Surface Area of a Prism Formula

Surface Area of Prism

The surface area of a 3-dimensional solid prism depends upon the shape of its base. The surface area of a prism is the total area occupied by the faces of the prism. A prism is a polyhedron with flat faces. It has no curves.

The surface area of a prism refers to the amount of total space occupied by the flat faces of the prism. Finding the surface area of a prism means calculating the total space occupied by all the faces of that respective type of prism or the sum of the areas of all faces (or surfaces) in a 3D plane.

Surface Area of a Prism Formula

To find the surface area of any kind of prism we use the general formula. The total surface area of a prism is the sum of lateral surface area and area of two flat bases. Let us look at the surface area of the prism formula

The lateral area is the area of the vertical faces, in case a prism has its bases facing up and down. Thus, the lateral surface area of prism = base perimeter × height
The total surface area of a Prism = Lateral surface area of prism + area of the two bases = (2 × Base Area) + Lateral surface area or (2 × Base Area) + (Base perimeter × height).

There are various types of prisms. The bases of different types of prisms are different so are the formulas to determine the surface area of the prism. See the table below to understand this concept behind the surface area of various prism:

Shape	Base	Surface Area of Prism = (2 × Base Area) + (Base perimeter × height)
Triangular Prism	Triangular	Surface area of triangular prism = bh + (s1 + s2 + b)H
Square Prism	Square	Surface area of square prism = 2a2 + 4ah
Rectangular Prism	Rectangular	Surface area of rectangular prism = 2(lb + bh + lh)
Trapezoidal Prism	Trapezoidal	Surface area of trapezoidal prism = h (b + d) + l (a + b + c + d)
Pentagonal Prism	Pentagonal	Surface area of pentagonal prism = 5ab + 5bh
Hexagonal Prism	Hexagonal	Surface area of hexagonal prism = 6b(a + h) Surface area of regular hexagonal prism = 6ah + 3√3a2
Octagonal Prism	Octagonal	Surface area of octagonal prism = 4a2 (1 + √2) + 8aH

Check out types of prisms to get more details about various prisms.

Let us calculate the surface area of the triangular prism given below with a base “b”, the height of prism “h”, and length “L”.

The given prism has two triangular bases. Therefore, according to the surface area of the prism formula (2 × Base Area) + (Base perimeter × height). Here the base is triangular so the base area A = ½ bh, and the base perimeter = the sum of three sides of the triangle let’s say (a + b + c). On substituting the respective values in the formula we have, the surface area of a triangular prism = bh + (a + b + c)H = .(2A + PH)

The steps to determine the surface area of the prism are:

• Step 1: Note down the given dimensions of the prism.
• Step 2: Substitute the dimensions in the surface area of prism formula (2 × Base Area) + (Base perimeter × height).
• Step 3: The value of the surface area of the prism is obtained and the unit of the surface area of the prism is placed in the end (in terms of square units).

Example: Find the surface area of a prism given above whose base area is 12 square units, the base perimeter is 18 units and the height of the prism is 6 units.

Solution: As we know, the surface area of the prism is given as
Surface Area of Prism = (2 × Base Area) + (Base perimeter × height)
Base area = 12 square units
Base perimeter = 18 units
Height of the prism = 6 units
Thus, Surface Area of Prism = (2 × 12) + (18 × 6)
⇒ S = 132 units²
∴ The surface area of prism is 132 square units.

Solved Examples

Triangular Prism

A prism with a triangular base is referred to as a triangular prism. A triangular prism consists of three inclined rectangular surfaces and two parallel triangle bases. Let “H” be the height of the triangular prism; “a, b, and c” are the sides’ lengths, and “h” is the height of the triangular bases.

The perimeter of a triangular base (P) = Sum of its three sides = a + b + c

The area of a triangular base (A) = ½ × base × height =  ½ bh

We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.

By substituting all the values in the general formula we get,

The lateral surface area of a triangular prism = (a + b +c)H square units

where,

a, b, c are sides of triangular base

H is height of  triangular prism

We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.

By substituting all the values in the general formula we get

The total surface area of the triangular prism = (a + b + c)H + 2 × (½ bh)

The total surface area of the triangular prism = (a + b + c)H + bh square units

where,

a, b, c are sides of triangular base

H is height of  triangular prism

h is height of triangle

Rectangular Prism

A prism with a rectangular base is referred to as a rectangular prism. A rectangular prism consists of four rectangular surfaces and two parallel rectangular bases. Let the prism’s height be “h” and its rectangular bases’ length and width be “l” and “w,” respectively.

The perimeter of a rectangular base (P) = Sum of its four sides = 2 (l + w)

The area of a rectangular base (A) = length × width =  l × w

We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.

By substituting all the values in the general formula we get,

The lateral surface area of a rectangular prism = 2h(l + w) square units

where,

l is length

w is width

h is height

We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.

