Area and Volume

1Summary 重點摘要

Rectangle, triangle, circle area; cuboid, cylinder volume.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. Rectangle areaA rectangle's area = length × width. Example: length 5 cm, width 3 cm — area = 5 × 3 = 15 cm². Note the square in the unit, because multiplying two lengths multiplies their units. A life example: an A4 sheet is about 30 cm by 21 cm, area about 630 cm². Two common errors: writing the unit as cm instead of cm²; or computing the perimeter 2 × (length + width) instead of the area. Sketch the rectangle, label length and width, then pick the formula — no more mixing.
2. 2. Triangle areaA triangle's area = base × height ÷ 2. Example: base 6 cm, height 4 cm — area = 6 × 4 ÷ 2 = 12 cm². Why divide by 2? A parallelogram with the same base and height has area base × height, and the triangle is exactly half of it. Note that 'height' is the perpendicular distance to the base, not the length of a slanted side. Common errors: forgetting to divide by 2, giving 24 cm²; or using a slanted side as the height. In an obtuse triangle the height can fall outside — extend the base when drawing it.
3. 3. Circle areaA circle's area = πr², with r the radius and π ≈ 3.14. Example: radius 3 cm — area = π × 3² = π × 9 ≈ 28.27 cm². The point: square the radius first, then multiply by π. If the question gives the diameter, halve it first — a diameter of 10 cm means r = 5 cm and area = π × 25 ≈ 78.54 cm². Two common errors: substituting the diameter as if it were the radius; or computing (πr)² — multiplying by π before squaring — which makes the result about three times the correct value. On seeing 'diameter', divide by 2 first.
4. 4. Cuboid volumeA cuboid's volume = length × width × height. Example: 5 cm by 3 cm by 4 cm — volume = 5 × 3 × 4 = 60 cm³. The cubic in the unit comes from multiplying three lengths. A life example: a shoebox 30 cm by 20 cm by 10 cm holds 30 × 20 × 10 = 6000 cm³, i.e. 6 litres. Common errors: writing the unit as cm²; or dropping one of the three dimensions. Count the three dimensions of the box, write each down, multiply, and finish by checking the unit is cubic.
5. 5. Cylinder volumeA cylinder's volume = base area × height, with base area = πr². Example: base radius 2 cm, height 10 cm — base area = π × 2² = 4π ≈ 12.57 cm², and volume = 12.57 × 10 ≈ 125.7 cm³. Picture the cylinder as a stack of circular cards: each card has area πr², stacked as tall as the height. Common errors: using the diameter for the base area without halving; or computing πr instead of πr². Write it in two stages — base area first, then times the height — and mistakes fade.
6. 6. Unit conversionArea and volume conversions are exam favourites. Length: 1 m = 100 cm. Area: 1 m² = 100 cm × 100 cm = 10000 cm² — converting m² to cm² multiplies by 10000. Volume: 1 m³ = 100 cm × 100 cm × 100 cm = 1000000 cm³ — volume multiplies by 1000000. Examples: 2 m² = 20000 cm²; 3 m³ = 3000000 cm³. A common error is multiplying area by 100 or volume by 1000; remember, the exponent of the unit tells you the power of 100 — squared means 100 squared, cubed means 100 cubed.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Rectangle area = length × width
• Triangle area = base × height ÷ 2
• Circle area = πr²
• Cuboid volume = length × width × height
• Cylinder volume = base area × height