Lesson 5 of 5

Volume.

About 20 minutes · 5 practice questions

Volume counts unit cubes: the area of the base times the height, for a box, a prism and a cylinder, and why a cube twice as big holds eight times as much.

Step 1 of 5 · Counting cubes

Step 1 of 5

Counting cubes

Volume is how much space a solid takes up. It is measured in cubes: a cube with edge $1$ ⁠ is one unit of volume. The board shows the base of a box and its outline. Work out how many cubes fill it, then drag Look inside to build it layer by layer.

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Step 2 of 5

Layer on layer

Each layer covers the base, so it holds as many cubes as the base area, and the number of layers is the height. Hence the rule: volume equals the base area times the height. Here Length and Width are locked and the volume is hidden: work out how many layers you need, then set Height.

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Step 3 of 5

A prism

The rule works for any solid whose cross-section is the same all the way up, even when the base is not a rectangle. In the prism on the board the base is a right triangle, half the base of the box.

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Step 4 of 5

A cylinder

A cylinder is also made of equal layers, but its base is a circle, with area $\pi r^2$ ⁠. So the volume of a cylinder is $\pi r^2 \times h$ ⁠, where $h$ ⁠ is its height.

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Step 5 of 5

A cube twice as big

In the lesson on similarity you saw that when every length grows by $k$ ⁠, area grows by $k^2$ ⁠. Predict what happens to the volume when every edge of a cube is doubled.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Calculate

A box is $4$ ⁠ long, $3$ ⁠ wide and $2$ ⁠ tall. How many unit cubes fill it?

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Practice · Reach the target

The box’s base is $4 \times 3$ ⁠. Set Height so that the volume is $36$ ⁠, then press Check.

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Practice · Calculate

The prism’s base is a right triangle with legs $4$ ⁠ and $3$ ⁠, and the prism is $5$ ⁠ tall. What is its volume?

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Practice · Calculate

The cylinder has radius $3$ ⁠ and height $4$ ⁠. What is its volume? Round to one decimal place.

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Practice · Predict, then look

A cube with edge $1$ ⁠ holds one unit cube. Double every edge, to $2$ ⁠. How many unit cubes will the new cube hold?