How to Find the Surface Area of a Triangular Prism Easy

Surface Area of Triangular Prism

The surface area of triangular prism is the total area of all its faces. A triangular prism is a prism that has two congruent triangular faces and three rectangular faces that join the triangular faces. It has 6 vertices, 9 edges, and 5 faces. Let us learn more about the surface area of a triangular prism.

1.	What is the Total Surface Area of Triangular Prism?
2.	Formula for Surface Area of Triangular Prism
3.	How to Find the Surface Area of a Right Triangular Prism?
4.	FAQs on Surface Area of a Triangular Prism

What is the Total Surface Area of a Triangular Prism?

The surface area of a triangular prism is also referred to as its total surface area. The total surface area of a triangular prism is the sum of the areas of all the faces of the prism. A triangular prism has three rectangular faces and two triangular faces. The rectangular faces are said to be the lateral faces, while the triangular faces are called bases. If the bases of a triangular prism are placed horizontally, they are referred to as the top and the bottom (faces) of the prism, respectively. The surface area of triangular prism is expressed in square units, like, m², cm², in² or ft², etc.

Formula for Surface Area of Triangular Prism

The formula for the surface area of a triangular prism is formed by adding up the area of all the rectangular and triangular faces of a prism. Observe the following figure of a triangular prism to know the dimensions that are considered to frame the formula.

Surface Area of Triangular Prism formula

The formula for the surface area of triangular prism is:

Surface area = (Perimeter of the base × Length of the prism) + (2 × Base Area) = (S₁ +S₂ + S₃)L + bh

where,

b is the bottom edge of the base triangle,
h is the height of the base triangle,
L is the length of the prism and
S₁, S_2,and S₃ are the three edges (sides) of the base triangle
(bh) is the combined area of the two triangular faces [2 × (1/2 × bh)] = bh

Lateral Surface Area of Triangular Prism

The lateral surface area of any solid is the area without the bases. In other words, the lateral surface area of a triangular prism is calculated without considering the base area. When a triangular prism has its bases facing up and down, the lateral area is the area of the vertical faces. The lateral surface area of a triangular prism can be calculated by multiplying the perimeter of the base by the length of the prism. The perimeter of the base is the total length of the edges of the base triangle, while the length of the prism is its height. Observe the following figure to understand the lateral surface and the base of a triangular prism.

Lateral Area of Triangular Prism Formula

Thus, the lateral surface area of triangular prism is:

Lateral Surface Area = (S₁ + S₂ + S₃) × l = (Perimeter × Length) or LSA = p × l

where,

l is the height (length) of a prism
p is the perimeter of the base

How to Find the Surface Area of a Right Triangular Prism?

A right triangular prism has two parallel and congruent triangular faces and three rectangular faces that are perpendicular to the triangular faces. The surface area of a right triangular prism can be calculated by representing the 3-d figure into a 2-d net, which makes it easier to understand. After expanding this 3-d shape into the 2-d shape we get two right triangles and three rectangles. Observe the following figure which shows a right triangular prism. The following steps are used to calculate the surface area of a right triangular prism:

Surface area of a right triangular prism

Step 1: Find the area of the top and the base triangles using the formula, Area of the two base triangles = 2 × (1/2 × base of the triangle × height of the triangle) which simplifies to 'base × height' (bh).
Step 2: Find the product of the length of the prism and the perimeter of the base triangle which will give the lateral surface area = (S₁ + S₂ + h) × l.
Step 3: Add all the areas together to get the total surface area of a right triangular prism in square units. This means, total surface area of a right triangular prism = (S₁ + S₂ + h) × l + bh

Example: Find the total surface area of a right triangular prism which has a base area of 60 square units, the base perimeter of 40 units, and the length of the prism is 7 units.

Solution: Given, base area = 60 square units, base perimeter = 40 units and length of prism = 7 units

Thus, the surface area of the right triangular prism, Surface Area = (Perimeter of the base × Length of the prism ) + (2 × Base Area)
⇒ SA = (40 × 7) + (2 × 60)
⇒ SA = (280 + 120)
⇒ SA = 400 square units

Thus, the surface area of the right triangular prism is 400 square units.

☛ Related Articles

Surface Area Formulas
Surface Area of Prism
Surface Area of Cylinder
Surface Area of Cube
Surface Area of Cone
Lateral Area of a Right Triangular Prism

Surface Area of Triangular Prism Examples

Example 1: Find the surface area of the right triangular prism shown below.

Solution: Given, base (b) = 5 units, in this case, S₁and base is the same, the height of the triangle (h) = 12 units, length of a prism = 11 units, and the hypotenuse of the right triangle = 13 units.

The surface area of a right triangular prism = bh + (S₁+ S₂ + h)L

On substituting the values, we get
SA = (5 × 12) + [(5 + 13 + 12) × 11]
⇒ SA = 60 + (30 × 11)
⇒ SA = 390 square units

Therefore, the total surface area of a right triangular prism is 390 square units.
Example 2: Find the surface area of a triangular prism in which the area of the top and base triangles is 30 square units each, the perimeter of the triangle is 11 units, and the length of the prism is 25 units.

