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).
Surface Area: This is the total area of all the surfaces (or faces) of a three-dimensional (3D) object. Imagine you want to paint the outside of a box; the surface area tells you how much total paint you'd need for all its sides. It's also measured in square units (e.g., cm², m²).
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 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.
Key Skill: Using π as 22/7
Using the fraction 22/7 for π is very helpful when the radius or diameter is a multiple of 7. It allows for "cross-cancellation" to simplify the calculation.
Example: A circle has a diameter of 49cm. Find the circumference.
1. Start with the formula: C = π × d
2. Substitute the values: C = (22/7) × 49
3. Cross-cancel: Divide 49 by the 7 on the bottom of the fraction. 49 ÷ 7 = 7.
4. The calculation becomes much simpler: C = 22 × 7
5. Final Answer: C = 154cm.
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.
Key Skill: Finding Height with Pythagoras
If 'h' isn't given for an isosceles triangle, find it by making a right-angled triangle:
- Split the triangle in half.
- The new base is ½ the original base.
- The slant side is the hypotenuse.
- Use Pythagoras:
h² + (½ base)² = slant².
Example: Base 6cm, slant 5cm.
h² + 3² = 5² → h² = 16 → h = 4cm.
Area = (6×4)/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.
L-Shape (Composite Shape)
This shape doesn't have a direct formula. We treat it as a composite shape by splitting it into simpler shapes (in this case, two rectangles).
Strategy: Split the L-shape into two rectangles (A and B), calculate their individual areas, and then add them together.Total Area = Area of A + Area of B
Example (Horizontal Split):
1. Split into a top rectangle (A) and a bottom rectangle (B).
2. Height of A = Total Height - Height of B = 8m - 3m = 5m.
3. Area of A = width × height = 4m × 5m = 20m².
4. Area of B = width × height = 10m × 3m = 30m².
5. Total Area = 20m² + 30m² = 50m².
Calculating Surface Area
Surface area is the total area of all the faces of a three-dimensional object. Just like 2D area, it's measured in square units (e.g., cm², m²).
Example: Surface Area of a Cuboid
Let's calculate the surface area of a cuboid with the following dimensions:
- Length (l) = 6 cm
- Width (w) = 4 cm
- Height (h) = 5 cm
A cuboid has 6 faces, and opposite faces are identical. The formula for the surface area (SA) of a cuboid is:
SA = 2 × (length × width + length × height + width × height)
Calculation:
- Calculate the area of each unique pair of faces:
- Front and Back Faces:
2 × (Length × Height) = 2 × (6cm × 5cm) = 2 × 30cm² = 60cm² - Top and Bottom Faces:
2 × (Length × Width) = 2 × (6cm × 4cm) = 2 × 24cm² = 48cm² - Left and Right Faces:
2 × (Width × Height) = 2 × (4cm × 5cm) = 2 × 20cm² = 40cm²
- Front and Back Faces:
- Add the areas of all faces together:
Total Surface Area =60cm² + 48cm² + 40cm² = 148cm².
The surface area of the cuboid is 148 cm².
Example: Surface Area of a Cylinder
Let's calculate the surface area of a cylinder with the following dimensions:
- Height (h) = 10 cm
- Diameter (d) = 12 cm (which means Radius (r) = 6 cm)
- Use π ≈ 3.14
The surface area (SA) of a cylinder consists of two circular bases and one curved rectangular side. The formula is:
SA = 2 × π × radius² + 2 × π × radius × height
Calculation:
- First, determine the radius from the given diameter:
Radius (r) = Diameter / 2 =12cm / 2 = 6cm. - Calculate the area of the two circular bases:
Area of 2 Bases =2 × 3.14 × (6cm)² = 2 × 3.14 × 36cm² = 2 × 113.04cm² = 226.08cm². - Calculate the area of the curved rectangular side (circumference × height):
Area of Curved Side =2 × 3.14 × 6cm × 10cm = 376.8cm². - Add the areas of the bases and the curved side together:
Total Surface Area =226.08cm² + 376.8cm² = 602.88cm².
The surface area of the cylinder is approximately 602.88 cm².
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)
Workings:
a) Perimeter = 2 × (length + width)
Perimeter = 2 × (12m + 8m) = 2 × 20m = 40m.
b) Area = 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)
Workings:
Diameter (d) = 4m, so Radius (r) = d/2 = 2m. π ≈ 3.14.
a) Circumference = 3.14 × 4m = 12.56m.
b) 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
Workings:
- Calculate the area of the triangular wall:
Area =(base × height) / 2 = (5m × 3m) / 2 = 15m² / 2 = 7.5m². - Determine the number of tins needed:
The area to cover is 7.5m². One tin covers 10m². Since you cannot buy part of a tin, Maria needs to buy 1 tin.
4. Painting Prisms
Maria paints some boards. Each board is in the shape of a triangular prism. The prism is 200cm long. The triangular faces have a base of 60cm and two equal slant sides of 50cm.
The area of each triangular face is 0.15 m². Maria needs to cover all five faces of 2 of these boards with paint. She has a tin of paint that covers 6 m². Does she have enough paint? Show all your workings.
Answer: No, she does not have enough paint.
Workings:
- Convert dimensions to metres.
The paint coverage is in m², so first convert the lengths from cm to m.
Length =200cm ÷ 100 = 2m
Base =60cm ÷ 100 = 0.6m
Slant side =50cm ÷ 100 = 0.5m - Find the total area of the rectangular faces of one prism.
The prism has one bottom face and two identical slanted side faces.
Area of bottom rectangle =length × width = 2m × 0.6m = 1.2m².
Area of one slanted rectangle =length × slant height = 2m × 0.5m = 1.0m².
Area of both slanted rectangles =2 × 1.0m² = 2.0m².
Total rectangular area =1.2m² + 2.0m² = 3.2m². - Calculate the total surface area of one prism.
The area of the two triangular faces is given. Total area of triangles =2 × 0.15m² = 0.3m².
Total Surface Area = (Area of triangles) + (Area of rectangles)
Total Surface Area =0.3m² + 3.2m² = 3.5m². - Calculate the total area for two prisms.
Total area to paint =2 × 3.5m² = 7.0m². - Conclusion.
Maria needs to cover7.0m², but her tin of paint only covers6m². Therefore, she does not have enough paint.
5. Fencing a Trapezoid Field
A field is in the shape of a quadrilateral as shown below. The two parallel sides are 50m and 80m long. The perpendicular height between them is 40m. A new fence needs to be built along the slanted edge. Calculate the length of this fence.
Answer: 50 metres
Workings:
- Visualise the shape. To find the length of the slanted side, we can split the trapezoid into a rectangle and a right-angled triangle. We can draw a vertical line down from the top-left corner to the base.
-
Find the dimensions of the triangle.
- The height of this triangle is the same as the perpendicular height of the trapezoid, which is40m.
- The base of this triangle is the difference between the long bottom side and the short top side:80m - 50m = 30m. -
Use Pythagoras' Theorem. Now we have a right-angled triangle with sides 30m and 40m. The fence is the hypotenuse (the longest side).
The formula isa² + b² = c².
30² + 40² = c²
900 + 1600 = c²
2500 = c²
c = √2500
c = 50m -
Conclusion.
The length of the fence needed for the slanted edge is 50m.
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 composite shapes (like an L-shape), break them down into simpler shapes to calculate area. The perimeter is still the length of all the outside edges added together.