# Formula for Surface Area of Cuboid – Derivation and Examples

Here you will learn formula for surface area of cuboid (total surface area and lateral surface area), its derivation and examples based on it.

Let’s begin –

## What is Cuboid ?

A cuboid is a three dimensional box. It is defined by the virtue of its length l, breadth b and height h. It can be visualised as a room which has length, breadth and height differerent from each other.

## Formula for Surface Area of Cuboid

#### (a) Total Surface Area of Cuboid (T.S.A)

T.S.A = 2(lb + bh + hl)

#### (b) Lateral Surface Area of Cuboid (L.S.A)

L.S.A = 2(l + b)h

### Derivation :

The outer surface of a cuboid is made up of six rectangles (rectangular regions, called the faces of the cuboid), whose areas can be found by multiplying length by breadth for each of them seperately and then adding the six areas together.

Now, if we take the length of cuboid as l, breadth as b and the height as h, then

The sum of areas of the six rectangles is:

Area of rectangle 1 (= l × h) + Area of rectangle 2 (= l × b) + Area of rectangle 3 (= l × h) + Area of rectangle 4 (= l × b) + Area of rectangle 5 (= b × h) + Area of rectangle 6 (= b × h)

= 2(l × b) + 2(b × h) + 2(l × h)

= 2(lb + bh + hl)

This gives us: Total Surface Area of a Cuboid = 2(lb + bh + hl)

Suppose, out of six faces of a cuboid, we only find the area of four faces, leaving the bottom and top faces. Then in such case, the area of these four faces is called the lateral surface area of the cuboid. So, lateral surface area of a cuboid of length l, breadth b and height h is equal to 2lh + 2bh or 2(l + b)h.

Then  Lateral Surface Area = 2(l + b)h

Example : If we have a cuboid whose length, breadth and height are 15 cm, 10 cm and 20 cm respectively, then find its total surface area.

Solution : We have l = 15 cm , b = 10 cm and h = 20 cm

Total Surface Area = 2[(15 $$\times$$ 10) + (10 $$\times$$ 20) + (20 $$\times$$ 15)] $$cm^2$$

Total Surface Area = 2(150 + 200 + 300) $$cm^2$$

= 2 × 650 $$cm^2$$

= 1300 $$cm^2$$