By substituting all the values in the general formula we get

The total surface area of the rectangular prism = 2h(l + w) + 2(l × w)

= 2 lh + 2 wh + 2 lw

The total surface area of the rectangular prism = 2 (lh + wh + lw) square units

where,

l is length

w is width

h is height

Square Prism

A prism with a square base is referred to as a square prism. A square prism consists of four rectangular surfaces and two parallel square bases. Let the prism’s height be “h” and its square bases’ lengths be “s”.

The perimeter of a square base (P) = Sum of its four sides = s + s + s + s = 4s

The area of a square base (A) = (length of the side)² =  s²

We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.

By substituting all the values in the general formula we get,

The lateral surface area of a square prism = 4sh square units

where,

s is side of square base

h is height of square prism

We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.

By substituting all the values in the general formula we get

The total surface area of the square prism = [4sh + 2s²] square units

where,

s is side of square base

h is height of square prism

Pentagonal Prism

A prism with a pentagonal base is referred to as a pentagonal prism. A pentagonal prism consists of five inclined rectangular surfaces and two parallel pentagonal bases. Let “h” be the height of the pentagonal prism; “a and b” be the apothem length and side lengths of the pentagonal bases.

The perimeter of a pentagon base (P) = Sum of its five sides = 5b

The area of a pentagon base (A) = 5/2 x (apothem length) x (length of the side) = 5ab

We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.

By substituting all the values in the general formula we get,

The lateral surface area of a pentagonal prism = 5bh square units

where,

b is side of pentagonal base

h is the height of pentagonal prism

We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.

By substituting all the values in the general formula we get,

The total surface area of the pentagonal prism = [5bh + 5ab] square units

where,

b is side of pentagonal base

a is apothem length.

h is the height of pentagonal prism

Hexagonal Prism

A prism with a hexagonal base is referred to as a hexagonal prism. A hexagonal prism consists of six inclined rectangular surfaces and two parallel hexagonal bases. Let “h” be the height of the hexagonal prism; “a” be the side lengths of the hexagonal bases.

The perimeter of a hexagon base (P) = Sum of its six sides = 6a

The area of a hexagon base (A) = 6 x (Area of an equilateral triangle) 

A = 6 x (√3a²/4) ⇒  A = 3√3a²/2

We know that the general formula for the lateral surface area of a right prism is L. S. A. = PH, where P is the base perimeter, and A is the base area.

By substituting all the values in the general formula we get,

The lateral surface area of a hexagonal prism = 6ah square units

where,

a is side of hexagonal base

h is height of hexagonal base

We know that the general formula for the total surface area of a right prism is T. S. A. = PH+2A, where P is the base perimeter, A is the base area, and H is the height of the prism.

By substituting all the values in the general formula we get

The total surface area of the hexagonal prism =  [6ah +3√3a² ]  square units

where,

a is side of hexagonal base

h is height of hexagonal base

Surface Area of Prism Formula

Shape	Base of the prism	Lateral surface area  [Base perimeter × height]	Total Surface Area [(2 × Base Area) + (Base perimeter × height)]
Triangular Prism	Triangle	(a + b +c)H square units	(a + b + c)H + bh square units
Rectangular Prism	Rectangle	2h(l + w) square units	2 (lh + wh + lw) square units
Square Prism	Square	4sh square units	[4sh + 2s2] square units
Pentagonal Prism	Pentagon	5bh square units	[5ab + 5bh] square units
Hexagonal Prism	Hexagon	6ah square units	[3√3a2 + 6ah] square units

Sample Problems

Problem 1: What is the height of a prism whose base area is 36 square units, its base perimeter is 24 units, and its total surface area is 320 square units?

Solution:

Given data,

Base area = 36 square units

Base perimeter = 24 units

The total surface area of the prism = 320 square units

We have,

The total surface area of the prism = (2 × Base Area) + (Base perimeter × height)

⇒ 320 = (2 × 36)+ (24 × h)

⇒ 24h = 248 ⇒ h = 10.34 units

Hence, the height of the given prism is 10.34 units.

Problem 2: Find the total surface area of a square prism if the height of the prism and the length of the side of the square base are 13 cm and 4 cm, respectively.

Solution:

Given data,

The height of the square prism (h) = 13 cm

The length of the side of the square base (a) = 4 cm

We know that,

The total surface area of a square prism = 2a² + 4ah

 = 2 × (4)² + 4 × 4 × 13

= 32 + 208 = 240 cm²

 Hence, the total surface area of the given prism is 240 sq. cm.

Problem 3: Determine the base length of a pentagonal prism if its total area is 100 square units and its height and apothem length are 8 units and 5 units, respectively.

Solution:

Given data,

The total surface area of the pentagonal prism = 100 square units

The height of the prism (h) = 8 units

Apothem length (a) = 5 units

We know that,

The total surface area of the pentagonal prism = 5ab + 5bh

⇒ 100 = 5b (a+ h)

⇒ 100/5 = b (5 + 8)

⇒ 20 = b × (13) ⇒ b = 25/16 = 1.54 units

Hence, the base length is 1.54 units

Problem 4: Determine the height of the rectangular prism and the total area of a rectangular prism if its lateral surface area is 540 sq. cm and the length and breadth of the base are 13 cm and 7 cm, respectively.

