Rectangular Prism Notes
Date: ____________________________________
Rectangular Prism: _____________________________________________________________
___________________________________________________________________________
Total Area: ___________________________________________________________________
____________________________________________________________________________
Lateral Area: _________________________________________________________________
6cm
How do we find Total Area?
Example 1
Find the area of each face:
Front: ___________ Back: ____________
Top: ____________
8cm
Bottom: __________
Left Side: ________ Right Side: _______
10cm
Total: ___________
How do you find the Lateral Area? _________________________________________
Formula for the Lateral Area: ________________________________ Total Area: ___________
Formula for the Total Area of a Rectangular Prism: ________________ Lateral Area: __________
Example 3
Example 2
Find the lateral area: ________________
Find the total area: _______________
6m
Find the lateral area: ________________
Find the total area: _______________
9in
9in
6m
20m
9in
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: __________________
Other Right Prisms - Notes
Date: ____________________________________
Volume (V) = Area of the Base x height of the prism
(V = Bh)
Lateral Area (L.A) = Perimeter of Base x height
(L.A = ph)
Perimeter of Base (p)
height (h)
Total Area (T.A) = Lateral Area + 2(Area of the Base) (T.A. = L.A. + 2B)
Example 1 – Triangular Right Prism
12m
12m
8m
7m
8m
4m
Area of Base (B)
Example 2 – Triangular Right Prism
10m
7m
10m
16m
14m
12m
6m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Example 3 – Trapezoidal Right Prism
30m
10m
40m
14m
Example 4 – Hexagonal Right Prism
6cm
Base is a
Regular
Hexagon
12cm
6cm
18m
8m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Unit 10, Worksheet 1
Date: ____________________________________
Find the Volume, Lateral Area, and Total Area of each figure.
1.
2.
100m
42m
42m
24m
20m
36m
24m
28m
18m
12m
18m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
3.
4.
100m
75m
70m
60m
16m
24m
40m
28m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Unit 10, Worksheet 2
Date: ____________________________________
Find the Volume, Lateral Area, and Total Area of each figure.
1.
8cm
Base is a Regular Hexagon
2.
5m
10m
6m
15cm
7.5m
12.5m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
3.
4.
4in
70m
16in
20m
24m
22m
28m
50m
Base is a square.
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Cylinder Notes
Date: ____________________________________
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 - ____________
Radius – ___________ Height - ____________
Area of Base – _________________________
Area of Base – _________________________
Circumference of Base – __________________
Circumference of Base – __________________
Volume – _____________________________
Volume – _____________________________
Lateral Area - __________________________
Lateral Area - __________________________
Total Area - ___________________________
Total Area - ___________________________
Unit 10, Worksheet 3
Date: ____________________________________
1.
2.
2yd
47ft
24ft
All answers must be in feet.
13ft
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
3.
4.
10ft
23ft
25m
10m
22m
23ft
13ft
25ft
15m
15m
15m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
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 – ___________ 2. Circumference of the base - _______________
3. Height - ____________________
4. Slant height - __________________________
Example 1:
Radius - ______________________________________________
Area of the base – ______________________________________
Circumference of the base – _______________________________
10m
Height - ______________________________________________
Slant height – _________________________________________
Lateral Area - _________________________________________
6m
Total Area - __________________________________________
Volume - _____________________________________________
Example 2
Radius - _________________________________________
Area of the base – _________________________________
Circumference of the base – __________________________
Height - _________________________________________
24cm
Slant height – _____________________________________
Lateral Area - ____________________________________
Total Area - ______________________________________
Volume - _________________________________________
26cm
Pyramid Notes
Date: ____________________________________
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 – ______________________________________
10in = l
Slant Height – _________________________________
Area of the base – ______________________________
Perimeter of the base - __________________________
12in = e
Lateral Area - _________________________________
Total Area - ___________________________________
Volume - ______________________________________
Example 2 –
Base Edge - ___________________________________
Height – ______________________________________
Slant Height – _________________________________
Area of the base – ______________________________
Perimeter of the base - __________________________
16ft
15ft
Lateral Area - _________________________________
Total Area - ___________________________________
Volume - ______________________________________
Unit 10, Worksheet 4 – Cones & Pyramids
1.
Date: ___________________________
2.
24ft
17in
20ft
8in
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
3.
4.
61m
25in
60m
14in
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Unit 10, Worksheet 5 – Mixed Review
1.
Area of square base is
324mm2.
Date:___________________________
2.
Circumference of
Base is 60pft.
40ft
12mm
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
3.
4.
12in
6in
18in
10in
A = 36pin2
7in
7in
8in
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Sphere Notes
Date: ____________________________________
Sphere - _______________________________________________________________
______________________________________________________________________
Volume - ____________________________________
Total Area - _________________________________
Example 1 – Find the Total Area and Volume of the Sphere
Radius - __________________________________
6in
Volume - _________________________________
Total Area - _____________________________
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 - _____________________________
Unit 10, Worksheet 6 – Mixed Review
1.
Date: __________________________
2.
The area of the circular
base is 64pm.
4m
17m
Volume: ________________
3.
Volume: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
The circumference
of the circular base
is 8pcm. The height
is twice the length
of the diameter.
4.
12m
5m
11m
13m
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
Unit 10, Worksheet 6 - Mixed Review - Continued Date: ___________________________
1.
2.
3in
2in
6m
7m
Area = 70m2
6in
1.5in
6in
Volume: ________________
Volume: ________________
Lateral Area: ________________
Lateral Area: ________________
Total Area: ________________
Total Area: ________________
3.
