ________.
The Lateral Area and Total Area is calculated in a similar manner. However we must replace “perimeter of base” with ____________________________________________, use _________
Therefore, to find the Total Area and Volume of a cylinder you must still calculate the same three pieces of information:
1. ________________ of the base – ______________
2. ________________ of the base – _____________
3. Height of the object – given
Example 1
Example 2
14m 7m
10in
4in
Radius – ___________ Height - ____________
Area of Base – _________________________
Circumference of Base – __________________
Volume – _____________________________
Lateral Area - __________________________
Total Area - ___________________________
Radius – ___________ Height - ____________
Area of Base – _________________________
Circumference of Base – __________________
Volume – _____________________________
Lateral Area - __________________________
Total Area - ___________________________

8. All answers must be in feet.
Unit 10, Worksheet 3 Date: ____________________________________
page 8 1. 2.
2yd
47ft
24ft
All answers must be in feet.
13ft
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________ 3. 4.
25m
23ft
10ft
10m
22m
23ft
13ft
15m
15m
25ft
15m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________

9. Day 3: Cones Notes Date: ____________________________________
Volume - _______________________________________________________________
Lateral Area - __________________________________________________________
Total Area - ____________________________________________________________
Therefore, now we need to find the four key pieces of information first:
1. Area of the base – ___________ Circumference of the base - _______________
3. Height - ____________________ Slant height - __________________________
Example 1:
Radius - ______________________________________________
Area of the base – ______________________________________
Circumference of the base – _______________________________
Height - ______________________________________________
Slant height – _________________________________________
Lateral Area - _________________________________________
Total Area - __________________________________________
Volume - _____________________________________________
Example 2
Radius - _________________________________________
Area of the base – _________________________________
Circumference of the base – __________________________
Height - _________________________________________
Slant height – _____________________________________
Lateral Area - ____________________________________
Total Area - ______________________________________
Volume - _________________________________________
10m 6m
24cm
26cm
page 9

10. Day 3: Pyramid Notes Date: ____________________________________
page 10
We will be looking at square pyramids only.
Lateral Area - _________________________________________________________________
Total Area - __________________________________________________________________
Volume - _____________________________________________________________________
Therefore, we need to find the following four pieces of information for each problem:
1. Area of the base – A = e2 2. Perimeter of the base – P = 4e
3. Height – h 4. Slant height - l
Example 1 – Base Edge - ___________________________________
Height – ______________________________________
Slant Height – _________________________________
Area of the base – ______________________________
Perimeter of the base - __________________________
Lateral Area - _________________________________
Total Area - ___________________________________
Volume - ______________________________________
Example 2 – Base Edge - ___________________________________
10in = l
12in = e
15ft
16ft

11. Unit 10, Worksheet 4 – Cones & Pyramids
Unit 10, Worksheet 4 – Cones & Pyramids Date: ___________________________
page 11 1. 2.
24ft
17in
20ft
8in
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________ 3. 4.
61m
25in
60m
14in
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________

12. Unit 10, Worksheet 5 – Mixed Review. - Day 4
Unit 10, Worksheet 5 – Mixed Review - Day 4 Date:___________________________
page 12 1.
Area of square base is 324mm2. 2.
Circumference of Base is 60pft.
40ft
12mm
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________ 3. 4.
12in
6in
18in
10in
A = 36pin2
7in
7in
8in
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________

13. Day 5: Sphere Notes Date: ____________________________________
______________________________________________________________________
Volume - ____________________________________
Total Area - _________________________________
Example 1 – Find the Total Area and Volume of the Sphere
Radius - __________________________________
Volume - _________________________________
Total Area - _____________________________
6in
Example 2 – Find the Total Area and Volume of the spherical model of a planet – The length of the equator is 30pmm.
Radius - __________________________________
Volume - _________________________________
Total Area - _____________________________
page 13

14. The area of the circular base is 64pm2.
Unit 10, Worksheet 6 – Mixed Review Date: __________________________
page 14 1. 2.
The area of the circular base is 64pm2. 4m
17m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Total Area: ________________ 3.
The circumference of the circular base is 8pcm. The height is twice the length of the diameter. 4. 5m
12m
11m
13m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________

