Volume And Surface Area

IX C

CONTENTSCUBOID

CUBE

CYLINDER

CONE

SPHERE

HEMISPHERE

CUBOID

To find the surface area of a shape, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The top and the bottom of the cuboid have the same area.

Surface area of a cuboid

To find the surface area of a shape, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The front and the back of the cuboid have the same area.

Surface area of a cuboid

To find the surface area of a shape, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The left hand side and the right hand side of the cuboid have the same area.

Surface area of a cuboid

We can find the formula for the surface area of a cuboid in the way which is shown below.

Surface area of a cuboid =

Formula for the surface area of a cuboid

h

l b

2 × lb Top and bottom

+ 2 × hb Front and back

+ 2 × lh Left and right side

= 2lb + 2hb + 2lh= 2( lb + bh + lh )

10 cm4 cm

6 cm

VOLUME OF A CUBOIDLook at this cuboid. Now imagine it is full of cubic centimetres.

Can you see that there are 10 4 = 40 cubic centimetres on the bottom layer?

There are 6 layers of 40 cubes making 40 6 = 240 cm3

1 cm3

10 cm4 cm

6 cm

VOLUME OF A CUBOID

lengthbreadth

height

When we worked out the volume, we multiplied the length by the breadth and then by the height.

Volume of a cuboid = length breadth heightor

V = l b h

10 cm4 cm

6 cm

V = l b h

= 10 4 6 cm3

= 240 cm3

Lets us look again at the same cuboid and this time try the formula.

CUBE

How can we find the surface area of a cube of length a?

Surface area of a cube

x

All six faces of a cube have the same area.

The area of each face is a × a = a2

Therefore,

Surface area of a cube = 6a2

Volume of a cube Volume of a cube is calculated by A cube is a cuboid with all the edges (a)

equal. Volume of a cube = lbh

Volume of a cube = a3

V = (4 x 4) x 4 = 64 m3

Cylinder

This will happen if we unrolled and removed the end of a cylinder….

h

2Πr2

C.S.A = Area of the rectangle = 2Πr2 X h = 2Πr2h

Notice that we had formed 2 circles and a 1 rectangle….

The 2 circles serves as our bases of our cylinder and the rectangular region represent the body

This is the formula in order to find the surface area of a cylinder.

T.S.A. = Area of 2 circular bases + Area of the rectangle

T.S.A = 2πr2 + 2πrh T.S.A = 2πr(r+h)

Volume of A cylinder

Volume of a cylinder = Area of the base area x height

= πr2 x h = πr2h

CONE

A cone has a circular base and a vertex that is not in the same plane as a base.

In a right cone, the height meets the base at its center.

The height of a cone is the perpendicular distance between the vertex and the base.

The slant height of a cone is the distance between the vertex and a point on the base edge.

Height

Lateral Surface

The vertex is directly above the center of the circle.

Baser

Slant Height

r

Surface Area of a Cone

C.S.A = πrl T.S.A = πr2 + πrl

r

h

Comparing Cone and Cylinder

• Use plastic space figures.• Fill cone with water.• Pour water into cylinder.• Repeat until cylinder is full.

r r

h

Volume of Cone

• 3 cones fill the cylinder, so…

• Volume = ⅓ Base Area x height

Volume = 1/3 πr2h

=

SPHERE

Surface Area of a Sphere

Surface Area of a Sphere= 4πr2

Volume of a Sphere

Using relational solids and pouring material we noted that the volume of a cone is the same as the volume of a hemisphere (with corresponding dimensions)

Using math language Volume (cone) = ½ Volume (sphere)

Therefore 2(Volume (cone)) = Volume (sphere)

=OR+

2(Volume (cone)) = Volume (sphere)

2( ) (height) /3= Volume (sphere)

2( )(h)/3= Volume (sphere)

BUT h = 2r 2(r2)(2r)/3 = Volume(sphere)

4 ( r3)/3 = Volume(sphere)

Volume of a Sphere

Area of Base

r2

2 X =

hr

r

HEMISPHERE

Surface area of hemisphere

Total surface area of a hemisphere:

3Пr 2

Curved surface area of a hemisphere:

2Пr 2

Volume of hemisphere

Volume of a hemisphere:2/3 Пr 3

CONVERSION OF UNITS

Conversion of units• 1cm =10mm• 1m = 100cm• 1km = 1000m

Conversions of Units1 cm2 = 10 mm x 10 mm =100 mm2

1 m2 = 100 cm x 100 cm = 10 000 cm2

1 m2 = 1000 mm x 1000 mm = 10 00 000 mm2

CUBIC UNITS• 1 cm3

• = 1cm x 1cm x 1cm• = 10 mm × 10 mm × 10 mm• = 1000 mm3

• 1 m3

• = 1m x 1m x 1m• = 100 cm × 100 cm × 100 cm• = 1 000 000 cm3

VOLUME• The volume of

three-dimensional figure is the amount of space within it.

• Measured in cubic unit.

CAPACITY• Capacity is the

amount of material usually liquid) that a container can hold.

• Measured in millilitres, litres and kilolitres.

Volume and capacity are related.

How does Volume relate to Capacity?

• 1000 mL = 1 L• 1000 L = 1 kL• 1 cm3 = 1 mL• 1000cm3 = 1000ml = 1L

• 1 m3 = 1000 L = 1 kL

Examples of Capacity

Achieved byViolet Black