13m
The area of
the square
base is 100m2.
4.
The circumference is 22pft.
Volume: ________________
Lateral Area: ________________
Volume: ________________
Total Area: ________________
Total Area: ________________
Geometry/Trig 2
Name: ____________________________________
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:
Scale Factor: _________________
6
Ratio of Total Areas: ___________
10
Cubes
Ratio of Volumes: ______________
All cubes are _________________.
Example 2:
Similar Cylinders
Scale Factor: _________________
Ratio of Total Areas: ___________
3
2
Ratio of Volumes: ______________
Example 3:
All spheres are ________________.
8
Scale Factor: _________________
2
Ratio of Total Areas: ___________
Ratio of Volumes: ______________
Spheres
Geometry/Trig 2
Ratios Practice – Unit 9 & 10
Name: __________________________
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 27:2744. 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?
Word Problems
Name:____________________________________
Draw a picture for each scenario – then solve the problem. **Don’t forget labels!
1. A cylinder has a volume of 1728π m3. If the height equals the radius, find the total area of the cylinder.
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 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?
4. 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.)
5. Mr. Bloop has a cylindrical water tank on his farm. It is nine feet long and 2.2 feet in diameter. Water
flows out a valve in the bottom of the tank at a rate of 3.1 cubic feet per minute. At that rate, how long
will it take to empty the tank when the tank is full?
Geometry
Name: ___________________________________
APPLICATIONS
Date: ____________________________________
Complete each word problem. Show work. Label answers.
1. A manufacturer needs to fit 6 soccer balls (spheres) in a box (rectangular prism). The
balls are organized in 3 columns of 2 balls each. The radius of each soccer ball is 3 inches.
If the box is created to be as small as possible, determine the volume of the box and the
amount of wasted space in the box.
Volume of Box: _________________
Wasted Space: _________________
2. A manufacturer wants to pack tennis balls into a box. Assume the tennis balls are perfect
spheres with a radius of 4cm. (The box will have 5 rows of 6 tennis balls for a total of 30.)
Determine the dimensions of the box that would designed to fit as closely as possible around
the tennis balls. Find the amount of wasted space in the box.
Volume of Box: _________________
Wasted Space: _________________
Geometry
Name: __________________________
Applications
Date: __________________________
3. Three tennis balls are packed into a cylindrical container. The radius of one tennis ball is
5cm.
a. Draw a diagram and label all dimensions (radii, height).
b. Find the volume of all three tennis balls.
c. Find the volume of the cylindrical container.
d. Find the volume of the wasted space.
4. Given the diagram below that consists of a cylinder and hemisphere (half sphere) answer
each question.
a. What is the volume of the entire solid.
50ft
b. What is the total area of the entire solid.
12ft
c. Imagine the solid is a silo barn that you must paint. Find the area to be painted.
f. If one gallon of paint can cover approximately 350ft2 of area, how many cans of paint
would be required to paint the barn?
Geometry
Name: ____________________________________
Applications
Date: ____________________________________
Complete each application problem. Draw a diagram for each scenario.
5. The diameter of Pluto is 2274km. Assuming that Pluto is a perfect sphere, calculate the
volume and total area of Pluto.
Volume: __________________
Total Area: ____________________
6. If a cube with a 5-in. side length is sliced in half, what is the total area of the two
pieces?
7. If you have five 5-in. by 5-in. aluminum cubes and superglue them together in a row, what
is the total area of the resulting shape made by the five cubes?
8. If the lateral area of a cone equals 125π in2 and the slant height is 25 in, find the radius.
9. A water tank has been purchased for a farm. It will be used to water cattle. It is
cylindrical metal container that is 2.6 feet tall. the area of the bottom of the tank is 9.3
square feet. If the cattle drink two hundred four cubic feet of water a day, how many times
per day will the tank have to be filled?’
Geometry
Name: __________________________
Volume Applications - HW
Date: __________________________
1. A driveway 30 m long and 5 m wide is to be paved with blacktop 3cm thick. How much will
the blacktop cost if it is sold at the price of $175 per cubic meter? (Hint: 1 cm = 0.01m)
2. A solid metal cylinder with a radius of 6cm and height of 18 cm is melted down and recast
as a solid cone with a radius of 9 cm. Find the height of the cone.
3. Which of the containers below holds more, the can or the bottle? How much more?
(Assume the top part of the bottle is a complete cone)
4. Water is pouring into a conical (cone-shaped) reservoir at the rate of 1.8 m3 per minute.
Find, to the nearest minute, the number of minutes it will take to fill the reservoir. (Use
π=3.14).
5. A solid metal ball with radius 8 cm is melted down and recast as a solid cone with the same
radius. What is the height of the cone?
6. Two similar cylinders have radii 3 and 4. Find the ratios of the following:
a. Heights
b. base circumferences
c. Lateral areas
d. Volumes
7. Two similar pyramids have heights 12 and 18. Find the ratios of the following:
a. Base areas
b. Lateral areas
c. Total areas
d. Volumes
8. Two similar cylinders have lateral areas 81 π and 144 π. Find the ratios of:
a. The heights
b. The total areas
c. The volumes
9. Two similar cones have volumes 8 π and 27 π. Find the ratios of:
a. The radii
b. The slant heights
c. The lateral areas
10. Two similar pyramids have volumes 3 and 375. Find the ratios of:
a. The heights
b. The base areas
c. The total areas