Prism can be defined as a polyhedron that has two polygon-shaped bases opposite to each other and some lateral surfaces. Prism has smooth polished surfaces which refract light. Bases of the prism are generally triangular in shape and lateral surfaces have rectangular or parallelogram shapes.
Properties of Prism
- The two bases of a prism are parallel to each other and are mostly triangular in shape.
- Faces other than bases (base and top) are called lateral faces.
- Prism has the property of refracting light.
- White light can be split into seven rainbow colors by using a prism at certain angles.
- Lateral surfaces are of a parallelogram or rectangular shape.
Prism Formula
Surface area of Prism = (2×BaseArea) + Surface Area of Lateral Surfaces
Types of Prism
There are different types of prism:
Triangular Prism
It is the simplest type of prism with two triangular faces which can be called base and top, and three lateral faces that are rectangular in shape.
Area of base = 1/2 × (base) × (height) = 1/2 × (b) × (h)
Total Surface Area = Area of two bases + Area of 3 lateral surfaces
= 2 × (1/2×(base)×(height)) + 3 × length×breadth
= b × h + 3 × a × b
Rectangular Prism
It is a type of prism with two rectangular faces which can be called a base and top, and four lateral faces which are rectangular in shape. It looks like a cuboid in shape.
Area of base = (breadth) × (height)
Total Surface Area = Area of two bases + Area of 4 lateral surfaces
= 2×(breadth)×(height) + 2×length×breadth + 2×length×height
Pentagonal Prism
It is a type of prism with two pentagonal faces which can be called base and top, and five lateral faces which are rectangular in shape.
Area of base = 5/2×a×b
Total Surface Area = Area of two bases + Area of 5 lateral surfaces
= 2×(5/2×a×b) + 5×b×h
= (5×a×b) + 5×b×h
Hexagonal Prism
It is a type of prism with two hexagonal faces which can be called base and top, and six lateral faces which are rectangular in shape.
Area of base = 3×a×b
Total Surface Area = Area of two bases + Area of 6 lateral surfaces
= 2×(3×a×b) + 6×b×h
= (6×a×b) + 6×b×h
Sample Questions
Question 1: Find the area of the triangular prism which has a length of 10 cm, a breadth of 6 cm and a height of 2 cm.
Solution:
Given length (a) = 10 cm, breadth (b) = 6 cm, and height (h) = 2 cm.
Area of triangular prism
Total Surface Area = Area of two bases + Area of 3 lateral surfaces
= 2×(1/2×(base)×(height)) + 3×length×breadth
= b×h + 3×a×b
= 6×2 + 3×10×6
= 12 + 180
= 192 cm2
Question 2: Find the area of the rectangular prism which has a length of 10 cm, a breadth of 6 cm and a height of 2 cm.
Solution:
Given length (a) = 10 cm, breadth (b) = 6 cm, and height (h) = 2 cm.
Area of the rectangular prism
Total Surface Area = Area of two bases + Area of 4 lateral surfaces
= 2×(breadth)×(height) + 2×length×breadth + 2×length×height
= 2×6×2 + 2×10×6 + 2×10×2
= 24 + 120 + 40
= 184 cm2
Question 3: Find the area of the pentagonal prism which has a = 5 cm, breadth of 6 cm and a height of 10 cm.
Solution:
Given (a) = 5 cm, breadth (b) = 6 cm, and height (h) = 10 cm.
Area of a pentagonal prism
Total Surface Area = Area of two bases + Area of 5 lateral surfaces
= 2×(5/2×a×b) + 5×b×h
= (5×a×b) + 5×b×h
= (5×5×6) + 5×6×10
= 150 + 300
= 450 cm2
Question 4: Find the area of the hexagonal prism which has a = 5 cm, breadth of 6 cm and a height of 10 cm.
Solution:
Given (a) = 5 cm, breadth (b) = 6 cm, and height (h) = 10 cm.
Area of a hexagonal prism
Total Surface Area = Area of two bases + Area of 6 lateral surfaces
= 2×(3×a×b) + 6×b×h
= (6×a×b) + 6×b×h
= (6×5×6) + 6×6×10
= 180 + 360
= 540 cm2
Question 5: Find the lateral surface area of hexagonal prism which has a breadth of 6 cm and a height of 10 cm.
Solution:
Given breadth (b) = 6 cm, and height (h) = 10 cm.
Area of 6 lateral surfaces = 6×b×h
= 6×6×10
= 360 cm2
Practice Problems on Prism Formula
1. Find the volume of a triangular prism with a base area of 24 square units and a height of 10 units.
2. A rectangular prism has dimensions of 4 units by 6 units by 8 units. What is its volume?
3. Calculate the surface area of a pentagonal prism with a base perimeter of 30 units, a base area of 50 square units, and a height of 12 units.
4. Find the volume of a hexagonal prism with a base area of 18 square units and a height of 15 units.
5. A rectangular prism has a length of 5 units, a width of 3 units, and a height of 9 units. Find the surface area.
6. Calculate the surface area of a triangular prism with a base perimeter of 20 units, a base area of 30 square units, and a height of 10 units.
7. A prism has a volume of 360 cubic units and a base area of 40 square units. What is its height?
8. A prism has a base area of 25 square units, a base perimeter of 15 units, and a height of 8 units. Find its surface area.
9. A triangular prism has a base length of 6 units, a base height of 4 units, and a height of 12 units. Calculate the volume.
10. Calculate the surface area of a rectangular prism with dimensions 7 units, 3 units, and 5 units.
Conclusion
Therefore, a prism is a geometric shape which has two parallel and congruent bases connected by rectangular or parallelogram shaped lateral faces. The type of prism is determined by the shape of the base, which can be rectangular, triangular, or pentagonal. The properties of prisms as well as calculation of their volumes and surface areas is necessary and quite useful in various fields like mathematics, engineering, and architecture for solving real life practical problems.
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