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Volume and Total Surface Area of RIGHT Prisms and CYLINDERS

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Volume

- The amount of space occupied by an object.
- How many 1by1by1 unit cubes that will fit inside.

Example:

The VOLUME of this cube is all the space contained by the sides of the cube, measured in cube units (units3).

Volume

- Once we know the area of the base (lxw), this is then multiplied by the height to determine the VOLUME of the prism (how many cubes will fit inside.

h

w

IMPORTANT! We know that:

Volume of a prism =

(Area of Base)x(Height of the prism)

Not so important: Volume of this rectangular prism = (l x w) x H

Because… the base area IS length times width

l

Volume

Formula:

V = B x H

Where B is the area of the BASE

H is the height of the PRISM

- Find the volume of this prism…

7 cm

- BASE area: a 5 by 4 rectangle
- B=20 square cm

4 cm

5 cm

V = B x H

V = 20 x 7

V = 140 cubic cm

Total Surface Areaa totaling of the surface areas

Label the sides

Find each area and add them up:

5x7 = 35 sq. cm

5x7 = 35 sq. cm

4x7 = 28 sq. cm

4x7 = 28 sq. cm

5x4 = 20 sq. cm

5x4 = 20 sq. cm

total =

- Draw each surface

7

7

5

5

7

7

7 cm

4

4

4 cm

5 cm

4

4

5

5

Volume

- Volume of a Triangular PRISM
- (area of the BASE) x (Height of the prism)

Volume

- The same principles apply to the triangular prism.

To find the volume of the triangular prism, we must first find the area of the triangular base (shaded in yellow).

h

b

Volume

- To find the area of the Base…

Area (triangle) = b x h

2

This gives us the Area of the Base (B).

h

b

Volume

- Now to find the volume…

We must then multiply the area of the base (B) by the height (H) of the prism.

This will give us the Volume of the Prism.

B

H

Volume

Volume

BASE area = (8 x 4) = 16 sq. cm

2

the Height of the prism is 12 cm

V = B x h

V =16 x 12

V =_____ cubic cm

Total Surface Areaa totaling of the surface areas

Label the sides

- Draw each surface

12

12

12

??

??

8

Find the missing length using the right triangle inside the isosceles triangle and the Pythagorean Theorem…

4

4

8

8

Find each area and add them up:

Volume

Volume of a Cylinder

A cylinder is like a prism with a circle base, so we can use the SAME VOLUME formula

V = B x H

Where B is the area of the base (circle)

And H is the height of the cylinder

H

r

r

H

VOLUME

Formula for Area of Circle

A= r2

= x 32

= x 9 = 9

28.27 square units

H = 6 units

VOLUME = B x H

= 9x 6 = 54

_____ cubic units

The Soup Can

Think of the Cylinder as a soup can.

You have the top and bottom lid (circles) and you have the label (a rectangle – wrapped around the can).

The lids and the label are related.

The circumference of the lid

is the same as

the length of the label.

Area of the Circles and rectangle

Area of the Circles

Area of the RECTANGLE

TOTAL SURFACE AREA = circle + circle + rectangle

_______ square units

Area of the Circles and rectangle

Area of the Circles

A= r2 =9 28.27 (from before)

Circle with radius of 3

Circle with radius of 3

Area of the Circles and rectangle

Area of the RECTANGLE

A=(circumference)(Height of the cylinder)

=( d)(H) = ( 6)(6) = 36 113.10

Circle with radius of 3

Height of cylinder

6 units

CIRCUMFERENCE OF THE CIRCLE

Diameter times pi

6 times

Area of the Circles and rectangle

Area of the Circles

A= r2 =9 28.27 (from before)

Area of the RECTANGLE

A=(circumference)(Height of the cylinder)

=( d)(H) = ( 6)(6) = 36113.10

Circle with radius of 3

Height of cylinder

6 units

CIRCUMFERENCE OF THE CIRCLE

6 times

Circle with radius of 3

TOTAL SURFACE AREA = circle + circle + rectangle

_______ square units

r

b

Area Rectangle

(and parallelograms)

= base x height

Area Circle = π x r2

a

h

h

b

b

Area Trapezoid

= ½ x (a + b) x h

Area Triangle

= ½ x base x height

For all this you need:Area Formulas
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