Surface Area of a Prism – Explanation & Examples (2024)

Surface Area of a Prism – Explanation & Examples (1)The total surface area of a prism is the sum of areas of its lateral faces and its two bases.

In this article, you will learn how to find the total surface area of a prism by using the surface area of a prism formula.

To recall, a prism is a 3-dimensional polyhedron with two parallel and congruent bases, which are connected by lateral faces. A prism is named according to the shape of the polygonal bases. In a prism, the lateral faces, which are parallelograms, are perpendicular to the polygonal bases.

How to Find the Surface Area of a Prism?

  • To find the total surface area of a prism, you need to calculate the area of two polygonal bases, i.e., the top face and bottom face.
  • And then calculate the area of lateral faces connecting the bases.
  • Add up the area of the two bases and the area of the lateral faces to get the total surface area of a prism.

Total surface area of a prism formula

Since we know the total surface area of a prism is equal to the sum of all its faces, i.e., the floor, walls, and roof of a prism. Therefore, the surface area of a prism formula is given as:

Total surface area of a prism = 2 x area of the base + perimeter of the base x Height

TSA = 2B + ph

Where TSA = Total surface area of a prism

B = Base area

p = perimeter of the base

h = height of the prism

Note: The formula to find the base area (B) of a prism depends on the base’s shape.

Let’s solve a few example problems involving the surface area of different types of prisms.Surface Area of a Prism – Explanation & Examples (2)

Example 1

The dimensions of a triangular prism are given as follows:

Apothem length of the prism, a = 6 cm

Base length = 4 cm

height of the prism, h = 12 cm

The other two sides of the triangular base are 7 cm each.

Find the total surface area of the triangular prism.

Solution

By the formula,

TSA = 2 x area of the base + perimeter of the base x Height

Since the base is a triangle, then the base area, B =1/2 ba

=1/2 x 4 x 6

= 12 cm2.

Perimeter of the base, p = 4 + 7 + 7

= 18 cm

Now substitute the base area, height, and perimeter in the formula.

TSA = 2B + ph

= 2 x 12 + 18 x 12

= 24 + 216

= 240 cm2

Therefore, the total surface area of the triangular prism is 240 cm2.

Example 2

Find the total surface area of a prism whose base is an equilateral triangle of side 8 cm and height of the prism is 12 cm.

Solution

Given:

Height of the prism, h = 12 cm

The base is an equilateral triangle of side 8 cm.

By Pythagorean theorem, the apothem length, a of the prism is calculated as:

a = √ (82 – 42)

= √ (64 – 16)

= √ 48 = 6.93

Thus, the apothem length of the prism is 6.93 cm

Base area, B = ½ b a

= ½ x 8 x 6.93

= 27.72 cm2

Perimeter of the base = 8 + 8 + 8

= 24 cm

TSA = 2B + ph

= 2 x 27.72 + 24 x 12

= 55.44 + 288

= 343.44 cm2.

Hence, the total surface area of the prism is 343.44 cm2.

Example 3

The apothem length, base length, and height of a pentagonal prism are 10 cm. 13 cm, and 19 cm, respectively. Find the total surface area of the pentagonal prism.

Solution

The formula for the total surface area of a pentagonal prism is given by;

TSA = 5ab + 5bh

Where

By substitution, we have,

TSA = 5 x 10 x 13 + 5 x 13 x 19

= 650 +1235

= 1885 cm2

Thus, the total surface area of the pentagonal prism is 1885 cm2

Example 4

A rectangular prism of dimensions, length = 7 in, width = 5 in and height = 3 in is to be painted. If the painting cost is $50 per square inch, find the total cost of painting all faces of the prism.

Solution

First, calculate the total surface area of the prism

Surface area of a rectangular prism = 2h (l +b)

= 2 x 3 (7 + 5)

= 6 x 12

TSA = 72 in2

The total cost of painting the prism = TSA x cost of painting

= 72 x 50

= $3,600

Thus, the cost of painting the rectangular prism is $3,600

Example 5

Find the total surface area of a hexagonal prism whose apothem length, base length, and height are given as 7 m, 11 m, and 16 m, respectively.

Solution

The total surface area formula for a hexagonal prism is given as:

TSA = 6ab + 6bh

Substitute.

