Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have an irregularly shaped 3D object. I know the areas of the cross-sections in regular intervals. How can I calculate the volume of this object?

The object is given as a set of countours in 3D space. For each contour I can calculate the area using the surveyor's formula and the spacing between the contour lines is known.

share|improve this question
Can you provided an explicit example the object? –  matt Jan 13 '12 at 14:16
add comment

2 Answers

up vote 3 down vote accepted

Take two consecutive cross sectional areas $S_{i-1}$ and $S_i$ and connect them to create a finite volume. If their separation is $h=x_{i}-x_{i-1}$ then the finite volume is (using the trapezoidal rule)

$$\Delta V_i = h\;\frac{S_i+S_{i-1}}{2}$$

The total volume is then

$$ V = \sum_{i=1+1}^N \Delta V_i = \frac{h}{2}\left(S_1+S_N\right)+h\sum_{i=2}^{N-1}S_i $$

This is very similar to numerical integration techniques for a function $y=f(x)$ approximated by a series of points $y_1$, $y_2$ $\ldots$ $y_N$, but instead of $y_i$ you have sectional areas $S_i$.

share|improve this answer
thanks.that's it! –  stef Jan 20 '12 at 14:29
add comment

The obvious approximation would be to add up the areas of the cross sections, and multiply by the regular distance between the (parallel) cross sections

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.