Content Curator
In mathematical terms, a prism is a polyhedron that has two congruent parallel bases joined together laterally by parallelogram or rectangular faces. The number of lateral surfaces a prism has is equal to the number of sides of the base polygon. In other words, for a base polygon with sides, the prism will have rectangular faces. Hence, the total number of surfaces for such a prism is and it’s. And the total surface area can be defined as: The Total Surface Area of a Prism = Lateral Surface area + Area of the two Bases
| Table of Content |
Key Terms: Area, prism, polygon, base, polyhedron, formula, faces, lateral surface area, perimeter
What is Surface Area of a Prism?
[Click Here for Sample Questions]
A prism is a polyhedron that has no curves, only flat faces. The surface area of a three-dimensional solid prism can be defined as the total area occupied by the faces of the prism. Hence, it depends upon the shape of its base. To find the surface area of a prism, 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 must be calculated.
Formula for Surface Area of a Prism
[Click Here for Sample Questions]
There is a general formula for calculating the surface area of a prism.
But since there are various types of prisms with different kinds of bases, there are different formulas for determining their surface area.
| Lateral Surface Area = Base Perimeter × Height |
And the Total Surface Area = Lateral Surface Area of a Prism + Area of the Two Bases = (2 × Base Area) + Lateral Surface Area or (2 × Base Area) + (Base Perimeter × Height)
But since there are various types of prisms with different kinds of bases, there are different formulas for determining their surface area.
| Particulars | Details |
|---|---|
| Shape | Surface Area of Prism = (2 × Base Area) + (Base perimeter × height) |
| Triangular Prism | The surface area of a triangular prism |
| Square Prism | The surface area of a square prism a2 |
| Rectangular Prism | The surface area of a rectangular prism |
| Trapezoidal Prism | Surface area of a trapezoidal prism |
| Pentagonal Prism | The surface area of a pentagonal prism |
| Hexagonal Prism | The surface area of a hexagonal prism The surface area of a regular hexagonal prism |
| Octagonal Prism | The surface area of an octagonal prism |
Consider a triangular prism with b, height h and length L.
The general formula for calculating the surface area of a prism states,
| Total Surface Area = (2 × Base Area) + (Base Perimeter × Height) |
The base perimeter of a triangular prism is (a + b + c) and the base area is ½ bh, where a, b and c are the sides of the triangular base. Substituting the values in the general formula we have,
Total Surface Area = bh + (a + b + c)L
The video below explains this:
Prism Formula Detailed Video Explanation:
Things to Remember
[Click Here for Sample Questions]
- A prism is a polyhedron that has two congruent parallel bases joined together laterally by parallelogram or rectangular faces.
- The number of lateral surfaces a prism has is equal to the number of sides of the base polygon.
- To find the surface area of a prism, 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 must be calculated.
- Lateral Surface Area of a Prism= Base Perimeter × Height
- 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).
- Since there are various types of prisms with different kinds of bases, there are different formulas for determining their surface area.
Sample Questions
Ques. Find the surface area of a prism whose base area is 12 square units, the base perimeter is 18 units and the height of the prism is 6 units. (3 Marks)
Ans. Given, base area, A = 12 square units,Perimeter,P= 18 units, and height h = 6 units.
Total Surface Area = (2×Base Area)+(Base perimeter×height)
Substituting the given values in the equation, we get,
Total Surface Area=(2 × 12) + (18 × 6) = 132 square units
Ques. Find the surface area of a triangular prism whose apothem length, base length, and height are 7 cm, 10 cm, and 18 cm respectively. (3 Marks)
Ans. Given,
a = 7 cm
b = 10 cm
h = 18 cm
We know that the surface area of a triangular prism,
S = ab + 3bh
Substituting the given values in the above equation, we get,
S = (7 cm × 10 cm) + (3 × 10 cm × 18 cm) = 70 (cm)2+ 540 (cm)2 = 610 (cm)2
Ques. Find the surface area of the following prism. (3 Marks)
Ans. From the given values in the figure we can calculate,
Area of the Base A = 10×10×1.73/4 = 43.25 (cm)2 and Perimeter of the Base P = 10+10+10 = 30 cm
Surface area of the prism S = 2A+Ph
Substituting the calculated values in the above equation, we get,
S = 2×43.25+30×14=506.5(cm)2
Ques. Find the surface area of a rectangular prism with base width 3 cm, base length 5 cm and height 4 cm. (3 Marks)
Ans. To find the area of a rectangular prism first flatten it creating the net, and then work out the total surface area by adding the areas of the individual rectangles.