15. The area of the square base is 100m2. 4. The circumference is 22pft.
Unit 10, Worksheet 6 - Mixed Review - Continued Date: ___________________________
page 15 1. 2.
3in
2in 6m 7m
Area = 70m2
6in
1.5in
6in
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Lateral Area: ________________
Total Area: ________________ 3.
The area of the square base is 100m2. 4.
The circumference is 22pft.
13m
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Volume: ________________
Total Area: ________________

16. Geometry/Trig Name: ____________________________________
Day 6; Similar Solids Date: ____________________________________
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 total areas is a2:b2.
3. The ratio of their volumes is a3:b3
Given the solids, determine the ratio of their totals areas and their volumes.
Example 1:
Example 2:
Example 3:
Scale Factor: _________________
Ratio of Total Areas: ___________
Ratio of Volumes: ______________ 6 10
Cubes
All cubes are _________________.
Similar Cylinders
Scale Factor: _________________
Ratio of Total Areas: ___________
Ratio of Volumes: ______________ 3 2
All spheres are ________________.
Scale Factor: _________________
Ratio of Total Areas: ___________
Ratio of Volumes: ______________ 8 2
Spheres
page 16

17. Geometry/Trig Name: __________________________
Ratios Practice – Unit 9 & 10 Date: ___________________________
1. Two similar polygons have a scale factor of 3:5. What is the ratio of the perimeters? What is the ratio of the areas?
2. Two circles have radius lengths 4 and 12. What is the scale factor? What is the ratio of circumferences? What is the ratio of areas?
3. The ratio of areas of two squares is 9:16. What is the ratio of their perimeters? What is the ratio of their side lengths?
4. The ratio of areas of two similar polygons is 9:625. The perimeter of the smaller polygon is 12. What is the perimeter of the larger polygon?
5. The ratio of perimeters of two similar polygons is 4:15. The area of the smaller polygon is 64. What is the area of the larger polygon?
6. Two similar solids have a scale factor of 3:7. What is the ratio of their lateral areas? What is the ratio of their total areas? What is the ratio of their volumes?
7. Two spheres have diameters with lengths 10 and 20. What is the ratio of their radii? What is the ratio of their circumferences? What is the ratio of their total areas? What is the ratio of their volumes?
8. The ratio of volumes of two cubes is 27: What is the scale factor? What is the ratio of lateral areas? What is the ratio of total areas?
9. The ratio of lateral areas of two similar solids is 25:36. What is the ratio of their volumes?
10. The ratio of total areas of two similar solids is 64:81. If the volume of the smaller solid is 1024, what is the volume of the larger solid?
11. The ratio of volumes of two similar solids is 1:729. The lateral area of the larger solid is 324, what is the lateral area of the smaller solid?
page 17

18. Word Problems Date: ____________________________________
Draw a picture for each scenario. Leave all answers in terms of p unless otherwise indicated.
1. A cylinder has a volume of 1728p. If the height equals the radius, find the total area of the cylinder.
2. If the lateral area of a cone equals 125p and the slant height is 5, find the radius.
3. If the ratio of areas of two similar prisms is 4:169, find the ratio of volumes.
4. The total area of a cube is 2400m2. Find the volume.
5. A solid metal cylinder with radius 3 and height 3 is recast (melted down and then remolded) as a solid cone with radius 3. Find the height of the cone.
6. A solid metal ball with radius 4 cm is melted down and recast as a solid cone with the same radius. What is the height of the cone?
7. A manufacturer wants to pack cans of soup into a box. Assume the cans are cylinders and the box is a rectangular prisms and that the box is designed to fit as closely as possible around the cans. The manufacturer would like to fit 12 cans into one box (3 rows of 4 cans each). The diameter of each can is 6cm and the height of each can is 9cm. Determine the size of the necessary box and the amount of wasted space in the box. (Convert answer of wasted space into a decimal and round to the nearest tenth.)
page 18

19. 8. The surface area (total area) of a sphere is pcm2
8. The surface area (total area) of a sphere is pcm2. Find the diameter of the sphere.
9. Two similar cylinders have lateral areas 81p and 144p. Find the ratio of the heights. If the height of the smaller cylinder is 12, find the height of the larger cylinder.
10. A regular square pyramid has a base edge of 3cm and a volume of 135cm2. Find the height.
11. Find the volume and total area of a sphere that has a circumference of 122p.
12. A cylinder with radius 7cm has a total area of 168pcm2. Find its height.
13. A cone has a diameter of 18 cm and a slant height of 15 cm. How much water can fit in this cone? (1cm3 = 1mL)
14. If two similar cones have a ratio of lateral areas of 144:25, find the ratio of the volumes. If the volume of the larger cone is 6912, find the volume of the smaller cone.
15. If two similar prisms have a ratio of volumes of 8:2197, find the ratio of total areas.
page 19
Suggested Book Word Problems:
p. 479 #23-28; p. 493 #18-21; p. 494 #22-24; p. 495 Challenge; p. 501 #19-20, 23, 24; p. 511 #1-8