Solution: Given, area of the top triangle = 30 square units, the area of the base triangle = 30 square units, the perimeter of the right triangle = 11 units, and length of triangle = 25 units

The combined area of the top and base triangles = (30 + 30) = 60 square units
The perimeter of the triangle = 11 units
The length of the prism = 25 units.

Surface area of triangular prism = Combined area of the top and base triangles + (Perimeter of the triangle × length of the prism)

Substituting the values,
Surface area of a triangular prism = 60 + (11 × 25) = 335 square units

Therefore, the total surface area of the triangular prism is 335 square units.
Example 3: State true or false.

a.) The surface area of a triangular prism is the total area of all its faces.

b.) The lateral surface area of a triangular prism is calculated along with the base area.

c.) A right triangular prism has two parallel and congruent triangular faces and three rectangular faces that are perpendicular to the triangular faces.

Solution:

a.) True, the surface area of a triangular prism is the total area of all its faces.

b.) False, the lateral surface area of a triangular prism is calculated without considering the base area.

c.) True, a right triangular prism has two parallel and congruent triangular faces and three rectangular faces that are perpendicular to the triangular faces.

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Practice Questions on Surface Area of Triangular Prism

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FAQs on Surface Area of Triangular Prism

What is the Surface Area of Triangular Prism?

The surface area of a triangular prism is defined as the sum of the areas of all the faces or surfaces of the prism. A triangular prism has three rectangular faces and two triangular faces. The rectangular faces are said to be the lateral faces, while the triangular faces are called bases.

What is the Formula for the Surface Area of Triangular Prism?

The formula that is used to calculate the surface area of a triangular prism is, Surface area = (Perimeter of the base × Length of the prism) + (2 × Base Area) = (S₁ + S₂+ S₃)L + bh; where 'b' is the bottom edge of the base triangle, 'h' is the height of the base triangle, L is the length of the prism and S₁, S₂ and S₃are the edges of the base triangle.

How to find the Surface Area of a Right Triangular Prism?

The surface area of a right triangular prism can be calculated in the same way as we calculate the surface area of a triangular prism. The only difference is that in a right triangular prism the 3rd side of the base becomes the height of the prism since it forms a right-angled triangle. Let us understand this using the following steps:

Step 1: Find the area of both the base triangles using the formula, Area of 2 base triangles = 2 × (1/2 × base of the triangle × height of the triangle) which becomes 'base × height'.
Step 2: Multiply the length of the prism with the perimeter of the base triangle which will give the lateral surface area.
Step 3: Then, add all the areas together to get the surface area of a right triangular prism. This can be expressed as, Surface area = bh + (S₁+ S₂ + h)L; where 'bh' is the combined base area of the two triangles, S₁, S₂ , and hare the sides of the base triangle, and 'L' is the length of the prism.

How to find the Lateral Surface Area of a Triangular Prism?

The lateral surface area of a triangular prism is the area of the prism without considering the bases. We use the following steps to calculate the lateral surface area of a triangular prism :

Step 1: Identify the length of the prism from the given dimensions.
Step 2: Find the perimeter of the base of the prism.
Step 3: Now, multiply them to get the lateral surface area which is expressed in square units. This can be expressed with the formula, Lateral Surface Area = (Perimeter of the base × Length of the prism)

How to find the Base Area if the Surface Area of a Triangular Prism is Given?

We use the following steps to calculate the base area if the surface area of a right triangular prism is given:

Step 1: Identify the given dimensions.
Step 2: Now using the formula, Surface Area = (Length of the prism × Perimeter of the base) + (2 × Base Area), substitute the known values and solve for the base area.

How to find the Surface Area of a Triangular Prism without the Height?

The surface area of a triangular prism can be calculated even when the height of the prism is not given. In this case, the base area of the 2 triangular faces is calculated with the help of the formula, Area = \(\sqrt {s(s - a)(s - b)(s - c)} \) where a, b, c are the sides and 's' is the semi-perimeter, s = (a+b+c)/2. Then, the same formula of the total surface area is used. Surface Area = (Perimeter of the base × Length of the prism) + (2 × Base Area) = (Side₁ + Side₂+ Side₃)L + [2 × \(\sqrt {s(s - a)(s - b)(s - c)} \)]

How to find the Total Surface Area of a Triangular Prism?

The surface area of a triangular prism is also known as the total surface area. Therefore, it is calculated using the formula, Total Surface Area = (Perimeter of the base × Length) + (2 × Base Area); or TSA = (S₁ + S₂ + S₂)L + bh; where:

b is the bottom edge of the base triangle,
h is the height of the base triangle,
L is the length of the prism and
S₁, S_2,and S₃ are the three edges (sides) of the base triangle
(bh) is the combined area of the two triangular faces [2 × (1/2 × bh)] = bh

What is the Unit for the Surface Area of a Right Triangular Prism?

The surface area of a right triangular prism is expressed in square units, for example, m², cm², in² or ft², and so on.

What Happens to the Surface Area of a Right Triangular Prism if the Length of the Prism is Doubled?

The surface area of the right triangular prism increases in magnitude when the length of the prism is doubled. However, the surface area of the right triangular prism does not double as the surface area of a right triangular prism does not depend entirely on the length of the prism.

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Source: https://www.cuemath.com/measurement/surface-area-of-a-right-triangular-prism/