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A prism is a solid structure, having flat rectangular side faces, identical bases and same cross-section all along its length. Prisms are of different types and each of them are classified based on the shape of their base. A triangular prism is a polyhedron consisting of three rectangular faces and two parallel triangular bases. The volume of a triangular prism refers to the space occupied by it from all three dimensions.
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Key Takeaways: Triangular Prism, Triangle, bases, faces, parallel, volume
Triangular Prism
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A triangular prism is a polyhedron having two triangular bases and three rectangular faces. It can also be considered a pentahedron, as it has five faces. The edges and vertices of the prism base are joined with one another via the three rectangular sides. The two triangular bases of the prism are parallel and congruent to each other. If the sides of the prism are rectangular, it is called a right triangular prism and otherwise it is called an oblique triangular prism. A prism is called a regular or uniform triangular prism if its sides are squares and bases are equilateral. The height (h) of the triangular prism is the perpendicular distance between the centres of the two parallel bases.
Volume of Triangular Prism
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The volume of a triangular prism is the space occupied by it from all the three dimensions. In simple words, the volume of a triangular prism refers to the space inside it. It is measured in cubic units such as cm3, m3 etc. The volume of a triangular prism can be calculated by taking the product of the height of the prism and the area of the triangular base.
Volume of prism= base area x height
Formula of Volume of Triangular Prism
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Volume of a triangular prism = area of base triangle x height
Since the prism base is in triangular shape,
the base area = ½ bh, where b is the base length and h is the height of the triangle.
therefore, the
Volume of triangular prism = ½ x b x h x l = ½ bhl
Here b is the base length, h is the height of the triangle and l is the length between the triangular bases.
Formulas to find the Base Area of different triangles
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Following are the formulas used to find the base area of different types of triangles.
- If the triangle base is equilateral or the prism is called an equilateral triangular prism with each side 'a', then its area is √3a2/4 square units.
- If the triangle's height 'h' and base 'b' are given, then its area is (1/2) bh square units.
- If the base triangle is a right-angled triangle or the prism is called a right triangular prism, with two legs 'b' and 'h' then its area is (1/2) bh square units.
- If the base triangle is an isosceles triangle with its sides to be 'a', 'a', and 'b' then its area is (b/4) × √(4a2 - b2) square units.
- If the type of base triangle is scalene, where all three sides 'a', 'b', and 'c' are given, then its area is calculated using √[s(s-a)(s-b)(s-c)] square units
(Where s = (a + b + c)/2).
The same formula can be applied for an isosceles triangle or an equilateral triangle. The formula is also known as Heron's formula.
- If the base triangle's two sides 'a' and 'b' and the included angle 'θ' are given, then its area is found using the formula 1/2 ab sin θ square units.
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Points to Remember
Following are some important points:
- A prism is a solid structure and they are of different types.
- A triangular prism is a polyhedron having three rectangular sides and two triangular bases.
- A triangular prism has got six corners and nine edges in total.
- The volume of a triangular prism is the space inside the prism or the space occupied by it.
- Formula for measuring the volume of a triangular prism is the product of the area of the base triangle and the height of the prism,i.e., V= ½ bhl
Sample Questions
Ques: Find the volume of the triangular prism with base is 10 cm, height is 5 cm, and length is 15 cm. (2 Marks)
Ans: Volume of Triangular Prism = ½ × b × h × l
V = ½ × 10 × 5 × 15
Volume, V = 375 cm3
Ques: The height of the prism is 4cm and the length of the side of the equilateral triangular base is 6cm. Find the area of the prism for the above example. Given, the base of the prism is 5 cm and its height is 10cm. (3 Marks)
Ans: We have,
Base = 5cm, height of the base= 10cm, Height of the prism = 4cm
The base is an equilateral triangle, which means that all its sides will be equal.
Hence, a = b = c = 6cm.
By the formula,
Area of the Triangular Prism = (bh + (a + b + c) H)
Area = (5 × 10 + (6 + 6 + 6)4)
= (50 + (18)4) = 50 + 72
= 122 cm2
Ques: A triangular prism is such that one of the sides of its triangular faces and its corresponding height are equal in length (say x cm). The length and the volume of the prism are respectively equal to 10 cm and 20 cm3. Find the value of x. (3 Marks)
Ans: Base area of prism (BA) = (1/2) ×x × x = x2/2
Volume = BA × l
BA = V/ l
Therefore, x2/2 = V / l
That is, x2/2 = 20/10
X2/2 = 2
X2= 4
X = 2 cm
Ques: Calculate the volume of the triangular prism whose height is 15 cm and whose base is an equilateral triangle of side 4 cm. (2 Marks)
Ans: The base triangle is an equilateral triangle with its side as a = 4 cm. So its area is found using the formula,
√3a2/4 = √3(4)2/4 = 4√3 square inches.
Height (h) of the prism = 15 cm.
Volume of the given triangular prism = base area × height
That is, Volume V = 4√3 × 15 = 60√3 cm3.
Ques: Determine the volume of a triangular prism whose height is 20 m and whose triangle's base is 6 m and height is 5 m. (2 Marks)
Ans: The base of the triangle is, b = 6 m, and its height is h = 5m.
So, the base area = 1/2 x b x h = (1/2) (6× 5) = 15 m2.
The height of the prism is H = 20 m.
Using the volume of triangular prism formula,
Volume of the given triangular prism = base area × height = 15 x 20 = 300 in3.
Ques: Find the volume of a right triangular prism, whose triangle base is 12 ft and its height is 8 ft. The height H of the prism is 10 ft. (2 Marks)
Ans: The base of the triangle is, b = 12 ft and its height h = 8 ft.
So, the base area = ½ x b x h = (1/2) (12 × 8) = 48 square feet.
Height of the prism is, H = 10 ft.
Using the volume of the triangular prism formula,
Volume of the given triangular prism = base area × height = 48 x 10 = 480 ft3.
Ques: What is the surface area of a triangular prism? (2 Marks)
Ans: surface area of a triangular prism is the total area covered by its surface in a three-dimensional plane. The formula is:
Surface area of triangular prism= b x h + (a + b + c) H
Here a, b, c are the sides of the triangular bases and H is the height of the prism.
Ques: the volume of a triangular prism is 60 cm3. The base area of the triangle is 12 cm2, and the length of the side is 4 cm. Find the height H of the prism? (3 Marks)
Ans: Volume V of the triangular prism = 60 cm3
Area of the base triangle = 12 cm2
Height H = ?
Using the formula, volume of the given triangular prism = base area × height
That is, 60 = 12 x H
Therefore, H = 60 / 12
H = 5 cm.
Ques: Find the volume of a triangular prism of height 8 cm, whose base of the triangle is 6 cm and its height is 4 cm. (2 Marks)
Ans: base of triangle = 6 cm, its height h = 4 cm
Therefore, area of the triangle base = ½ x b x h = ½ x 6 x 4 = 12 cm2.
Volume of a triangular prism = base area x height of the prism
V = 12 x 8 = 96 cm3.
Ques: If the height of the prism is 5 cm and the length of the side of the equilateral triangular base is 8 cm. Then find the area of the prism for the above example. Given the base of the prism is 4 cm and its height is 10 cm. (3 Marks)
Ans: We have,
Prism base = 4cm, height of the base h = 10cm, Height of the prism H = 5cm
The base is an equilateral triangle, which means that all its sides will be equal.
Hence, a = b = c = 8 cm.
By the formula,
Area of the Triangular Prism = (b x h + (a + b + c) H)
Area = (4× 10 + (8+ 8 + 8)5)
= (40+ (24)5) = 40 + 120
= 160 cm2
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