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Volume of a Cone

Volume of a Cone

The volume of a cone is:


Volume of a cone

NOTE that `h` is the vertical height of the cone.

A way of remembering this is `frac(1)(3) xx text(vertical height ) xx text( base area)`

Example 1

A cone has a vertical height of 12cm and a radius of 5 cm. What is the volume of the cone, to 3 significant figures?

For a cone: Volume `= pir^2frac(h)(3`
Substitute `= pi(5)^2 xx frac(12)(3)`

Answer: 314 cm3

Example 2

The shape shown below consists of a hemisphere placed on a cone. The total volume of the shape is 25 cm3. The overall width of the shape is `x` cm, and the total vertical height of the shape is `3x` cm.

What is the value of `x`? Give your answer to 1 decimal place.

Volume of a cone and hemisphere

The radius of the hemisphere is `frac(x)(2)`.

The vertical height of the cone is the height of the shape minus the radius, or `3x - frac(x)(2)`

Hemisphere: Volume `= frac(1)(2) xx frac(4)(3)pir^3`
`V_h` `= frac(1)(6) xx frac(4)(3)pifrac(x)(2)^3`
`= frac(1)(12)pix^3`
Cone: Volume `= pir^2h`
`V_c` `= pifrac(x)(2)^2(3x - frac(x)(2))`
`= pifrac(x)(2)^2(frac(6)(2)x - frac(x)(2))`
`= pifrac(5)(8)x^3`
Total `= V_h + V_c`
`25` `= frac(1)(12)pix^3 + pifrac(5)(8)x^3`
`25` `= frac(17)(24)pix^3`
`110.8797` `= x^3`
`4.804` `= x`

Answer: 4.8 cm