Area and Perimeter

Key Concepts

• Perimeter is the total distance around the boundary of a closed shape.

  • It is measured in units of length, such as cm, m, or km.
  • To find perimeter, add up the lengths of all the sides.

• Area is the amount of surface inside a flat 2-dimensional shape.

  • It is measured in square units, such as cm², m², or km².
  • The word “square” means the unit is multiplied by itself, for example:
    • $1 \text{ cm} \times 1 \text{ cm} = 1 \text{ cm}^2$

• A square is a shape with:

  • 4 equal sides
  • 4 right angles

• A rectangle is a shape with:

  • opposite sides equal
  • 4 right angles

• A triangle is a shape with 3 sides.

  • To find its area, we need its base and height.
  • The height must be perpendicular to the base.

• A composite figure is a shape made by joining 2 or more simple shapes, such as rectangles, triangles, semicircles, or squares.

  • To find the area, split the figure into known shapes and add or subtract areas.
  • To find the perimeter, add only the outer boundary. Do not include internal lines.

• A parallelogram is a 4-sided shape where:

  • opposite sides are parallel
  • opposite sides are equal in length

• A trapezium is a 4-sided shape with one pair of parallel sides.

• A circle is a round plane figure.

  • It has no corners and no straight sides.
  • Important parts:
    • Radius: distance from the centre to the circle
    • Diameter: distance across the circle through the centre
    • Circumference: distance around the circle

• Relationship in circles:

  • Diameter = 2 × radius
  • Radius = diameter ÷ 2

• Use $\pi$ for circle formulas.

  • In Primary 6, $\pi$ is often taken as:
    • 3.14, or
    • $\frac{22}{7}$ when suitable

• Formula for perimeter of a square:

  • $P = 4 \times \text{side}$

• Formula for area of a square:

  • $A = \text{side} \times \text{side}$

• Formula for perimeter of a rectangle:

  • $P = 2(\text{length} + \text{width})$

• Formula for area of a rectangle:

  • $A = \text{length} \times \text{width}$

• Formula for area of a triangle:

  • $A = \frac{1}{2} \times \text{base} \times \text{height}$

• Formula for area of a parallelogram:

  • $A = \text{base} \times \text{height}$

• Formula for area of a trapezium:

  • $A = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$

• Formula for circumference of a circle:

  • $C = \pi d$
  • or $C = 2\pi r$

• Formula for area of a circle:

  • $A = \pi r^2$

• Semicircle formulas (half a circle):

  • Area of a semicircle: [ A = \frac{1}{2} \pi r^2 ]
  • Perimeter of a semicircle (curved arc + diameter): [ P = \pi r + 2r ]
  • Common mistake: forgetting to add the diameter (2r) when finding the perimeter of a semicircle — the perimeter includes both the curved part and the straight edge across the middle.

  Worked Example: Semicircle

  A semicircle has radius 7 cm. Find its area and perimeter. (Use $\pi \approx 3.14$)

  • Area: [ A = \frac{1}{2} \times \pi \times 7^2 = \frac{1}{2} \times \pi \times 49 = 24.5\pi \approx 76.97 \text{ cm}^2 ]
  • Perimeter: [ P = \pi \times 7 + 2 \times 7 = 7\pi + 14 \approx 35.98 \text{ cm} ]

• Shaded region questions usually involve:

  • finding the area of a large shape
  • finding the area of the smaller unshaded shape(s)
  • subtracting:
    • Shaded area = Whole area − Unshaded area

• Always check:

  • Are all units the same?
  • Is the question asking for area or perimeter?
  • Is the answer in the correct unit?

Important Definitions

• Perimeter: the total length of the boundary of a closed figure.

• Area: the amount of surface enclosed inside a 2-dimensional figure.

• Square: a 4-sided figure with all sides equal and all angles 90°.

• Rectangle: a 4-sided figure with opposite sides equal and all angles 90°.

• Triangle: a 3-sided figure.

• Base: the side chosen as the bottom of a figure, often used when finding area.

• Height: the perpendicular distance from the base to the opposite side or vertex.

• Perpendicular: meeting at a right angle of 90°.

• Composite figure: a figure formed by combining two or more simple shapes.

• Parallelogram: a 4-sided figure with both pairs of opposite sides parallel.

• Trapezium: a 4-sided figure with one pair of parallel sides.

• Circle: a plane figure where every point on the boundary is the same distance from the centre.

• Centre: the fixed point in the middle of a circle.

• Radius: a straight line from the centre of a circle to its boundary.

• Diameter: a straight line passing through the centre of a circle and joining two points on the boundary.

• Circumference: the distance around a circle.

• Shaded region: the part of a figure that is coloured or marked, usually found by subtracting one area from another.

• Square unit: a unit used to measure area, such as cm² or m².

Worked Examples

Example 1: Area and perimeter of a rectangle

A rectangle has length 12 cm and width 5 cm. Find:

1. its perimeter
2. its area

Step 1: Write the formulas

• Perimeter of rectangle = $2(l+w)$
• Area of rectangle = $l \times w$

Step 2: Substitute the values

• $l = 12$ cm
• $w = 5$ cm

Step 3: Find the perimeter

Step 4: Find the area

Answer

• Perimeter = 34 cm
• Area = 60 cm²

Example 2: Area of a trapezium

A trapezium has parallel sides of 8 m and 14 m. Its height is 6 m. Find its area.

