Total Surface Area

Lateral Surface Area (LSA) includes only the side faces.

Total Surface Area (TSA) = all side faces + base(s) + top (if any).

For example:

• A cylinder’s LSA includes only the curved surface, while TSA includes the curved surface plus both circular ends.
• A cuboid’s LSA includes only four side walls, while TSA includes top and bottom too.

A cuboid has length \(l\), breadth \(b\), and height \(h\). It has 6 faces:

• 2 faces of size \(l \times b\)
• 2 faces of size \(b \times h\)
• 2 faces of size \(h \times l\)

So, its total surface area is:

A cube has 6 square faces, each of side \(a\). So the area of one face is \(a^2\).

Total surface area of a cube:

This represents the area of all six identical square faces.

A cylinder has:

• two circular ends, each of area \(\pi r^2\)
• a curved surface of area \(2\pi rh\)

So its total surface area is:

A cylinder has radius \(r = 4\,\text{cm}\) and height \(h = 7\,\text{cm}\). Its total surface area is:

Using \(\pi \approx 3.14\):

A right circular cone has:

• base radius \(r\)
• slant height \(l\)

The total surface area includes:

• curved surface area = \(\pi rl\)
• base area = \(\pi r^2\)

So TSA is:

If a frustum has radii \(R\) and \(r\) and slant height \(l\):

• CSA = \(\pi (R + r) l\)
• Areas of the two bases = \(\pi R^2 + \pi r^2\)

Total surface area:

A right circular cone has radius \(r = 5\,\text{cm}\) and slant height \(l = 13\,\text{cm}\).

Total surface area:

A sphere with radius \(r\) has a total surface area of:

A hemisphere has:

• curved surface area = \(2\pi r^2\)
• base area = \(\pi r^2\)

So the total surface area becomes: