Area of Prism

Introduction

The area of a prism is defined as the total amount of space that the prism encloses in a three-dimensional space . The region that describes the substance that will be utilised to cover a geometric solid is known as the surface area. When calculating the surface areas of a geometric solid, we add the areas of all the geometric forms that make up the solid. A figure's volume, which is measured in cubic units, indicates how much it can store. We can learn something about a figure's capacity from its volume. A prism is a solid shape with two bases—two parallel congruent sides—that are joined by the parallelogram-shaped lateral faces. Prisms come in both triangular and rectangular shapes. In this tutorial we discuss the area of prisms.

Prisms

• A prism is a polyhedron in geometry made up of an n-sided polygon base, a second base that is a rigidly translated copy of the first base, and n additional faces that must all be parallelograms and connect the corresponding sides of the two bases.

• The bases are translated into all cross-sections that are parallel to them. Prisms are given names based on their bases, such as pentagonal prism for a prism having a pentagonal base. Prismoids are a class of prisms.

• The word prism, which derives from the Greek (Prisma)'something sawed,' was first used in Euclid's Elements, along with many other fundamental geometric words. "A solid figure encompassed by two opposed, equal, and parallel planes, with the rest being parallelograms," is how Euclid defined the term in Book XI.

Lateral Surface Area of a Prism

• The whole area of a prism's lateral faces makes up the lateral surface area. The sum of a prism's lateral face and base surface areas equals the area of the prism as a whole. In most cases, you can presume the prism is a right prism if "right" or "oblique" are not indicated.

• The standard formula for a right prism's lateral surface area is

  L.S.A.=ph,

  where p stands for the base's perimeter and h for the prism's height.

Example:

A triangular prism with bases that are 6 inches, 8 inches, and 10 inches wide and an altitude of 12 inches should have a lateral surface area.

Answer:

Perimeter of the base =6+8+10=24 inches

Given that height of the prism is h=12 inches

Thus, the lateral surface area of the prism is given by;

$$\mathrm{LSA=ph=24×12=288\: sq. inches}$$

TSA of Prism

The total surface area of a prism is equal to the sum of its lateral surface area and the areas of its two bases, which is equal to either 2 Base Area + Lateral Surface Area or 2 Base Area + (Base perimeter height). Prisms come in a variety of forms. The formulas used to calculate the surface area of the prism vary, just as the bases of various forms of prisms do. To comprehend this idea behind the surface area of various prisms, see the table below −

Shape	Base	Surface area of prism=2 base area+base perimeter×height
Triangular Prism	Triangular	bh+(s1+s2+b)H
Square Prism	Square	2a2+4ah
Rectangular Prism	Rectangle	2(lb+bh+lh)
Trapezoidal prism	Trapezoidal	h(b+d)+l(a+b+c+d)
Pentagonal Prism	Pentagonal	5ab+5bh
Hexagonal Prism	Hexagonal	6b(a+h)
Octagonal Prism	Octagonal Prism	4a2 (1+√2)+8aH

Solved Examples

1)Find the total surface area of an isosceles trapezoidal prism whose parallel base edges are 5 cm and 10 cm, whose base legs are each 4 cm, and whose base is 3cm in height.

Answer:

The lengths of the sides add up to the base's perimeter.

$$\mathrm{p=5+10+4+4=23cm}$$

Since the base is an isosceles trapezoid, its area is

$$\mathrm{=\frac{1}{2} h(b_1+b_2)=\frac{1}{2}×3×(5+10)=\frac{45}{2}\: sq. cm}$$

2)Find the surface area of the given prism, whose height is 8 units, base area is 16 square units, and base perimeter is 24 units.

Answer:

Given that base area of prism =16, base perimeter of prism =24, height of the prism =8

As we know that the surface area of prism is given by the formula

Surface area of prism=2 base area+base perimeter×height

Surface area of prism=2(16)+24×8=32+192=224 sq. units

3)Find the total surface area of an isosceles trapezoidal prism whose parallel base edges are 7 cm and 14 cm, whose base legs are each 6 cm, and whose base is 5cm in height.

Answer:

The lengths of the sides add up to the base's perimeter.

$$\mathrm{p=7+14+6+6=33cm}$$

Since the base is an isosceles trapezoid, its area is

$$\mathrm{=\frac{1}{2} h(b_1+b_2)=\frac{1}{2}×5×(7+14)=\frac{105}{2}\: sq. cm}$$

4)Find the surface area of the given prism, whose height is 10 units, base area is 20 square units, and base perimeter is 30 units.

Answer:

Given that base area of prism =20, base perimeter of prism =30, height of the prism =10

As we know that the surface area of prism is given by the formula

Surface area of prism=2 base area+base perimeter×height

Surface area of prism=2(20)+30×10=40+300=340 sq. units

5)A triangular prism with bases that are 12 inches, 16 inches, and 20 inches wide and an altitude of 12 inches should have a lateral surface area .

Answer:

Perimeter of the base =12+16+20=48 inches

Given that height of the prism is h=12 inches

Thus, the lateral surface area of the prism is given by

$$\mathrm{LSA=ph=48×12=576\: sq. inches}$$

Conclusion

• A prism is a polyhedron in geometry made up of an n-sided polygon base, a second base that is a rigidly translated copy of the first base, and n additional faces that must all be parallelograms and connect the corresponding sides of the two bases.

• There are various types of prisms based on the shape of bases as discussed in this tutorial.

• Surface area of prism=2 base area+base perimeter×height

FAQs

1. How do you calculate a prism's surface area?

SA=2B+Ph, where SA stands for surface area, B for area of the prism's base, P for its perimeter, and h for its height, can be used to calculate a prism's surface area.

2. What are the two methods for determining a prism's surface area?

Surface area is the total area occupied by a prism's sides. One of two methods can be used to determine the surface area. Utilizing a lateral area equation is one approach. The alternative method entails adding up the areas of all the sides.

3. Which prism has the smallest area of surface?

The cube has the least surface area amongst all types of prisms

4. What three types of prisms are there?

Based on Base Shape, Prism has following types, three-sided prism (has triangular bases) cube (has square bases) cuboid (has rectangular bases), etc.

5. What constitutes a prism's base?

A solid with two parallel and congruent faces is called a prism. These are referred to as the prism's bases. Any cross section of a prism that is parallel to those bases will have a cross section that resembles those bases if you cut through it in the same direction.