Surface Area Of A Cube

To derive the formula of the surface area of a cube, you will need to start with a cube as shown below and call the length of one side a.

A cube with side a

In order to make a cube like the one shown above, you basically use the following cube template:

Template of a cube

Looking at the cube template, it is easy to see that the cube has six sides and each side is a square

The area of one square is a × a = a2

Since there are six sides, the total surface area, call it SA is:

SA = a2 + a2 + a2 + a2 + a2 + a2

SA = 6 × a2

Example 1:

Find the surface area if the length of one side is 3 cm.

Surface area = 6 × a2

Surface area = 6 × 32

Surface area = 6 × 3 × 3

Surface area = 54 cm2

Surface Area of a Cuboid

The total surface area (TSA) of a cuboid is the sum of the areas of its 6 faces, which is given by:

TSA = 2 (lw + wh + hl)

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Remember the surface area is the total area of all the faces of a 3D shape.

The lateral surface area of a cuboid is given by:

LSA = 2 (lh + wh) = 2 h (l + w)

Find the surface area of a cuboid of dimensions 4.8 cm, 3.4 cm and 7.2 cm.

Solution:

Area of Face 1: 4.8 × 7.2 = 34.56 cm²

Area of Face 2: 3.4 × 7.2 = 24.48 cm²

Area of Face 3: 4.8 × 3.4 = 16.32 cm²

Adding the area of these 3 faces gives 75.36 cm², since each face is duplicated on the opposite side, the total surface area of the cuboid will be:

TSA = 2 (75.36) = 150.72 cm²

Surface area of a cylinder

Definition: The number of square units it takes to exactly cover the surface of a cylinder. Given by the formula:

area=2πr2+2πrh

where:

π is Pi, approximately 3.142

r is the radius of the cylinder

h height of the cylinder

The surface area of a cylinder can be found by breaking it down into three parts:

  • The two circles that make up the ends of the cylinder.
  • The side of the cylinder, which when "unrolled" is a rectangle

Combining these parts we get the formula:

area=2πr2+2πrh