20. Geometry/Trig Name: __________________________
Day 7:Total Area and Volume Application Questions Date: ____________
page 20
Directions: Work with your group to solve each application problem. You will be assigned one problem to present to the class. Be sure to label all answers and round any values appropriately.
1. A manufacturer needs to fit 6 soccer balls in a box for shipping. The balls are organized in 3 columns of 2 balls each. Each soccer ball has a radius of 4 inches. If the box fits around the balls as closely as possible, find the volume of the box and the wasted space inside the box. How much cardboard (in square inches) is required to create the box? (Assume the soccer balls are perfect spheres.)
2. A spherical scoop of ice cream with a diameter of 6cm is placed in an ice cream cone with a diameter of 5cm and a height of 12cm. Is the cone big enough to hold all of the ice cream if it melts? If it is, how much extra space will be available? If it is not, how much more space would be needed to hold all of the ice cream?
3. A solid metal cone with radius 3cm and height 2cm is melted down and recast as a solid cylinder with height of 1.5cm. Find the radius of the cylinder. What is the total area of the original cone and the new cylinder?
Volume of Box: ___________________
Wasted Space: ___________________
Amount of Cardboard: _____________
Is the cone big enough? _________________
Extra Space or Needed Space: ____________
Radius of Cylinder: ________________
Total Area of Cone: _______________
Total Area of Cylinder: _____________

21. Geometry/Trig Name: __________________________
Total Area and Volume Application Questions Date: ____________________
page 21
4. A manufacturing company has constructed 10,000 solid metal spherical balls with a circumference of 16p cm for use in a product. After feedback from their customers, they have determined that a cone shaped solid with the same volume and same radius would be more effective. If each solid metal ball is melted down and recast as a solid cone, what is the height of each new cone?
Customers have also requested that the cones be painted yellow to match the product. Each quart of special paint costs $43.98 and advertises that it covers 12m2. How many quarts will be needed? What will the cost be to paint the newly constructed set of solid cones?
5. You are going to paint a barn. The barn is made up of two solids: a square pyramid sitting on a rectangular prism. The base of the pyramid and rectangular prism are congruent. The base edge of the square pyramid is 24 feet. The height of the rectangular prism is 10 feet. The height of the entire barn is 15 feet. Find the total area to be painted. (Assume there are no windows and that you paint the roof.) One gallon of paint covers approximately 350 square feet of space. How many gallons of paint will you need? If each gallon of paint costs $24.75, what will be the total cost for paint?
Height of Cone: ____________
Number of Quarts: __________
Cost: ____________________
Total Area to be Painted: _________________
Number of Gallons Required: ______________
Cost: ________________________________

22. Geometry/Trig Name: __________________________
Total Area and Volume Application Questions Date: ____________________
page 22
6. A silo barn consists of a cylinder and a hemisphere (half of a sphere). (The base of the cylinder and hemisphere are congruent). Find the volume and total area of the silo if the height of the cylinder is 20ft and the height of the silo barn is 25ft.
You want to paint the barn. (Assume there are no windows and that you do paint the roof.) One gallon of paint covers approximately 350 square feet of space. What is the total area that needs to be painted? How many gallons of paint will you need? If each gallon of paint costs $22.50, what will be the total cost for paint?
7. Water is pouring into a conical (cone-shaped) container at the rate of 1.8 m3 per minute. Find the volume of the cone. Find, to the nearest minute, then number of minutes it will take to fill the container.
The container has been emptied. A hole has been punctured in the bottom of the container and is allowing the water to escape at a rate of 1.2 m3 per minute. Now how long will it take to the nearest minute to fill the container?
Volume; ____________________
Total Area: _________________
Total Area to be Painted: _____________
Gallons of Paint: ____________________
Total Cost: ________________________
Volume of the Cone: _______________
Number of Minutes: ______________
Number of Minutes: ______________