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Surface Area of Isosceles Triangular Prism
The surface area of isosceles triangular prism refers to the sum total of the area of all the faces of an isosceles triangular prism. An 'isosceles' triangular prism has two isosceles triangles as bases facing each other and the 3 rectangular faces joining the corresponding sides of the two triangles. Since in an isosceles triangle, any two sides are equal, therefore, in an isosceles triangular prism, any two of the rectangular faces must be congruent. In this minilesson, we will learn about the surface area of an isosceles triangular prism. Stay tuned to learn more!!!
What is Surface Area of Isosceles Triangular Prism?
The surface area of an isosceles triangular prism is defined as the total area of all the faces of an isosceles triangular prism. An isosceles triangular prism is a polyhedron with polygons as its faces. It has 6 vertices, 5 faces, and 9 edges. Out of the 5 faces, isosceles triangles form the two parts at the base facing each other and rectangles form the vertical faces. Since the base parts are isosceles triangles facing each other, any two rectangles must be congruent. This means that 2 isosceles triangles and 3 rectangles out of which 2 rectangles are congruent, forms the isosceles triangular prism. The surface area of the isosceles triangular prism can be expressed as m^{2}, cm^{2}, in^{2}, etc depending upon the given units.
Formula for Surface Area of Isosceles Triangular Prism
The surface area of an isosceles triangular prism refers to the sum total of the area of all the faces of an isosceles triangular prism. To find the surface area of an isosceles triangular prism, we will have to add the areas of the 2 isosceles triangles at the base facing each other and the area of the rectangles formed by the corresponding sides of the two congruent triangles.
The surface area of an isosceles triangular prism is found as SA = Sum of areas of all the faces
⇒ SA = Sum of areas of 2 isosceles triangles + Sum of the areas of the 3 rectangles
⇒ SA = (bh) + (2la + lb)
Lateral Area refers to the total area of the lateral or vertical faces of any solid. Since we know that the vertical faces in the case of an isosceles triangular prism are rectangles, therefore, to find the lateral area we will have to find the areas of all the vertical faces and then add them up.
The lateral area of an isosceles triangular prism is found as Lateral area, LA = Sum of the areas of all the vertical faces = Sum of the areas of the three rectangles
⇒ LA = 2la + lb
Derivation of Surface Area of Isosceles Triangular Prism
The surface area of the isosceles triangular prism is found as SA = Sum of areas of 2 isosceles triangles at the bases + Sum of the areas of the 3 rectangles. Let us consider an isosceles triangle with the equal sides be "a" units, the base of each of the triangle be "b" units and the height of the triangle is "h"
Area of an isosceles triangle = (1/2 × base × height) = 1/2 × b × h
⇒ Area of two isosceles triangles = 2 × 1/2 × b × h = b × h
For rectangles:
Since we already know that in an isosceles triangular prism, there are 2 congruent rectangles. So let's first find the area of the 2 congruent rectangles:
Let the length of the congruent rectangles is "l" units and the breadth of the congruent rectangles is "a" units.
Thus, the area of the two congruent rectangles = 2 × l × a
Let the length of the third rectangle is "l" units and the breadth of the third rectangle = 'b' units
Thus, the area of the third rectangle = l × b
Therefore, the total area of the three rectangles = 2la + lb
Thus, the lateral surface area of an isosceles triangular prism is LSA = 2la + lb.
Now, the surface area of an isosceles triangular prism is, SA = Area of the two isosceles triangles + Area of the three rectangles
⇒ SA = (bh) + (2la + lb)
How to Find the Surface Area of Isosceles Triangular Prism?
We can find the surface area of the isosceles triangular prism using the following steps:
 Step 1: Identify the given dimensions of the isosceles triangular prism.
 Step 2: Find the surface area of the isosceles triangular prism using the formula, SA = (bh) + (2la + lb)
 Step 3: Represent the obtained answer with square units.
