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Volume of a Triangular Prism

A triangular prism has an uniform cross-section of a triangle.

volume of a triangular prism_1Here the area of cross-section can be found using area of triangle and perpendicular height of prism is provided. For clarity purposes, let us take the base of the triangular cross-section to be ‘b’ and the perpendicular height of the triangle to be ‘h’. We will represent the height or length of the prism as ‘H’.

In the triangular prisms shown below-

volume of a triangular prism_2

Volume of a triangular prism = Area of triangular face × height of prism

Area of triangular face = 1/2 × base × perpendicular height of triangle

= 1/2 × b × h

Perpendicular height of prism = H

So V = (1/2 × b × h) × H

It is simpler, easier and clearer to find the area of triangle first and then multiply it by the height of prism.

However we can find the volume of a triangular prism in one go by multiplying the base of triangle, height of the triangle and height of prism and then halving it.

 

volume of a triangular prism_3Example 1: Calculate the volume of the triangular prism.

Area of triangular face = 1/2 × base × perpendicular height of triangle

= 1/2 × 12 × 9

= 54 cm²

Height of the prism = H = 10 cm

Volume of the triangular prism = Area of triangular base × Perpendicular height of prism

Volume of the triangular prism = 54 × 10 = 540 cm³

 

volume of a triangular prism_4Example 2: Calculate the volume of the triangular prism.

Area of the triangular cross-sectional face = 1/2 × base × perpendicular height of triangle

1/2 × 10 × 12

= 60 cm²

Perpendicular height of prism = 24 cm

Volume of the triangular prism = Area of cross-section × Perpendicular height of prism

= 60 × 24 = 1440 cm³

 

volume of a triangular prism_5Example 3: Find the volume of the triangular prism.

Note that here perpendicular height of the prism is 0.3 m. We first have to make it also in cm. Hence the perpendicular height of the prism is 0.3 m = 30 cm.

Area of triangular cross-section of prism = 1/2 × base × height of triangle

1/2 × 20 × 10

= 100 cm²

Perpendicular height of prism = 30 cm

Volume of the triangular prism = Area of triangular cross-section × Perpendicular height of prism

= 100 × 30 = 3000 cm³

NOTE: We could have found the volume of the triangular prism using V = (1/2 × b × h) × H

= (1/2 × 20 × 10) × 30 = 3000 cm³

 

volume of a triangular prism_6Example 5: Calculate the volume of the triangular prism.

Note that the face of the triangular prism is an isosceles triangle, hence the perpendicular height bisects the base. We will use Pythagoras theorem to the find the perpendicular height of the triangle.

volume of a triangular prism_7h = sqrt{10^2~−~8^2}

sqrt{36}

= 6 cm

Area of triangular cross-section = 1/2 × base × height of triangle

=  1/2 × 16 × 6 = 48 cm²

Perpendicular height of prism = 16 cm

Volume of the prism = Area of cross-section × height of prism

= 48 × 16 = 768 cm³