Surface Area & Volume

Prism & Cylinders

• A polyhedron (3D shape) has faces that are polygons (2D shape)- Poly means many; hedron means faces

• Two faces meet at an edge• Three or more edges meet at a vertex

Edge

Vertex

A net is a two-dimensional figure that, when folded, forms a three-dimensional figure

Nets of a Cube

• Prism: A solid object that has two identical ends (bases) and all flat sides

Bases

Triangular Prism

Height

Triangular prismRectangular prism

Base

Height

Base

Prism Names

Name the prism

Side 2

Bottom

Back

Top

Side 1Front

Base 2

Bottom

Back

Top

Base 1Front

Length (L)Base (B)

Height (H)

Rectangular Prism

Example: Rectangular Prism Net

This is a rectangular prism. The l, w and h have different values.l

w

h

Surface Area - the total area of all the faces (surfaces) of an object.

To find the surface area of an object we can add up the areas of the separate faces. Note: some faces have the same size and shape

C

D

A

Surface Area of this prism:

Add the areas of the two brown sides (A) to the two green sides (D) and to the two red sides (C).

Surface Area

• Find the surface area of the rectangular prism• We should use a table to tabulate the various areas.

Face Area Number of

Sides

Total Area

A 12cm2 2 24cm2

D 15cm2 2 30cm2

C 20cm2 2 40cm2

TOTAL

6 94cm2

Area of rectangle = l x w

3 cm5 cm

CD

A

4 cm

Volume

• The number of cubic units needed to fill the space occupied by a solid.

2 cm 5 cm

3 cm

Volume = Base area x height

= (l x w) x height

= (3 cm x 2 cm) x height

= 6 cm2 x 5 cm

= 30 cm3

VOLUME OF A PRISM

The volume V of a prism is the area of its base B times its height h.

V = BhNote – the capital letter stands for

the AREA of the BASE not the linear measurement.

B B Bh h h

Area of rectangle = l x w

Prism Volume Example

4 ft

3 ft

8 ft

V = BhFind area of the base 1st

= (4ft x 3ft)h= (12 ft2)h

Multiply it by the height

= (12 ft2) x 8 ft= 96 ft3

B

Area of rectangle = l x w

What is the surface area and volume of the prism?

12 cm

10 cm

22 cm

V = Bh= (12 cm x 10 cm) x h= (120 cm2) x 22 cm= 2640 cm3

SA rectangle = l x w

SA (rect1) = 12 cm x 10 cm = 120 cm2 x 2 = 240 cm2

SA (rect2) = 12 cm2 x 22 cm= 264 cm2 x 2 = 528 cm2

SA (rect3) = 10 cm x 22 cm2 = 220 cm2 x 2 = 440 cm2

Total SA = SA (rect1) + SA (rect2) + SA(rect3)= 240 cm2 + 528 cm2 + 440 cm2

= 1208 cm2

Note Units

B

Example Nets For Triangular Prisms

Surface Area of a triangular prism

15ft

Area Triangle 1 & 2 =

=

=

= 90 ft2

Area Rect. 1 = l x w

= 12 ft x 25 ft

= 300 ft2

Area Rect. 2 & 3 = 25 ft x 20 ft

= 500 ft2

SA = 1480 ft2

SA=triangle1 + triangle2 + rectangle1 + rectangle2 + rectangle3

SA=90 + 90 + 300 + 500+ 500

2 area of same triangle + 2 areas of 3 rectangles, which 2 are the same

Volume of a triangular prism

15ft

Area Triangles =

=

=

= 90 ft2V= Bh

Find area of the base

= (90 ft2)hMultiply it by the height

= (90 ft2) x 25 ft= 2250 ft3

Classwork

• Page 186-188 #4, 5, 6a, 7a• Page 192-193 #4,6,8• Page 198 #4,6a• Page 205 #4, 5

Cylinder

Bases

Cylinder

SA of a Cylinder

Formula for Area of CircleA= r2

= 3.14 x 32

= 3.14 x 9= 28.26

But there are 2 of them so28.26 x 2 = 56.52 units squared

Find the circumference to determine the length of the rectangleC = x d

= 3.14 x 6 (radius doubled)= 18.84

Now use that as your base.A = b x h

= 18.84 x 6 (the height given)= 113.04 units squared

Now add the area of the circles and the area of the rectangle together.

56.52 + 113.04 = 169.56 units squared

The total Surface Area!

Volume of a Cylinder

Formula for Area of CircleA= r2

= 3.14 x 32

= 3.14 x 9= 28.26 unit2

V= Bh

Find area of the base

= (28.26)hMultiply it by the height

= (28.26) x 6= 169.56 unit3

Calculate the SA and Volume of the cylinderBe sure you know the difference between a radius and a diameter!

V = Bh

The radius of the cylinder is 5 m, and the height is 4.2 m; therefore B = A = r2 = 3.14 · 52 = 78.54 m2

V = BhV = 78.54 · 4.2

V = 329.9 m3

Formula for Area of CircleA= r2

= 3.14 x 52

= 3.14 x 25= 78.54

But there are 2 of them so78.54 x 2 = 157.075 units squared

C = x d= 3.1415 x 10 (radius doubled)= 31.15

Now use that as your base.A = b x h

= 31.15 x 4.2 (the height given)= 130.83 units squared

SA = 157.075 + 130.83 = 287.905 m2

Classwork

• Page 211-212 #4,8,9