# Surface Area of a Prism

The surface area, also known as the total surface area (TSA) of a prism, is the total space occupied by its flat faces. The surface area is measured in square units such as m2, cm2, mm2, or in2.

## Formulas

The general formula to find the total surface area of a prism is:

Total Surface Area (TSA) = 2 Ã— Base Area + Base Perimeter Ã— Height, here, the height of a prism is the distance between the two bases.

The total surface area of a prism is the combined area of the 2 bases and the areas of the lateral faces.

Since Lateral Surface Area (LSA) of a prism = Base perimeter x height

The lateral surface area is the area of all the faces except the bases.

We can also write:

Total Surface Area (TSA) = 2 Ã— Base Area + LSA

However, there are specific formulas to calculate the surface area of different prisms. They are given below:

Some formulas have additional labeling for particular prisms.

In triangular, rectangular, and trapezoidal prisms, â€˜lâ€™ (or length) stands for the distance between the bases, and â€˜hâ€™ stands for the height of the polygonal base.  â€˜lâ€™ is the length for a square prism, and â€˜aâ€™ represents the four congruent base edges. For pentagonal and hexagonal prisms, ‘a‘ is the apothem, and â€˜bâ€™ is the base edge.

Let us solve some examples involving the above formulas to understand the concept better.

## Solved Examples

Find the total surface area of a triangular prism whose base edges are 4 cm, 4 cm, 3 cm, height is 3.71 cm, and length is 6 cm.

Solution:

As we know,
Total Surface Area (TSA) =Â bÂ Ã—Â hÂ + (aÂ +Â bÂ +Â c) Ã—Â l, hereÂ aÂ =Â cÂ = 4 cm,Â bÂ = 3 cm,Â hÂ = 3.71 cm,Â lÂ = 6 cm
âˆ´Â TSA = 3 Ã— 3.71 + (4 + 3 + 4) Ã— 6
= 77.13 cm2

Find the total surface area of a hexagonal prism whose base edge is 3 cm, apothem is 2.6 cm and height is 4 cm.

Solution:

As we know,
Total Surface Area (TSA) =Â 6ab + 6bh, hereÂ aÂ = 6.5 cm, b = 3 cm,Â hÂ = 4 cm
âˆ´Â TSA = 6 Ã— 2.6 Ã— 3 + 6 Ã— 3 Ã— 4
= 6 Ã— 3(2.6 + 4)
= 118.8 cm2

Find the total surface area of a rectangular prism with a length of 7 cm, a width of 4 cm, and a height of 5 cm.

Solution:

As we know,
Total Surface Area (TSA) = 2(lwÂ +Â whÂ +Â lh), hereÂ lÂ = 7cm,Â wÂ = 4cm,Â hÂ = 5 cm
âˆ´Â TSA = 2(7 Ã— 4 + 4 Ã— 5 + 7 Ã— 5)
= 166 cm2

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