# Finding Volume from Its Slices

How to find the volume from its slices: formula, examples, and their solutions.

## Formula

The volume of a 3D figure is

the integral of its cross-sectional area: *S*(*x*).

(cross-sectional area = sliced area)

So *V* = ∫_{a}^{b} *S*(*x*) *dx*.

## Example 1

When the height is *h*,*S*(*h*) = *e*^{h} - 1.

And *h* changes in [0, 3].

So *V* = ∫_{0}^{3} (*e*^{h} - 1) *dh*.

Solve the integral.

Indefinite integration of *e*^{x}

## Example 2: Proof of the Formula (Volume of a Cone)

To find the cross-sectional area *S*(*y*),

find the radius of the cross-sectional circle (brown).

If the height is *h*,

then the radius of the cross-sectional circle is *r*.

If the height is (*h* - *y*),

then the radius of the cross-sectional circle is (brown).

Then the big right triangle and the small right triangle

are similar,

because they show the AA similarity:

(Angle) Both triangles have a right angle.

(Angle) Both triangles have the same top angle.

So set a proportion

between the sides of these two right triangles:

(*h* - *y*)/(brown) = *h*/*r*.

Similarity of sides in triangles

Then (brown) = *r*(1 - *y*/*h*).

*S*(*y*) = *π*⋅(brown)^{2}

Area of a circle

And (brown) = *r*(1 - *y*/*h*).

So *S*(*y*) = *π*[*r*(1 - *y*/*h*)]^{2}.

*S*(*y*) = *π**r*^{2}(1 - [2/*h*]⋅*y* + [1/*h*^{2}]⋅*y*^{2}).

And the height changes in [0, *h*].

So *V* = ∫_{0}^{h} *π**r*^{2}(1 - [2/*h*]⋅*y* + [1/*h*^{2}]⋅*y*^{2}) *dy*.

Solve the integral (with respect to *y*).

Definite integration of polynomials

Cancel the dark gray terms.

Then *V* = (1/3)*π**r*^{2}*h*.

So this is the proof of the volume of a cone formula.