Area & perimeter

1Summary 重點摘要

Perimeter is the distance around (cm); area is the surface covered (cm²). Different formulas, different units — don't mix them.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is area?Area measures how much flat surface a figure covers: how big a desktop is, how large the classroom floor is, how much wall needs painting. Area is measured in square units: 1 square centimetre (cm²) is the area of a square of side 1 cm. To compute it, imagine tiling the figure with 1 cm² squares and counting them — a 5 cm by 3 cm rectangle covers 15 squares, so 15 cm². Area and length are different kinds of quantity, and writing cm instead of cm² is the most common error.
2. 2. What is perimeter?Perimeter is the total length of a figure's boundary — the distance of one full walk along its edge: around a desk, the total fencing for a garden, or one lap of the playground. Perimeter uses length units, not square units. Practical uses are everywhere: buying fairy lights to frame a window, fencing a garden, or the length of one running lap. Simply add up every side; missing one side is the usual mistake, and the sides of irregular figures are especially easy to overlook.
3. 3. Rectangle areaThe rectangle area formula is area = length × width. A rectangle 5 cm long and 3 cm wide has area 5 × 3 = 15 cm². The formula comes from counting: each row holds 5 one-square-centimetre tiles, there are 3 rows, 15 tiles in all. Application: a blackboard 4 m by 1.5 m has area 4 × 1.5 = 6 m², so a paint purchase starts from the area. The common error is multiplying mixed units — one side in metres, the other in centimetres — so always convert to one unit first.
4. 4. Rectangle perimeterThe rectangle perimeter formula is perimeter = 2 × (length + width), because a rectangle has two lengths and two widths. For 5 cm by 3 cm: 2 × (5 + 3) = 2 × 8 = 16 cm. Application: ribbon around a rectangular photo frame is exactly the perimeter. Mind the brackets — add length and width first, then double; the error of dropping them gives 2 × 5 + 3 = 13, and 16 cm versus 13 cm differs by a single pair of brackets.
5. 5. Square areaA square has four equal sides, so area = side × side. A square of side 4 cm has area 4 × 4 = 16 cm². This is the special case of the rectangle formula where length equals width. Application: a square tile of side 30 cm covers 30 × 30 = 900 cm², so flooring 1 m² (10,000 cm²) takes about 12 tiles, with one or two extra bought against breakage. The product side × side may be written side², read as 'side squared': 4² = 16.
6. 6. Square perimeterThe square perimeter formula is perimeter = 4 × side, since all four sides are equal. For side 4 cm: 4 × 4 = 16 cm. Application: lace edging a square handkerchief or a square frame is exactly the perimeter. Note the coincidence that a 4 cm square has perimeter 16 cm and area 16 cm² — the digits match but the meanings and units are entirely different. For most figures the two numbers do not match at all, so never treat them as interchangeable.
7. 7. Area vs perimeterArea and perimeter are two different quantities: area measures the covered surface in square units such as cm², while perimeter measures the boundary length in units such as cm. A 5 cm by 3 cm rectangle has area 15 cm² and perimeter 16 cm — close numbers, completely different meanings. A common misconception is that a bigger area forces a bigger perimeter: a 15 cm by 1 cm rectangle has the same 15 cm² area as a 5 cm by 3 cm one, but a perimeter of 32 cm, double the 16 cm. The shape changes the relationship, so compute and answer them separately.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Area: space covered (unit cm²)
• Perimeter: boundary total length (unit cm)
• Rectangle area = length × width
• Rectangle perimeter = 2 × (length + width)
• Area and perimeter are different