Step 1: Write the formula

Step 2: Substitute the values

Step 3: Calculate

Answer

Area = 66 m²

Example 3: Shaded region involving a circle

A circle of radius 7 cm is drawn inside a square of side 14 cm. Find the shaded area inside the square but outside the circle. Use $\pi = \frac{22}{7}$.

Step 1: Find the area of the square

Step 2: Find the area of the circle

Formula:

Substitute:

Step 3: Find the shaded area

Answer

Shaded area = 42 cm²

Example 4: Composite figure (L-shape)

An L-shaped figure has a total width of 10 cm and a total height of 8 cm. A rectangle of width 4 cm and height 5 cm is removed from the top-right corner. Find the area of the L-shape.

Method 1 — Subtract the missing piece:

Step 1: Find the area of the whole rectangle

Step 2: Find the area of the removed piece

Step 3: Subtract

Method 2 — Split into two rectangles:

Split the L-shape into a top rectangle and a bottom rectangle:

• Bottom rectangle: $10 \times 3 = 30 \text{ cm}^2$ (full width, height = $8 - 5 = 3$ cm)
• Top-left rectangle: $6 \times 5 = 30 \text{ cm}^2$ (width = $10 - 4 = 6$ cm, height = 5 cm)

Answer

Area of L-shape = 60 cm²

Key idea: For composite figures, you can either (a) subtract the missing piece from the whole, or (b) split the shape into smaller rectangles and add the parts. Both methods give the same answer.

Common Mistakes to Avoid

• Mixing up area and perimeter.

  • Perimeter is around the shape.
  • Area is inside the shape.

• Forgetting to write the correct units.

  • Perimeter uses units like cm, m.
  • Area uses square units like cm², m².

• Using the wrong formula for triangles.

  • Correct formula: $\frac{1}{2} \times \text{base} \times \text{height}$
  • Do not use base × height without dividing by 2.

• Using the slanted side as the height in triangles or parallelograms.

  • Height must be perpendicular to the base.

• In trapeziums, adding all four sides in the area formula.

  • Only use the two parallel sides and the height.

• In composite figures, including internal lines when finding perimeter.

  • Perimeter includes only the outer boundary.

• Forgetting to find missing side lengths.

  • Some questions require subtraction before you can find area or perimeter.

• Using diameter instead of radius for area of a circle.

  • Area formula needs radius: $\pi r^2$

• Using radius instead of diameter in $C = \pi d$ without changing the formula.

  • If using radius, use $C = 2\pi r$

• Not using the correct value of $\pi$ stated in the question.

  • If not given, use the standard value expected by the question, usually 3.14 or $\frac{22}{7}$

• Rounding too early.

  • Calculate carefully first, then round only if needed.

• Not subtracting correctly in shaded region questions.

  • Decide clearly which part is the whole and which part must be removed.

Exam Tips

• Underline the key information:

  • lengths
  • widths
  • radii
  • diameters
  • heights
  • units

• Circle the command word:

  • find the area
  • find the perimeter
  • find the circumference
  • find the shaded region

• Always write the formula before substituting values.

  • This helps you earn method marks.

• Use labels in your working, such as:

  • “Area of rectangle = …”
  • “Area of circle = …”
  • “Shaded area = total area − inner area”

• For composite figures:

  • split the shape clearly
  • label each smaller part
  • add or subtract the areas carefully

• For perimeter:

  • trace the outside edge with your finger or pencil
  • check that no side is counted twice
  • make sure internal sides are not included

• For circle questions:

  • first decide whether you need radius or diameter
  • write:
    • $d = 2r$, or
    • $r = \frac{d}{2}$, if needed

• For shaded region questions, useful mark-earning phrases include:

  • “Find the area of the whole figure first.”
  • “Subtract the area of the unshaded part.”
  • “The shaded area is …”

• Check units at the end:

  • cm, m for perimeter or circumference
  • cm², m² for area

• Do a final check:

  • Is the answer reasonable?
  • Can the shaded area be bigger than the whole figure? No.
  • Can the perimeter be in square units? No.

Quick Summary

• Perimeter is the total distance around a shape.
• Area is the amount of space inside a shape.
• Square:
  • perimeter = $4 \times \text{side}$
  • area = $\text{side}^2$
• Rectangle:
  • perimeter = $2(l+w)$
  • area = $l \times w$
• Triangle:
  • area = $\frac{1}{2} \times \text{base} \times \text{height}$
• Parallelogram:
  • area = $\text{base} \times \text{height}$
• Trapezium:
  • area = $\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$
• Circle:
  • circumference = $\pi d$ or $2\pi r$
  • area = $\pi r^2$
• Diameter = 2 × radius.
• Composite figures can be split into simpler shapes to find area.
• For perimeter of composite figures, count only the outside edges.
• Shaded area usually = area of whole figure − area of unshaded part.