**VOLUME**

**Volume: Volume is the measure of the amount of space inside of a solid figure, like a cube, ball, cylinder or pyramid.**

Its units are always "cubic", that is, the number of identical cubes that fit inside the figure.

**For Example**

**l**

A= l x w

A= 8cm x 2cm

A= 16 cm

2

V= lwh

or

V = Bh

3 cm

3cm

3cm

V= s

V= 3cm x 3cm x 3cm

V= 27 cm

3

3

10 cm

6cm

4cm

V= lwh

V= 10cm x 4cm x 6cm

V= 240cm

3

V= Volume

l = length

w= width

h= height

B = base area

V= lwh

V= 2cm x 2cm x 8cm

V= 32 cm

3

Rectangular Pyramid

4cm

4cm

7cm

V= 1/3 lwh

V= 1/3 (7cm x 4cm x 4cm)

V= 37.3 cm

3

Cylinder

10m

6m

V= Bh B = πr

V=πr h

V= 3.14 x 3cm x 10cm

V= 282.6cm

2

2

3

**Cone**

V= 1/3 πr h

2

4km

7.5km

4km

8.5km

?

Pythagorean Theorem

A + B = C

2

2

2

C - A = B

8.5km - 4km = B

72.25 - 16 = 56.25

√ 56.25= 7.5

7.5= B

2

2

2

2

2

2

V= 1/3 πr h

V= 1/3 ( π x 4km x 7.5km)

V= 125.7 km

2

2

3

SPHERE

V= 4/3 πr

3

12m

V= 4/3 πr

V= 4/3 π x 12

V= 7238.2km

3

3

3

V= 7238.2km / 2

V= 3619.1km

2

2

Relations Between Shapes.

A right pyramid is related to a right prism.

A right cone and a cylinder.

A sphere and a Cylinder

8cm

2cm

8cm

2cm

2cm

Volume of Composite Objects.

To find the volume of this composite object, you will have to find the volume of the hemispheres first.

V= 4/3 πr

V= 4/3 (3.14 x 1)

V= 4.186'

2

3

V= πr h

V= 3.14 x 1 x 6

V= 18.84'

3

2

V= 23.026'

3

Then find the volume of the Cylinder

Then add them together.

First find the volume of the rectangular prism

3cm

V= lwh

V= 5cm x 5cm x 7cm

V= 175 cm

3

Then find the volume of the cone.

V= 1/3πr h

V= 1/3 (3.14 x 2.25cm x 5cm

V= 11.775cm

2

3

5cm

Now subtract the volume of the cone from the volume of the rectangular prism.

V= 175cm - 11.775 cm

V= 163.225 cm

3

3

3

A Mathematician says: Pi r squared

A Baker replies: Pi r round not squared, cake is squared.

A cube is a "box" where height, width and depth are all the same. We can compare volumes of figures when the "boxes" are the same size.

1 cm

1 cm

1 cm

This box is 1 cm on each side. We call it "one cubic centimeter". We write it as 1 cm^3.

What if the base of the prism is NOT a square or rectangle?

Find the area of the base and multiply it by the height of the prism.

B = lw which is the area of the base

Or:

Base area is 2 cm x 2 cm = 4 cm^2

Base Area x Height = Volume

4 cm^2 x 8 cm = 32 cm^3

4 cm

10 cm

height of triangle is 5 cm

Area of Triangle: 1/2 bh

=1/2 (4)(5)

= 10 cm^2

Volume of Triangular Prism

= Bh

= (10 cm^2)(10 cm)

= 100 cm^3

Slant Height

The distance from the

vertex to the circumference

of the base.

2

2

2

2

Ask questions.

Do the practice problems.

**Prisms and Cylinders (7-5)**

Pyramids and Cones (7-6)

Pyramids and Cones (7-6)

2

STOP

Do the practice problems for 7-5, Volume of Prisms and Cylinders

After you complete the mastery check for 7-5, return to here to learn about Volume of Pyramids and Cones.

Do this mini-lab at home.

Answers:

88.0 in^3

1,272.3 cm^3

Answer: 228 in^3

What did you learn about the volume of a pyramid?

What is the volume of this pyramid?

The volume of a pyramid is 1/3 the volume of the prism with the same base and height.

V = 1/3 Bh

Answer: 6.7 yd^3

Answer:

pi 4 * 7.5

2

3

= 125.67 km

3

The volume of a cone is 1/3 the volume of the cylinder with the same base and height.