What are Area and Perimeter?
Perimeter: The total distance around the outside of a two-dimensional (2D) shape. It's a one-dimensional measurement, representing length. Imagine walking all the way around the edge of a field; the total distance you walk is its perimeter. Perimeter is measured in units of length like centimetres (cm), metres (m), or kilometres (km).
Area: The amount of surface covered by a two-dimensional (2D) shape. It's a two-dimensional measurement. Imagine how much carpet you need to cover a floor, or how much paint is needed for a wall; that's its area. Area is measured in square units, such as square centimetres (cm²), square metres (m²), or square kilometres (km²).
Calculating Perimeter
To find the perimeter of most polygons (shapes with straight sides), you simply add up the lengths of all its sides. For curved shapes like circles, there's a specific formula for the circumference (which is its perimeter).
Square
All 4 sides are equal (length 's').
Perimeter = s + s + s + s = 4 × s
Example: If a square has a side of 5cm, Perimeter = 4 × 5cm = 20cm.
Rectangle
Two pairs of equal sides (length 'l' and width 'w').
Perimeter = l + w + l + w = 2l + 2w or 2(l + w)
Example: If length = 7m, width = 3m, Perimeter = 2(7m + 3m) = 2(10m) = 20m.
Triangle
Sides a, b, c. This formula applies to all triangles (scalene, isosceles, equilateral).
Perimeter = a + b + c
Example: If sides are 3cm, 4cm, 5cm, Perimeter = 3cm + 4cm + 5cm = 12cm.
Circle (Circumference)
The perimeter of a circle is called its circumference. 'r' is radius, 'd' is diameter (d=2r), π (pi) ≈ 3.14 or 22/7.
Circumference = π × d or 2 × π × r
Example: If radius = 5cm, Circumference = 2 × π × 5cm ≈ 2 × 3.14 × 5cm = 31.4cm.
Calculating Area
Area measures the surface inside a 2D shape. Different shapes have different formulas for calculating their area.
Square
Side length 's'.
Area = s × s = s²
Example: If side = 5cm, Area = 5cm × 5cm = 25cm².
Rectangle
Length 'l', width 'w'.
Area = l × w
Example: If length = 7m, width = 3m, Area = 7m × 3m = 21m².
Triangle
'b' is base, 'h' is perpendicular height.
Area = (base × height) / 2 or ½ × b × h
Example: If base = 6cm, height = 4cm, Area = (6cm × 4cm) / 2 = 24cm² / 2 = 12cm².
Parallelogram
'b' is base, 'h' is perpendicular height.
Area = base × height or b × h
Example: If base = 8m, height = 5m, Area = 8m × 5m = 40m².
Trapezium
'a' and 'b' are parallel sides, 'h' is perpendicular height.
Area = ½ × (a + b) × h
Example: If a=4cm, b=6cm, h=3cm, Area = ½ × (4cm + 6cm) × 3cm = ½ × 10cm × 3cm = 5cm × 3cm = 15cm².
Circle
'r' is radius, π (pi) ≈ 3.14 or 22/7.
Area = π × r²
Example: If radius = 5cm, Area = π × (5cm)² ≈ 3.14 × 25cm² = 78.5cm².
Exam-Style Area & Perimeter Problems
1. Fencing a Rectangular Garden
A rectangular garden is 12 metres long and 8 metres wide.
a) How much fencing is needed to go all the way around the garden?
b) What is the area of the garden?
Answers:
a) Perimeter: 40 metres
b) Area: 96 square metres (m²)
Explanation:
a) Perimeter of a rectangle = 2 × (length + width)
Perimeter = 2 × (12m + 8m) = 2 × 20m = 40m.
b) Area of a rectangle = length × width
Area = 12m × 8m = 96m².
2. Circular Pond Cover
A circular pond has a diameter of 4 metres.
a) What is the circumference of the pond? (Use π ≈ 3.14)
b) What is the area of the surface of the pond? (Use π ≈ 3.14)
Answers:
a) Circumference: 12.56 metres
b) Area: 12.56 square metres (m²)
Explanation:
Diameter (d) = 4m, so Radius (r) = d/2 = 2m. π ≈ 3.14.
a) Circumference = π × d
Circumference = 3.14 × 4m = 12.56m.
b) Area = π × r²
Area = 3.14 × (2m)² = 3.14 × 4m² = 12.56m².
3. Painting a Triangular Wall
A triangular section of a wall has a base of 5 metres and a perpendicular height of 3 metres. One tin of paint covers 10 m². How many tins of paint are needed to give the wall one coat?
Answer: 1 tin of paint
Explanation:
- Calculate the area of the triangular wall:
Area =(base × height) / 2 = (5m × 3m) / 2 = 15m² / 2 = 7.5m². - Determine the number of tins needed:
Since one tin covers 10m² and the area is 7.5m², only 1 tin is needed (as you can't buy a fraction of a tin and 7.5m² is less than 10m²).
Interactive Area & Perimeter Calculator
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Key Points for Area & Perimeter
- Perimeter is the distance around a 2D shape.
- Area is the space covered by a 2D shape.
- Know the formulas for common shapes (squares, rectangles, triangles, circles).
- For circles, the perimeter is called circumference.
- Units are important: length units for perimeter (cm, m), square units for area (cm², m²).
- For complex shapes, break them down into simpler shapes to calculate area or add all outer sides for perimeter.