Here, there are two rectangular faces with area 3×4= (12cm)2, two rectangular faces with area 3×5=(15cm)2 and two rectangular faces with area 4×5= (20cm)2.
Hence the surface area S = 2(12+15+20)= (94cm)2
Ques. How is the surface area of a prism calculated? (3 Marks)
Ans. First, the given dimensions of the prism are noted down. Then the dimensions are substituted in the surface area of the prism formula (2 × Base Area) + (Base Perimeter × Height).
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).
Ques. What is the formula for the lateral surface area of a prism? How is it related to the total surface area? (3 Marks)
Ans. The number of lateral surfaces a prism has is equal to the number of sides of the base polygon. In other words, for a base polygon with n sides, the prism will have n rectangular faces. Hence, the total number of surfaces for such a prism is n+2.
Lateral Surface Area of a Prism= Base perimeter × Height
Total Surface Area of a Prism = Lateral Surface Area of Prism + Area of the Two Bases = (2 × Base Area) + Lateral Surface Area
Ques. Define the surface area of a prism. How is it related to the base of the prism? (2 Marks)
Ans. A prism is a polyhedron that has no curves, only flat faces. The surface area of a three-dimensional solid prism can be defined as the total area occupied by the faces of the prism. Hence, it depends upon the shape of its base. To find the surface area of a prism, 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 must be calculated.
The surface area of a three-dimensional solid prism can be defined as the total area occupied by the faces of the prism. Hence, it depends upon the shape of its base.
Ques. Why are there different formulas derived from the general formula for calculating the surface area of a prism? (2 Marks)
Ans. The surface area of a three-dimensional solid prism can be defined as the total area occupied by the faces of the prism. Hence, it depends upon the shape of its base.
There are various types of prisms with different kinds of bases. For example, there are triangular prisms, rectangular prisms, etc. Hence, there are different formulas for determining their surface area.
Ques. Find the total surface area of the pentagonal prism if the apothem length, base length, and height of a pentagonal prism are 10 cm. 13 cm, and 19 cm, respectively.
Ans. The total surface area of a pentagonal prism,
S = 5ab + 5bh, where a is the apothem length, b is the base length and h is the height of the prism.
Here, a = 10,b =13 and h=19
=5 x 10 x 13 + 5 x 13 x 19= 650 +1235= 1885 (cm)2
Ques. Is the number of lateral surfaces of a prism greater than the number of sides of the base polygon? (2 Marks)
Ans. No. The number of lateral surfaces of a prism is equal to the number of sides of the base polygon. In other words, for a base polygon with n sides, the prism will have n rectangular faces. Hence, the total number of surfaces for such a prism is n+2.
Ques. Find the total surface area of a triangular prism whose bases are equilateral triangles of size 4 cm and the length of the prism is 6 cm. (3 Marks)
Ans. Given n=3 cm, b=4 cam and L=6 cm.
Since the base is an equilateral triangle, base area = √3/4 b2 = √3/4 42=4√3
Total Surface Area =2× Base Area +n × Area of One Rectangular Face = 2×4√3 + 3×4×6 = 8√3 +72 = 8(√3 + 9)
Ques. If two identical prisms with right-angled triangular bases are joined together like in the figure below, calculate the total surface area of the resulting rectangular prism. (3 Marks)
Ans. Given, length of the sides of the base are 12 cm and 5 cm and the height of the triangular prism is 15 cm. The two triangular prisms are joined together to form a rectangular prism. Hence, the length of the base of the rectangular prism is 12 cm,the breadth of the base of the rectangular prism is 5 cm and the height of the rectangular prism is 15 cm.
Therefore, Base Perimeter = 2(Length + Breadth) = 2 (12 + 5) cm
= 2 × 17 cm
= 34 cm
Lateral Surface Area = Height × Base perimeter
= 15 × 34 (cm)2
= 510 (cm)2
Base Area = Length × Breadth
= 5 × 12 (cm)2
= 60 (cm)2
Total Surface Area = Lateral Surface Area + 2 × Base area
= (510 + 2 × 60) (cm)2
= (510 + 120) (cm)2
= 630 (cm)2
Also check:
Comments