Prism Total Surface Area

A prism can have multiple quantities of the same shape, all touching each other and having the same or different shape at the top, parallel to the shape at the bottom, matching in size and area covered. A cube of ice, a cardboard box, your smartphone are all examples of prisms. The total surface area of a prism is thus considered as the total area of the sides plus the area covered by the top and bottom bases. Based on the shape of the prism, the units calculated can be different. However, the foundation formula remains the same.

The formula for the Total surface area (TSA) of a prism

The calculation for the total surface area of a prism

The prism has two main parts, as discussed above. The area of the shapes is covered by its sides. And the area is covered by the same shape at the top and bottom.

Considering the shapes at the side, we have to calculate the area of the boundaries plus the height. This will translate to Perimeter X Height.

The top and bottom shapes have the same area. This will translate to a 2 X base area.

So the formula for the total surface area of a prism can be defined as

(2 X base area) + (perimeter X height)

This formula can be changed depending on the base of the prism. The reason that the base of a prism changes is due to the existence of many shapes. Triangles, squares, trapezoids, pentagons, hexagons and octagons have different bases, and the formula also changes accordingly.

For example, a pentagon has five lines in its base and five sides to it. So the formula for the total surface area of a pentagon prism is 5 (base area) + 5 (perimeter  height)

Similarly, a square has two pairs of 2 lines at its base, and the sides have four corners. So the formula for the total surface area of a prism in the shape of a square would be 2(base area)² + 4 (perimeter X height)Type of prisms

To understand the complexity of prisms and how the formula is set up to be different for each shape, we need to understand the different types of prisms.

• Regular prism – The base of the prism represents a regular polygon
• Irregular polygon – The base of a prism is irregular in shape
• Right prism – The bases are completely flat and parallel

• Oblique prism – the bases are bent and appear tilted
• Shape – Prisms based on shapes mentioned above like triangles, squares, trapezoids, pentagons, hexagons and octagons create diversity in the formula for the total surface area of a prism.

Apart from the total surface area of a prism the volume of a prism can also be determined by simply multiplying the prism’s base with the height. The curved surface area of a prism can also determine the curvature of the area.

There is one type of prism that does not fit the accepted description and measure explained above. A twisted prism is one where the top and bottom bases are not parallel. They are of the same size and area; however, they do not face each other parallelly. This is because the prism sides are not always of the same size. More variants of the twisted prism can be star prism, a frustum, crossed prism or a toroidal prism.

Conclusion 

In the year 300 B.CA prominent Greek Mathematician named Euclid wrote a set of books known simply as Elements that would document all mathematical and topographical knowledge until that period. It was widely publicised and became the definitive book for mathematical history and derivatives that has helped scientists and mathematicians to this day to understand concepts and make discoveries in complex theories.

Euclid was the first person to coin the word Prism in his series of books and gave shape and purpose to this particular type of figure. It provides a rather unclear explanation, which has been changed and simplified in today’s world for better understanding.