Unit 28

PRISMS AND CYLINDERS: VOLUMES, SURFACE AREAS,

AND WEIGHTS

2

PRISMS A polyhedron is a three-dimensional

(solid) figure whose surfaces are polygons

A prism is a polyhedron that has two identical (congruent) parallel polygon faces called bases and parallel lateral edges

The volume of any prism is equal to the product of the base area and the altitude

heightbaseofarea

volumewhere

hAVhAV

B

B

3

PRISMS

Prisms are named according to the shape of their bases, such as triangular, rectangular, pentagonal, and octagonal

heightbaseofarea

volumewhere

hAVhAV

B

B

Triangular Prism Rectangular Prism Pentagonal Prism Octagonal Prism

4

VOLUME OF A PRISM Compute the volume of the fuel tank

shown in liters:

60 cm

25 cm

20 cm

• V = ABh• This prism has a rectangular base so AB = length width

AB = (60cm)(20cm) = 1200 cm2

• Now, V = ABh = (1200 cm2)(25 cm) = 30000 cm3

Literscm

Lx

cm30

1000

1

1

300003

3

5

CYLINDERS A circular cylinder is a solid that has

identical circular parallel bases The surface between the bases is called

the lateral surface. The altitude (height) of a circular cylinder,

is a perpendicular segment that joins the planes of the bases

Altitude Lateral Surface (Sides)

6

CYLINDERS The axis of a circular cylinder is a line that

connects the centers of the bases In a right circular cylinder the axis is

perpendicular to the bases The volume of right circular cylinders is the

same as prisms:V = ABh

Circular Cylinder Right Circular Cylinder

Axis

7

VOLUME OF A CYLINDER Find the volume of the soup can shown below

given that the bases have a radius of 8 centimeters:

16 cm

• V = ABh

• The cylinder has a circular base, so AB = r2 = (8)2 = 201.062 cm2

• Now, V = ABh = (201.062 cm2)(16 cm) = 3216.99 cm3 Ans

8

TRANSPOSING VOLUME FORMULAS

An engine piston has a height of 18.6 centimeters and a volume of 460 cubic centimeters. Find the radius of the piston:– The piston is a circular cylinder, so V = ABh = r2h

– Substitute the given measurements into the formula and solve for r:

V = r2h460 cm3 = r2(18.6 cm) r = 2.806 cm Ans

9

LATERAL AND SURFACE AREAS

The lateral area of a prism is the sum of the areas of the lateral faces. The lateral area of a cylinder is the area of the curved or lateral surface

The lateral area of a right prism equals the product of the perimeter of the base and height

The lateral area of a right circular cylinder is equal to the product of the circumference of the base and height

The surface area of a prism or a cylinder equals the sum of the lateral area and the two base areas

10

SURFACE AREA EXAMPLE Find the surface area of the

circular cylinder trash can below given that it has a radius of 8 inches:

40”

– The surface area is equal to the sum of the lateral area and the two base areas

– Lateral area = CBh= 2(8”)(40”)= 2010.62 in2

– Area of base = r2

= (8”)2 = 201.062 in2

– Surface area = 2010.62 + 201.062 + 201.062= 2412.744 in2 Ans

11

PRACTICE PROBLEMS

1. Find the volume of a mobile home (rectangular prism) with a length of 10 meters, width of 15 meters, and height of 20 meters.

2. Determine the lateral area of the mobile home in problem #1.

3. Compute the surface area of the mobile home is problem #1.

4. Find the volume of a triangular prism given that the triangular bases have sides of 8 inches, 10 inches, and 12 inches and that the prism has a height of 5 inches.

12

PRACTICE PROBLEMS (Cont)5. Compute the volume of the interior of a pen

(right circular cylinder) with a radius of 15 mm and a height of 25 mm.

6. Find the lateral area of the pen in problem #5.7. Find the surface area of the pen in problem

#5.8. Find the diameter of a circular culvert given

that it has a volume of 150 cubic feet and a height of 12.5 feet.

13

PRACTICE PROBLEMS (Cont)

9. A solid steel post 27.6 inches long has a square base. The post has a volume of 110 cubic inches. Compute the length of a side of the base.

10. Find the lateral area of a box with a length of 18 inches, width of 14 inches, and height of 12 inches.

14

PROBLEM ANSWER KEY

1. 3000 m3

2. 1000 m2

3. 1300 m2 4. 198.431 in3 5. 17671.46 mm3

6. 2356.19 mm2 7. 3769.91 mm2 8. 3.91 feet9. 1.996 inches10. 768 in2