VOLUME OF PYRAMIDS AND CONES

Consider the prism shown below.

The volume of the prism shown above is equal to Bh, where B is the area of the base and h is the height.

From the diagram shown below, it is clear that the volume  of the pyramid with the same base area B and the same height h must be less than the volume of the prism.

The volume of the pyramid is one third the volume of the prism.

Theorems

Theorem 1 (Volume of a Pyramid) :

The volume V of a pyramid is

V  =  1/3 ⋅ Bh

where B is the area of the base and h is the height.

Theorem 1 (Volume of a Cone) :

The volume V of a cone is

V = 1/3 ⋅ Bh

V = 1/3 ⋅ πr²h

where B is the area of the base, h is the height, and r is the radius of the base.

Finding the Volume of a Pyramid

Example 1 :

Find the volume of the pyramid with the regular base.

Solution :

The base can be divided into six equilateral triangles as shown below.

Using the formula for the area of an equilateral triangle,

√3/4 ⋅ s²

the area of the base B can be found as follows :

B  =  6(√3/4 ⋅ s²)

Substitute 3 for s.

B  =  6(√3/4 ⋅ 3²)

Simplify.

B  =  6(√3/4 ⋅ 9)

B  =  27√3 / 2  cm²

Formula for volume of a pyramid :

V  =  1/3 ⋅ Bh

Substitute 27√3 / 2 for B and 4 for h.

V  =  1/3 ⋅ (27√3 / 2)(4)

Simplify.

V  =  18√3

V  =  31.2

So, the volume of the pyramid is about 31.2 cubic centimeters.

Finding the Volume of a Cone

Example 2 :

Find the volume of the right circular cone shown below. 

Solution :

Formula for volume of a cone : 

V  =  1/3 ⋅ πr²h

Substitute 12.4 for r and 17.7 for h.

V  =  1/3 ⋅ π(12.4)²(17.7)

Simplify. 

V  =  907.184π

Use calculator. 

V  ≈  2850 mm³

So, the volume of the right circular cone is about 2850 cubic millimeters.

Example 3 :

Find the volume of the oblique circular cone shown below. 

Solution :

Formula for volume of a cone : 

V  =  1/3 ⋅ πr²h

Substitute 1.5 for r and 4 for h.

V  =  1/3 ⋅ π(1.5)²(4)

Simplify. 

V  =  3π

Use calculator. 

V  ≈  9.42 in³

So, the volume of the oblique circular cone is about 9.42 cubic inches.

Using Volume in Real-Life Problems

Example 4 :

A nautical prism is a solid piece of glass, as shown. Find its volume.

Solution :

To find the volume of the entire solid, we have to add the volumes of the prism and the pyramid.

The bases of the prism and the pyramid are regular hexagons made up of six equilateral triangles.

To find the area of each base, B, multiply the area of one of the equilateral triangles by 6. 

That is, 6(√3/4 ⋅ s²), where s is the base edge. 

Let V₁ be the volume of prism and V₂ be the volume of pyramid.

Volume of Prism :

V₁  =  Bh

V₁  =  6(√3/4 ⋅ s²) ⋅ h

Substitute 3.25 for s and 1.5 for h. 

V₁  =  6[√3/4 ⋅ (3.25)²] ⋅ 1.5

Use calculator and simplify. 

V₁  ≈  41.166

Volume of Pyramid :

V₂  =  1/3 ⋅ 6(√3/4 ⋅ s²) ⋅ h

Substitute 3 for s and 3 for h. 

V₂  =  1/3 ⋅ 6(√3/4 ⋅ 3²) ⋅ 3

Use calculator and simplify. 

V₁  ≈  23.38

Volume of the Nautical Prism : 

V  =  V₁ + V₂

V  ≈  41.16 + 23.38

V  ≈  64.54  in³

So, the volume of the nautical prism is about 64.54 cubic inches.

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