TSA = 6 x 7 x 11 + 6 x 11 x 16

= 462 + 1056

=1518 m2

Example 6

Calculate the total surface area of an isosceles trapezoid whose parallel sides of the base are 50 mm and 120 mm and legs of the base are 45 mm each, the height of the base is 40 mm, and the height of the prism is 150 mm.

Solution

The total surface area of a trapezoid prism = 2B + ph

Base area (B) of a trapezoid = 1/2h (b1 + b2)

= ½ x 40(50 + 120)

= 20 x 170

= 3400 mm2

Perimeter (p) of the base = 50 + 120 + 45 + 45

= 260 mm

Now, substitute into the formula.

TSA = 2 x 3400 + 260 x 150

= 6,800 + 39,000

= 45,800 mm2

Surface Area of a Prism – Explanation & Examples (2024)

FAQs

What is the surface area of the prism answer? ›

To find the surface area of a prism, use the formula SA=2B+ph, where SA stands for surface area, B stands for area of the base of the prism, p stands for the perimeter of the base, and h stands for the height of the prism. Since this is a triangular prism, substitute the area formula of a triangle for B.

What is the best way to explain surface area? ›

Surface area measures the space needed to cover the outside of a three-dimensional shape. Surface area is the sum of the areas of the individual sides of a solid shape. Surface area is measured in square units. There are formulas to find the surface area of different solid shapes.

What is surface area of prism grade 6? ›

The surface area of a prism is the sum of the areas of all its faces. A two-dimensional representation of a solid is called a net.

What is the rule for the surface area of a prism? ›

The general formula for the total surface area of a right prism is. S . A . = p h + 2 B where represents the perimeter of the base, the height of the prism and the area of the base.

How to find the surface area of a right prism? ›

The formula for finding the surface area is the same for all right prisms and is: Surface area = 2B + hP. In this formula, B is the area of one of the bases, while h is the prism's height, and P is the perimeter of the base.

Which is a correct method for finding the surface area of prisms? ›

To calculate the total surface area of a prism:
  • Find the area of the two end faces.
  • Work out the area of all the rectangular faces in one of two ways: Work out the area of each rectangle separately, length × width. Multiply the perimeter of the end face by the length of the prism.
  • Sum the areas of all the faces.

How to help students understand surface area? ›

You can provide student groups with a 3D box, they measure the dimensions (length, width, height), and then carefully cut to form the rectangular prism's net. They will see those same dimensions that form the box's net.

What is the basic formula for surface area? ›

Variables:
Surface Area FormulaSurface Area Meaning
SA=2B+PhFind the area of each face. Add up all areas.
SA=B+12sPFind the area of each face. Add up all areas.
SA=2B+2πrhFind the area of the base, times 2, then add the areas to the areas of the rectangle, which is the circumference times the height.
2 more rows
Sep 17, 2020

What is the surface area of a prism rectangular prism? ›

The surface area of a rectangular prism = 2 (lw+lh+hw) square units.

How do you find the surface area of a prism formula sheet? ›

Surface Area of Prism Formula
ShapeBase of the prismTotal Surface Area [(2 × Base Area) + (Base perimeter × height)]
Triangular PrismTriangle(a + b + c)H + bh square units
Rectangular PrismRectangle2 (lh + wh + lw) square units
Square PrismSquare[4sh + 2s2] square units
Pentagonal PrismPentagon[5ab + 5bh] square units
1 more row
Jun 4, 2024

What is the surface of the prism? ›

The surface area of a prism is given as S = (2 × Base Area) + (Base perimeter × height) where "S" is the surface area of the prism.

How do you find the surface area? ›

Surface area is total area on the surface of a three-dimensional shape. To find the surface area of a cuboid which has 6 rectangular faces, add the areas of all 6 faces. Or, you can label the length (l), width (w), and height (h) of the cuboid and use the formula: surface area (SA)=2lw+2lh+2hw.

What is the formula for a prism? ›

Formula for the volume of a prism:

Prism volume (V) = B × h, where, B is the area of the base and h is the height of the prism.

What is the answer to the surface area of a rectangular prism? ›

The surface area of a rectangular prism = 2 (lw+lh+hw) square units.

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