Aim: The Third Dimension: Volume – What is it?

Aim: Volume Course: Applied Geo.

Do Now: What is the area, in terms of of the shaded region in the figure below?

Aim: The Third Dimension: Volume – What is it?

sA. of sq. = s2

A. of circle = r2

A. of shaded region = A. of sq. - A. of circle

A. of s. r. = s2 - 1/2s)2 = s2 - (1/4 s2)= s2 - 1/4s2

r = 1/2 s


Perimeter & Area

PerimeterPerimeter - the distance around a polygon PP = 1 + 2 + 3 + 4

AreaArea - the space inside a polygon - measured in square units

1

1 sq. un.

1

1 sq. un.

1

1 sq. un.

1

1 sq. un.

1

1 sq. un.

1

1 sq. un.

11

12

3

4

units2, in2, miles2, etc.


Volume

VolumeVolume - the measure of the space inside a polyhedron -measured in cubic units.

V = Bh, where B = the area of the base

11 1

1 cu. unit

4

12

8 cubic units.

27cubic units.

3

3

3

Polyhedron - three dimensional figure whose surfaces are polygons

Edge – a segment that is the intersection of two faces

Vertex – point where edges meet


Volume Formulas

Prism – a polyhedron with two congruent, parallel bases. The other faces are lateral faces.

hBV Prism

wl

h

Rectangular B = l • w

Triangular B = 1/2 bh

B - area

of the baseb

Rectagular prism – V = l w h

A triangular prism is a solid whose base is a triangle.


Model Problems

Find the Volume of the following polyhedron: 13

17

10

V = l • w • h = 17 x 13 x 10 = 2210 units3

hwlV

Prism

wl

h



Model Problem

What is the volume of a rectangular box whose base measures 3 units by 10 units and whose height is 20 units?

Prism

wl

h


Rectangular prism – V = l w h

20

310

V = 10 3 20 = 600 units3


Model Problem

The volume of a rectangular box is 2,400 cu. units. The measurements of the base are 60 x 20. What is the height of the box?

wl

hBV = 2400 units2

2400 = 60 x 20 x h

2400 = 1200 x h

2 = h

60

h

20


Model Problem

The base of a triangular prism has an area of 15 cm2 and a height of 30 cm. What is the volume of the triangular prism?

Triangular prism – V = B h V = 15cm2 30 = 450 units3

hBV

Triangular B = 1/2 bh

B - area

of the baseb

30

15 cm2


Volume Formula

Prism – a polyhedron with two congruent, parallel bases. The other faces are lateral faces.

3eV

Cube

ee

e

What is the volume of a cube with a side of 10 units?

10

1010

3 310 1000V unit


Volume Formulas

hrV 2

Cylinderr

h

3

34 rV

Sphere

r


Model Problems

A cylindrical tank has a radius of 10 ft. and a height of 25 ft. What is its volume? Give your answer in terms of

hrV 2

Cylinderr

h

r

h 10 ft25 ft

210 25 2500V


Volume Formulas

h

B = area of base of pyramid

hBV 31

Pyramid

Pyramid – 3 dimensional figure with a single base and sides that are triangles.

Triangular pyramid

Rectangular pyramid

Pentagonal pyramid


Model Problems

Find the volume of the pyramid with a height of 9.

15

10

9

hBV 31

)91015(31 V

B = l • w )1350(31V

3450 unitsV


Volume Formula

hrV 2

31

Cone

Cone – a pyramid with a circle for a base

What is the volume of a cone whose height is 10 units and whose base has radius 5? Express in terms of

h = 10

r = 5

21 5 103V


Model Problems

The main tank at the Living Seas Aquarium at EPCOT Center in Florida is the largest enclosed tank in the world. It is a cylinder with diameter 203 ft. and height 25 ft. About how many million gallons of water does this tank hold? (1 gal. = 231 in3; 1728 in3 = 1 ft3)

25 ft.

203 ft.

Radius is 1/2 the diameter r = 203 2 = 101.5 ft.

hrV 2

h = 25’

)25()5.101(14.3 2V3625.726,808 ft

33

33 608,479,397,11728625.726,808 inft

inft

gallonsgalinin 273.695,049,6231608,479,397,1

33


A. Find the surface area of the Great Pyramid, including its base.

Model Problems

The Great Pyramid at Giza, Egypt, was built about 2580 B.C. as a final resting place for Pharoah Khufu. At the time it was built, its height was about 481 ft. Each edge of the square base was 756 feet long.

Height 481 ft.

756 ft 756’

B. Find the volume of the Great Pyramid.


Model Problems

Zia is planning to landscape her backyard. The yard is a 70ft-by-60ft rectangle. She plans to put down a 4- in. layer of topsoil. She can buy bags of topsoil at $2.50 per 3-ft3 bag, with free delivery. Or she can buy bulk topsoil for $25.00 per yd3, plus a $20 delivery fee. Which option is less expensive. Show your calculations and explanation. (1 yd3 = 9 ft3)

70 feet

60’4”


Model Problems

A cylinder has be cut out of the figure below. Find the volume of the remaining figure. Round your answer to the nearest tenth.

4”

6 in. 6 in.

6 in.


Volume Formulas

Aim: The Third Dimension: Volume – What is it?

Documents

Transcript of Aim: The Third Dimension: Volume – What is it?