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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.

1.	What is the Surface Area of Prism?
2.	How to Calculate the Surface Area of Prism?
3.	FAQs on Surface Area of a Rectangular Prism

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 = 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)

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 units²
∴ 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 units²
Therefore, the surface area of the given triangular prism is 408 units²
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 in²
S = (2 × Base Area) + (Base perimeter × height) where "S" is the surface area of the prism.
Base area = 250 in²
S = (2 × 250) + (15000)
S = 500 + 15000 = 15500 in²
Therefore, the lateral surface area of the given prism is 15,000 in^2, and the surface area of the prism is 15500 in².

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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 m², cm², in², or ft².

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.

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Surface Area of Triangular Prism Worksheets