什么是体积？

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Volume is the amount of space that a substance or object occupies, or that is enclosed within a container.

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Contest problems rarely look like the examples you studied. Solving one means finding your own way in. These are real problems from the AMC 8, AMC 12 and AIME, the national math competitions for middle and high school students, each with a worked solution on LIVE by Po-Shen Loh.

Volume problems 33 from AMC 8, AMC 12, AIME
3D geometry problems 25 from AMC 8, AMC 12, AIME

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Volume is the measure of the amount of 3D space taken up by an object, relative to a unit cube. It is a 3D analogue of two dimensional area.

What is the volume of a $2\times3\times4$ cuboid?

Video

By Alex Federspiel

(Video) Volume

by Math Antics

Here's a video by mathantics explaining what volume is. Volume is a quantity that all three dimensional objects have and it's a way to measure something in the third dimension. You also use cubic units are used to measure volume. The surface area is the area of the objects outer surface. It is important not to confuse this with volume.

Rectangular Prism

$\text{Volume}=\text{Length}*\text{Width}*\text{Height}\\ V=lwh$

Triangular Prism

$\text{Volume}=\frac12\text{Base}*\text{Length}*\text{Height}\\ V=\frac12blh$

Cylinder

$\text{Volume}=\pi\text{Radius}^2*\text{Height}\\ V=\pi r^2h$

Test Yourself: Find the volume of a box that has a length of six inches, a width of five inches, and a height of three inches.