Area and Volume

 

Perimeter

Perimeter = distance around the edge.

1

You could walk around the perimeter.

All dimensions must have the same units !

Don't mix cm with m.

Perimeter has plain units.

 

Example

an1

P = 5 + 2 + 2 + 3 + 9 + 3 + 2 + 2 cm
P = 28 cm

 

Area

Area = floor space covered

 1

 

You could paint an area.

All dimensions must have the same units !

Don't mix cm,2 with m2

1 m2 = 100cm x 100cm

= 10000 cm2

 

 

Area  has units2.

 

 

Area of a square

1

1

      

Example

Calculate the area of the square

5 98

Area of a rectangle

2

12

Examples

Calculate the area of the rectangles

1 99

 

7 100

   

 

Area of a triangle

Area of a triangle = ½ x base x perpendicular height

1 1

101

 

Examples

Find the area of the triangle below:

1 12

 

What is the length of the base of the triangle, if it has an area of 45 cm2 ?

1

103

 

 

Area of a circle

1

 

Area of kite

   

ani

 

  106

Example

Calculate the area of the following kite:

12

 

107


Area of Trapezium

 

4

Area of a trapezium = ½ x average of base x perpendicular height

104

 

Example

What is the area of this trapezium ?
(Each square represents 1 cm2 )

23

105

  

Area of a parallelogram

3

108

 

Example

Calculate the area of the parallelogram :

 

103

 

109

 

Area of a rhombus

12

10

 

Example

Calculate the area of the rhombus:

( The sizes are for the complete diagonals)

3

11

 

Volume

Volume = capacity held

1

You could fill a volume

All dimensions must have the same units !

Don't mix cm with m.

 

Volume  has  units3

Notice that for a cuboid

12

Example


Calculate the volume of the cuboid below:


23 12

 

Example

Converting 1m3 to litres

5

First, convert the units

12

But 1 cm3 = 1 ml and 1000 ml = 1litre
Divide cm3 by 1000 for litres.

So  1 000 000 cm3 = 1000 litres
           1 m3 = 1000 litres

 

Volume of a Sphere

A sphere has volume

  15

Where r is the radius of the sphere.

Examples


Calculate the volume of the following sphere.
Give your answer correct to 1 dp and also to 2 sig figs.

1  12

 

Calculate the volume of the following sphere.
Give your answer correct to 1 sig fig.

1 17

  

Calculate the diameter of a sphere which has a volume of 700cm3.
Give your answer correct to 1 dp.

1

 

  18

 

Volume of a Cone

 

A cone has volume

  1

Where r is the radius of the circular part of the cone  and h is the perpendicular height of the cone.

 

1

 

 

Example
Calculate the volume of an ice cream cone which has a diameter of 4cm and a height of 6cm. Give your answer correct to 1 dp.

120

 

How many of these cones can be filled from 1litre of ice cream ?
1000 cm3 = 1 l

1000 ÷ 25.1 = 39.84
So 39 cones can be filled from one litre of ice cream.

 

Example
Calculate the height an ice cream cone which has a diameter of 4cm and a volume of 35ml. Give your answer correct to 1 dp.

12

 

The cone is 8.4 cm tall.

 

Example
Calculate the diameter of an ice cream cone which has a height of 8cm and a volume of 90ml. Give your answer correct to 1 dp.

122

 

Volume of a prism

For a prism,  V=Ah

So Volume = Area x height   (or Area x Length if laying down)

    
Example

What is the volume of  a prism which has an area of 37 cm2    and a height of 4 cm ?

4

Volume of a cylinder

 A cylinder is a circular prism,

12

12

 

Example

Calculate the volume of a tin can which has a height of 0.8m and a diameter of 10 cm. Give your answer correct to 1 sigfig.

  124

Example
Calculate the diameter of a tin can which has a height of 8cm and a volume of 90ml. Give your answer correct to 1 dp.

125

 

Volume of a pyramid

The volume of any pyramid is given as

12

where A is the area of the base of the pyramid and h is its height.

Examples

What is the volume of this squared based pyramid ?

10 127

 

What is the volume of this rectangular based pyramid ?

12 12

 

What is the volume of this triangular based pyramid ?

12 129

 

Surface area

The surface area is the total external area
of the shape.

 

Example

Find the surface area of the cuboid :

1

This shape has 6 faces

2 faces have area    6cm x 4cm
2 faces have area    6cm x 2cm
2 faces have area    2cm x 4cm

    2 x 6cm x 4cm =  48 cm2
    2 x 6cm x 2cm =  24 cm2
    2 x 2cm x 4cm =  16  cm2
     Surface Area =   88  cm2

 

Surface Area ≠Volume

 

Composite area

Cut into convenient shapes
Find missing dimensions
Calculate individual areas
Calculate  total

 

            Remember
all dimensions must have the same units !

Example

1 

A shape = A 1 + A 2
A 1 = 5x2 = 10 cm2
A 2 = 3x9 = 27 cm2
A shape = 37 cm2

Composite Volume

Cut into convenient shapes
Find missing dimensions
Calculate individual areas
Calculate  total

Example

1111

 

1 2  5

© Alexander Forrest