Solved Examples on Surface Area of Isosceles Triangular Prism

Example 1: The surface area of an isosceles triangular prism is 720 cm^{2}. The base of the isosceles triangle is 12 cm, and the length of the prism is 24 cm. Find the height of the isosceles triangle at the base of the prism if the value of the equal sides is 5 cm.
Solution: Given, SA = 720 cm^{2}, b = 12 cm, l = 24 cm, a = 5 cm
Let's use the surface area formula of the isosceles triangular prism to find the height "h" of the isosceles triangles.
We know that surface area formula for an isoscleles triangular prism SA = bh + 2la + lb
Substituting the values, we get:
⇒ 720 = 12 × h + (2 × 24 × 5 + 24 × 12)
⇒ 720 = 12h + 24 × (10 + 12)
⇒ 720 = 12h + 24 × 22
⇒ 720 = 12h + 528
⇒ 12h = 192
⇒ h = 16Answer: The height of the isosceles triangles of the isosceles triangular prism is 16 cm

Example 2: Find the lateral surface area of the isosceles triangular prism if the base of the prism has sides 5 cm, 5 cm, and 10.4 cm and the length of the prism is 12 cm.
Solution: Given, perimeter of the isosceles triangle = 5 + 5 + 10.4 = 20.4 cm, length of the prism = 12 cm
To find the lateral area of an isosceles triangular prism, we can multiply the perimeter of the base by the length of the prism.
Lateral area, LA = Perimeter × Length
⇒ LA = 20.4 × 12
⇒ LA = 244.8 cm^{2}Answer: The lateral surface area of the isosceles triangular prism is 244.8 cm^{2}
Practice Questions on the Surface Area of Isosceles Triangular Prism
FAQs on the Surface Area of Isosceles Triangular Prism
What is the Surface Area of Isosceles Triangular Prism?
The surface area of an isosceles triangular prism is the number of unit squares that can fit into it. An isosceles triangular prism is a 3D polyhedron with isosceles triangles at the top and at the bottom along with rectangles at the vertical faces.
What Are the Units Used When You Find the Surface Area of an Isosceles Triangular Prism?
The unit of 'area' is "square units". For example, it can be expressed as m^{2}, cm^{2}, in^{2}, etc depending upon the given units.
What is the Formula of the Surface Area of Isosceles Triangular Prism?
The formula of the surface area of an isosceles triangular prism is given as SA = bh + 2la + lb where the isosceles triangle in the base have the equal sides be "a" units, the base of each of the triangle be "b" units, the height of the triangle is "h" units and length of the congruent rectangles is "l" units.
What is the Formula of the Lateral Surface Area of Isosceles Triangular Prism?
The formula of the lateral area of an isosceles triangular prism is given as LA = 2la + lb where the isosceles triangle in the base have the equal sides be "a" units, the base of each of the triangle be "b" units, and length of the congruent rectangles is "l" units.
How to Find the Surface Area of Isosceles Triangular Prism?
We can find the surface area of the isosceles triangular prism using the following steps:
 Step 1: Identify the given dimensions of the isosceles triangular prism.
 Step 2: Use the surface area of the isosceles triangular prism using the formula, SA = (bh) + (2la + lb) to determine its surface area where the isosceles triangle in the base have the equal sides be "a" units, the base of each of the triangle be "b" units, the height of the triangle is "h" units and length of the congruent rectangles is "l" units.
 Step 3: Write the obtained answer with square units.
How to Find the Surface Area of Isosceles Triangular Prism If Its Lateral Surface Area is Given?
We can find the surface area of the isosceles triangular prism if its lateral surface area is given using the following steps:
 Step 1: Identify the given dimensions of the isosceles triangular prism.
 Step 2: Use the formula, SA = (bh) + LA to determine its surface area where the base of each of the triangle be "b" units and the height of the triangle is "h" units
 Step 3: Write the obtained answer with square units.
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