This topic covers how to find surface area, cross sectional area, volume and capacity of common three dimensional shapes. These measurements describe different aspects of solid objects and are used in many practical situations.

 

 

Surface Area

The surface area of a 3-D shape is the total area of all its faces.

 

Surface area is measured in square units, such as \( cm^2 \) or \( m^2 \).

 

Surface area is found by:
• identifying all faces
• finding the area of each face
• adding the areas together

 

 

Surface Area and Volume of a Cube

A cube has six equal square faces.

 

If the side length is \( a \):

 

Surface area of a cube is:
$$
6a^2
$$

 

Volume of a cube is:
$$
a^3
$$

 

Volume measures how much space the cube occupies.

 

 

Surface Area and Volume of a Cuboid

A cuboid has rectangular faces.

 

If the length is \( l \), width is \( w \) and height is \( h \):

 

Surface area of a cuboid is:
$$
2(lw + lh + wh)
$$

 

Volume of a cuboid is:
$$
lwh
$$

 

 

Surface Area and Volume of a Prism

A prism has a constant cross section.

 

Volume of a prism is found using:
$$
volume = cross\ sectional\ area \times length
$$

 

Surface area of a prism includes:
• the areas of the two identical ends
• the areas of the rectangular faces joining them

 

The exact calculation depends on the shape of the cross section.

 

 

Cross-Sectional Area

The cross sectional area is the area of a shape formed by slicing straight through a solid.

 

For prisms and cylinders, the cross sectional area is the same throughout the length.

 

Cross sectional area is used directly when finding volume.

 

 

Composite Solids

A composite solid is made by combining two or more simple solids.

 

To find volume:
• split the shape into known solids
• find the volume of each part
• add or subtract volumes as required

 

To find surface area:
• include only the outer faces
• do not include internal faces

 

Careful identification of faces is essential.

 

 

Cylinder

A cylinder has:
• two circular ends
• one curved surface

 

If the radius is \( r \) and the height is \( h \):

 

Cross sectional area of a cylinder is:
$$
\pi r^2
$$

 

Volume of a cylinder is:
$$
\pi r^2 h
$$

 

Curved surface area of a cylinder is:
$$
2\pi rh
$$

 

Total surface area of a cylinder is:
$$
2\pi r^2 + 2\pi rh
$$

 

 

Volume and Capacity

Volume is measured in cubic units, such as \( cm^3 \) or \( m^3 \).

 

Capacity measures how much liquid a container can hold and is usually measured in litres or millilitres.

 

There is an important conversion:
$$
1\ cm^3 = 1\ ml
$$

 

$$
1\ m^3 = 1000\ litres
$$

 

This allows volume to be converted into capacity when needed.

 

 

Key Points to Remember

Surface area is the total area of all faces.
Volume measures the space inside a solid.
Prism volume is cross sectional area multiplied by length.
Cylinder volume is found using \( \pi r^2 h \).
Composite solids must be split into simpler shapes.
Volume and capacity are linked through unit conversions.

 

Understanding surface area, volume and capacity allows three dimensional shapes to be analysed and used confidently in mathematical and real life contexts.