Volume (I): cubic centimetre

1Summary 重點摘要

Fill with 1 cm³ cubes, count them.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Volume is occupied spaceCompare a basketball and a ping-pong ball: the basketball takes far more space, and this difference in how much space is taken is volume. The amount of space an object takes up is called its volume, which is the definition. Why measure with little cubes? Because cubes can fill the shape brick by brick, like building a wall with small bricks; count the bricks used and you know the space taken. Take care that volume is a three-dimensional measure and has nothing to do with weight; a big cotton pillow can have far more volume than a small steel ball.
2. Part 2: How big 1 cm³ isA little cube whose every edge is 1 cm has a volume of one cubic centimetre, written 1 cm³. A dice is roughly 1 cm³ in size; hold one in your hand and the unit stops being abstract. Remember: cm³ means three edge lengths multiplied together, which is different from the area unit cm². Take care that the 3 in cm³ is a small 3 in the top corner, read as cubic; it may not be swapped with square and may not be dropped, for without the 3 cubic centimetres become square centimetres, a completely different meaning.
3. Part 3: Building a cuboid with blocksBuild a cuboid 3 cm long, 2 cm wide and 1 cm high out of 1 cm³ blocks: the base holds 3 blocks in a row with 2 rows, so one layer is 3 × 2 = 6 blocks and the volume is 6 cm³. This cuboid has just one layer, so the volume equals the blocks in the base. Why exactly 3 in a row? The length is 3 cm, just enough room for three 1 cm blocks. Take care that length and width are both in cm, so the block count matches the cubic centimetres of one layer, block for block.
4. Part 4: Extra height means extra layersBuild the same cuboid higher, making it 3 × 2 × 2: each layer holds 6 blocks and there are 2 layers, so 6 × 2 = 12, and the volume is 12 cm³. Extra height means extra layers; every extra 1 cm of height adds one more layer of blocks. From this comes the formula: cuboid volume = length × width × height. Why does the formula hold? Length and width give the blocks per layer, height gives the number of layers, and multiplying them gives all the blocks. Take care that all three numbers are lengths in the same unit; only when all are in cm may they be multiplied directly.
5. Part 5: Counting a real box layer by layerA box measuring 4 × 3 × 2 inside stays tidy when counted layer by layer. Work out the base layer first: 4 blocks in a row and 3 rows, so 4 × 3 = 12; with 2 layers in all, 12 × 2 = 24, and the volume is 24 cm³. Thinking layer by layer, even a big box is counted without tangles: one layer first, then multiply by the number of layers, in a clear order. The usual error is adding the three sizes at once, 4 + 3 + 2 = 9, which is not volume at all; volume comes from multiplying, not adding, and 9 stands far from 24 the moment you check.
6. Part 6: Volume is not areaThe same 3 × 2 × 2 cuboid has volume 12 cm³, measuring how much space it takes, while one of its faces has area 3 × 2 = 6 cm², measuring a flat surface. cm² multiplies two edges; cm³ multiplies three; the two kinds of unit must never be mixed. Take care that area speaks of one face and volume of the whole solid: how much sticker covers the lid is area, how many blocks fit inside is volume; read which one the question wants before working, and the wrong formula will never be used.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Volume = occupied space, in cm³
• 1 cm³ is a cube with 1 cm edges
• Count by layers: per-layer blocks × layers
• Cuboid volume = length × width × height
• cm³ for volume, cm² for area — never mix them