Solution:

Given data,

The length of the rectangular base (l) = 13 cm

The width of the rectangular base (w) = 7 cm

The lateral surface area of the prism = 540 sq. cm

We have,

The lateral surface area of the prism = Base perimeter × height

⇒ 540 = 2 (l + w) h 

⇒ 2 (13 + 7) h = 540

⇒ 2 (20) h = 540 ⇒ h = 13.5 cm

We know that,

The total surface area of the rectangular prism = 2 (lw + wh + lh)

= 2 × (13 × 7 + 7 × (13.5) + 13 × (13.5))

= 2 × (91 + 94.5 + 175.5) = 722 sq. cm

Hence, the height and total surface area of the given rectangular prism are 13.5 cm and 722 sq. cm, respectively.

Problem 5: Determine the surface area of the regular hexagonal prism if the height of the prism is 12 in and the length of the side of the base is 5 in.

Solution:

Given data,

The height of the prism (h) = 12 in

The length of the side of the base (a) = 6 in

The surface area of a regular hexagonal prism = 6ah + 3√3a²

= 6 × 5 × 12 + 3√3(5)²

= 360 + 75√3

= 360 + 75 × (1.732) = 489.9 sq. in

Hence, the surface area of the given prism is 489.9 sq. in.

Problem 6: Calculate the lateral and total surface areas of a triangular prism whose base perimeter is 25 inches, the base length and height of the triangle are 9 inches and 10 inches, and the height of the prism is 14 inches.

Solution:

Given data,

The height of the prism (H) = 14 inches

The base perimeter of the prism (P) = 25 inches

The base length of the triangle = 9 inches

The height of the triangle = 10 inches

We know that,

The lateral surface area of the prism = Base perimeter × height

= 25 × 14= 350 sq. in

Area of the triangular base (A) = ½ × base × height = 1/2 × 9 × 10 = 45 sq. in

The total surface area of the triangular prism = 2A + PH 

= 2 × 45 + 25 × 14 = 90 + 350 = 440 sq. in 

Hence, the prism’s lateral and total surface areas are 350 sq. in and 440 sq. in, respectively.

Example 1:

Find the lateral surface area of a triangular prism with bases edges 3 inches, 4 inches and 5 inches and altitude 8 inches.

The perimeter is the sum of the bases.

p=3+4+5=12 inches

Lateral Surface Area =12(8)=96  inches2

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

There is no easy way to calculate the surface area of an oblique prism in general. The best way is to find the areas of the bases and the lateral faces separately and add them.

Example 2:

Find the total surface area of an isosceles trapezoidal prism with parallel edges of the base 6 cm and 12 cm, the legs of the base 5 cm each, the altitude of the base 4 cm and height of the prism 10 cm.

The perimeter of the base is the sum of the lengths of the sides.

p=6+5+12+5=28  cm

Solved Examples

Question 2: Calculate the surface area of a pentagonal prism with apothem length, base length and height as 12 cm, 15 cm, and 21 cm respectively ?
Solution:

Surface Area of a Rectangular Prism

The surface area of a rectangular prism is the measure of how much-exposed area a prism has. Surface area is expressed in square units. The total surface area of a rectangular prism is the sum of the lateral surface area (LSA) and twice the base area of the rectangular prism.

Total Surface Area of rectangular prism = LSA + 2 (Base area)       [Square units]

The lateral surface area of a rectangular prism is the sum of the surface area of all its faces without the base of the rectangular prism. The lateral surface area of any right rectangular prism is equivalent to the perimeter of the base times the height of the prism.

Therefore, the lateral surface area = P x h [Square units]

Where

P is the perimeter of a base

h be the height of the prism

The perimeter of the rectangular prism is,

Therefore, the lateral surface area (LSA) of a rectangular prism = 2 ( l + w ) h square units.

Hence,

TSA = LSA + 2 (Base Area) = 2 (l + w) h + 2 (l x w) = 2 lh + 2 wh + 2 lw   [Square units]

Therefore, the surface area of a rectangular prism formula is given as,

Solved Examples on Rectangular Prism

Question 1: Find the volume of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively.

Solution:

Given:

Length, l = 8 cm

Width, w = 6 cm

Height, h = 4 cm

The formula to find the volume of a rectangular prism is,

V = Length x Width x Height cubic units

V = 8 x 6 x 4 cm³

V = 192 cm³

Therefore, the volume of a rectangular prism is 192 cm³.

Question 2: Find the surface area of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively.

Solution: Given:

Length, l = 8 cm

Width, w = 6 cm

Height, h = 4 cm

The formula to find the area of a rectangular prism is,

A = 2 (lh +wh + lw )

A = 2 (8×4+6×4+8×6)

A = 2(32+24+48)

A = 2(104)

A = 208 sq.cm.

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