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.
What is the Surface Area of Prism?
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 = 2a^{2} + 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√3a^{2} |
Octagonal Prism | Octagonal | Surface area of octagonal prism = 4a^{2} (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)
How to Calculate the Surface Area of Prism?
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 units^{2}
∴ The surface area of prism is 132 square units.
Solved Examples
Example 1: What will be the surface area of the triangular prism if the base and height of a triangular prism are 8 units and 14 units respectively along with the height of the equilateral triangular bases being 9 units?
Solution: Given information is base = 8 units, height of the base = 9 units, length of each side of the base = 8 units, and height of the prism = 14 units
Surface area of a triangular prism = (bh + (a + b + c)H)
We know that all three sides of an equilateral triangle are equal. Therefore, a = b = c = 8 units
Surface area = (8 × 9) + (8 + 8 + 8) × 14
Surface area = 72 + 24 × 14
Surface area = 72 + 336
Surface area = 408 units^{2}
Therefore, the surface area of the given triangular prism is 408 units^{2}
Example 2: Find the lateral surface area of the prism given if the perimeter of the base is 200 inches, height of the prism is 75 inches. Also, find the surface area of the prism if the base area is 250 sq. inches.
Solution: The bases of the prism are polygons. Thus, the lateral surface area of the prism, L = perimeter of base × height of prism
Lateral surface area = 200 × 75 = 15,000 in^{2}
S = (2 × Base Area) + (Base perimeter × height) where “S” is the surface area of the prism.
Base area = 250 in^{2}
S = (2 × 250) + (15000)
S = 500 + 15000 = 15500 in^{2}
Therefore, the lateral surface area of the given prism is 15,000 in^{2,} and the surface area of the prism is 15500 in^{2}.
FAQs on Surface Area of Prism
What is the Definition of the Surface Area of Prism?
The amount of area occupied by a prism is referred to as the surface area of a prism. The surface area of the prism depends on the base area of the prism and the lateral surface area of the prism. The unit of the surface area of the prism is expressed in m^{2}, cm^{2}, in^{2}, or ft^{2}.
What is the Formula for Surface Area of Prism?
The formula for the surface area of a prism is obtained by taking the sum of (twice the base area) and (the lateral surface area 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 to Find the Surface Area of Prism?
We can find the surface area of the prism using the following steps:
- Step 1: Observe the pattern of the prism. Write down the given dimensions of the respective 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).
How Do You Find the Base Area of Prism If the Surface Area of Prism is Given?
The steps to determine the base area of the prism, if the surface area of the prism is given, is:
- Step 1: Write the given dimensions of the prism.
- Step 2: Substitute the given values in the formula S = (2 × Base Area) + (Base perimeter × height) where “S” is the surface area of the prism.
- Step 3: Now solve the equation for “Base Area by simplifying the equations”.
- Step 4: Once the value of the base area of the prism is obtained, write the unit of the base area prism in terms of square units.
What Happens to the Surface Area of Prism If the Base Area of Prism is Doubled?
The surface area of a prism depends on the base area of the prism and the lateral surface area of the prism. Let us substitute the value of the base area as 2B in the surface area of the prism formula. The final result we have, B’ = 2B, thus S’ = (4 × Base Area) + (Base perimeter × height).Thus, only the final value of the surface area of the prism will increase if the base area of the prism is doubled but the value of surface area will definitely not get doubled or quadrupled.
What Happens to the Surface Area of Prism When the Height of Prism is Doubled?
The surface area of the prism depends on the base area of the prism and the lateral surface area of the prism. This lateral surface area has an important parameter that is the height of the prism. Let us substitute the value of the height of prism as 2H in the surface area of the prism formula. The final result we have, H’ = 2H, thus S’ = (2 × Base Area) + (Base perimeter × 2H). Thus, only the final value of the surface area of the prism will increase if the height of the prism is doubled but the value of surface area will definitely not get doubled or quadrupled.
How Does the Surface Area of Prism Change If the Type of Prism Changes?
The surface area of the prism depends on the base area of the prism and the lateral surface area of the prism. Different types of prisms have different bases hence, as the type of prism changes, the base of the prism changes. This changes the base area of the prism changes which in turn changes the surface area of the prism.
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} = s^{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 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 + 2s^{2}] 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 (√3a^{2}/4) ⇒ A = 3√3a^{2}/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√3a^{2} ] 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 + 2s^{2}] square units |
Pentagonal Prism |
Pentagon |
5bh square units |
[5ab + 5bh] square units |
Hexagonal Prism |
Hexagon |
6ah square units |
[3√3a^{2} + 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 = 2a^{2} + 4ah
= 2 × (4)^{2} + 4 × 4 × 13
= 32 + 208 = 240 cm^{2}
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√3a^{2}
= 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
Solution:
a = 7 cm
b = 10 cm
h = 18 cm
Surface area of a triangular prism
= ab + 3bh
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:
a = 12 cm
b = 15 cm
h = 21 cm
Surface area of a pentagonal prism
= 5ab + 5bh
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,
P = 2 (l + w) |
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,
Surface Area of a rectangular prism = 2 (lh +wh + lw ) Square units. |
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 cm^{3}
V = 192 cm^{3}
Therefore, the volume of a rectangular prism is 192 cm^{3}.
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.
Frequently Asked Questions on Rectangular Prism
What is the right rectangular prism?
What is the difference between the right rectangular prism and an oblique rectangular prism?
What is an example of a rectangular prism?
What is the volume of a rectangular prism?
What is the surface area of a rectangular prism?
SA = 2 (lh +wh + lw ) Square units.
What is the rule for rectangular prism?
Is a rectangular prism also a cuboid?
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