POLYGONS

A polygon is a 2-dimensional shape made of straight lines.

Examples: (only a few)

Triangle Square Rectangle Pentagon Hexagon Octagon

In this lesson, among others, we are going to study SOLIDS and more specific PRISMS.

Definition:

A PRISM is a 3-dimensional solid with two identical bases.

The cross-section all along its length is the same everywhere. (uniform cross-section.)

A prism is named according to the shape of its base. (TRIANGULAR PRISM in this case.)

VOLUME FORMULAE:

1. CUBE:

X

3Volume x x x ; side V

3x

2. RECTANGULAR PRISM:

Volume b h

b

h

bh

VOLUME FORMULAE:

3. TRIANGULAR PRISM:

b

h

H

Volume area x H

b x h

x H2

h : height of

H : height (length) of prism

VOLUME FORMULAE:

h

b

H

4. CYLINDER:

Volume area of circle x h

2r x h

Did you notice that the volume of a right prism = area of the base x height?

VOLUME FORMULAE:

EXAMPLE 1

Determine the volume of the cube with side 2,1 m.

ANSWER

3Volume of cube (2,1)

2,1 x 2,1 x 2,1

39,261m

If approximation is not specified, it can be done to any decimal, otherwise as requested.

EXAMPLE 2

Determine the volume of the cylinder with 3h 2d

(Give your answer in terms of ).

Key:

h: height

d: diameter

ANSWER

3h 2(30)

60h

3

h 20 cm

2Volume of cylinder r x h

2(15) x 20

34500 cm

314137,2 cm

30r 15 cm

2

Answer approximated to one decimal numbers.

3h 2d

First determine h.

Volume is indicated as cubic units.

Answer is in terms of

12cm

21cm

EXAMPLE 3

Determine the volume of the triangular prism.

ANSWER

Volume of triangular prism area x H

12 x 8

x 212

48 x 21

31 008 cm

Height of is 8 cm.

EXAMPLE 4

A contractor has to build a solid staircase as indicated in the sketch.All the lines and planes meet at right angles and all the stairs have the same measurements.

90cm

17cm

30cm

Determine how many cubic metres concrete is needed to build the staircase.

ANSWERANSWER

STEP 1

STEP 2

STEP 3

STEP 4

90cm17cm

A

C

E

G

H

F

D

B

• Consider each step as a rectangular prism with and

h 17cm

b 90 cm.

• Length: STEP 1 = AB

STEP 2 = CD

STEP 3 = EF

STEP 4 = GH

3

Volume STEP 1

120 cm 90 cm 17 cm

1,2 m 0,9 m 0,17 m

0,1836 m

b h

3

Volume STEP 2

90 cm 90 cm 17 cm

0,9 m 0,9 m 0,17 m

0,1377 m

b h

3

Volume STEP 3

60 cm 90 cm 17 cm

0,6 m 0,9 m 0,17 m

0,0918 m

b h

3

Volume STEP 4

30 cm 90 cm 17 cm

0,3 m 0,9 m 0,17 m

0,0459 m

b h

3

Total volume : 0,1836 0,1377 0,0918 0,0459

0,459 m

The contractor needs half a cubic metre of concrete to build the staircase. (Approximated to one decimal figure).

A cylindrical hole is drilled through a solid rectangular metal block as indicated in the sketch.

x 4cm ; 3,14

2x

x

20cm

Calculate the volume of the remaining metal.

EXAMPLE 5

ANSWER

Volume rectangular prism b h

20 2x 2x

20 2(4) 2(4)

20 8 8 31 280 cm

2Volume of cylindrical hole r h

2(3,14) (2) 20

3,14 4 20

3251,2 cm

Volume of remaining metal 1 280 251,2

31 028,8 cm

x 4

r 22 2

20cm40cm

25cm

Determine the volume of the space inside the mailbox.

EXAMPLE 6

A mailbox consists of half a cylinder on top of a rectangular prism, withmeasurements as indicated in the sketch. = 3,142

ANSWER

20r 10

2

Volume of rec tangular prism 20 40 25

320 000 cm

2r hVolume of half acylinder

2

36 284 cm

23,142 10 40

2

Total volume 20 000 6 284

326 284 cm

EXERCISE 1

Determine the volume of each of the following solids:

(Approximate your answer to one decimal figure.)

1.

15cm

25cm

2.

15cm

25cmThe 2 base surfaces are quarter circles.

3,142

3.

Half a cylinder is cut from a rectangular prism.

4.

A cylindrical hole is cut out exactly in the middle of a cylinder.Diameter of inner cylinder is 10 mm.