Practice Maths

Perimeter and Area

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Key Terms

Perimeter
The total distance around the outside of a shape; measured in length units (mm, cm, m).
Area
The amount of 2D space enclosed by a shape; measured in square units (mm², cm², m²).
Composite shape
A shape formed by combining or subtracting two or more standard shapes; split into recognisable parts.
Annulus
The region between two concentric circles; Area = π(R² − r²) where R and r are outer and inner radii.
Sector
A “pie slice” of a circle; Arc length = (θ/360°) × 2πr; Area = (θ/360°) × πr².
Area conversion
1 cm² = 100 mm²; 1 m² = 10 000 cm²; 1 ha = 10 000 m².

Formula Reference

ShapePerimeterArea
RectangleP = 2(l + w)A = lw
TriangleP = a + b + cA = ½bh
ParallelogramP = 2(a + b)A = bh
TrapeziumP = a + b + c + dA = ½(a + b)h
CircleC = 2πr = πdA = πr²
SectorArc = (θ/360)×2πrA = (θ/360)×πr²

For a sector: θ is the angle at the centre in degrees; r is the radius. The full perimeter of a sector = arc length + 2r.

Rectangle l w Triangle a b c h Parallelogram b h Trapezium a b h Circle r Sector r θ

Worked Example 1 — Simple shapes

Find the area and perimeter of a trapezium with parallel sides 6 cm and 10 cm, height 4 cm, and non-parallel sides each 5 cm.

Area:

A = ½(a + b)h = ½(6 + 10) × 4 = ½ × 16 × 4 = 32 cm²

Perimeter:

P = 6 + 10 + 5 + 5 = 26 cm

Worked Example 2 — Composite shape

A rectangle is 8 m × 5 m with a quarter-circle (radius 5 m) cut from one corner. Find the area.

Rectangle area: Arect = 8 × 5 = 40 m²

Quarter-circle area: Aqc = ¼ × π × 5² = ¼ × π × 25 = 19.635 m²

Total area: 40 − 19.635 = 20.36 m² (to 2 d.p.)

Hot Tip: Always check units before calculating — if dimensions are mixed (e.g. some in cm, some in m), convert first. Area answers must be in square units (cm², m², etc.). When rounding, follow the question’s instruction; if not stated, give 2 decimal places for measurements.

Full Lesson

Perimeter is the total distance around the outside of a shape. Area is the amount of flat surface enclosed by the shape. These two measurements use different units — perimeter is linear (cm, m) while area is square (cm², m²).

Understanding the Formulas

Rectangle: Multiply length by width for area. Add all four sides (two pairs) for perimeter. The formula P = 2(l + w) adds one length and one width then doubles, which is the same as adding all four sides.

Triangle: Area = half base times height. The height must be perpendicular to the base — it is not the slant side. Perimeter = sum of all three sides.

Parallelogram: Area = base × height, where height is perpendicular distance between parallel sides. Opposite sides are equal, so perimeter = 2(a + b).

Trapezium: The two parallel sides are averaged, then multiplied by the height. Think of it as finding the area of a rectangle with the “average” width.

Circle: The circumference C = 2πr is the “perimeter” of a circle. Area = πr².

Sector: A sector is a “pie slice” of a circle. Its area and arc length are fractions of the full circle, determined by the angle θ.

Guide Example — Finding sector perimeter

Find the perimeter of a sector with radius 12 cm and angle 90°.

Arc length = (θ/360) × 2πr = (90/360) × 2 × π × 12 = ¼ × 24π = 6π ≈ 18.85 cm

Perimeter = arc + 2 radii = 18.85 + 12 + 12 = 42.85 cm

Notice the perimeter of a sector includes the two straight edges (radii), not just the arc.

Composite Shapes

A composite shape is made up of two or more basic shapes. To find its area:

  1. Identify which basic shapes make up the composite shape.
  2. Decide whether to add or subtract areas.
  3. Calculate each area separately.
  4. Combine the areas.

For perimeter of composite shapes, only count the outer boundary — internal lines are NOT included.

Guide Example — Composite perimeter

A rectangle 10 m × 4 m has a semicircle on one short end (diameter = 4 m). Find total perimeter.

Semicircle arc = ½ × 2πr = πr = π × 2 = 6.283 m

Perimeter = 10 + 4 + 10 + arc (the short side under the semicircle is replaced by the arc)

= 10 + 4 + 10 + 6.283 = 30.28 m

Note: We include the long sides (2 × 10), the far short side (4), and the semicircle arc, but NOT the short side where the semicircle sits.

Tip: Draw and label a diagram for every composite shape question. Mark which sections belong to which shape. This prevents missing parts or double-counting.

Mastery Practice

See Answers ➔

  1. Q1 — Perimeter and area of basic shapes

    Fluency

    (a) Find the perimeter and area of a rectangle with length 9 cm and width 4 cm.

    (b) A triangle has base 8 cm, height 5 cm, and sides of length 6 cm, 7 cm, and 8 cm. Find the perimeter and area.

    (c) A circle has radius 6 cm. Find the circumference and area, leaving answers in exact form (with π) then as decimals.

  2. Q2 — Areas of parallelogram, trapezium, and sector

    Fluency

    (a) Find the area of a parallelogram with base 12 cm and perpendicular height 7 cm.

    (b) Find the area of a trapezium with parallel sides 5 cm and 9 cm, and height 6 cm.

    (c) Find the area of a sector with radius 8 cm and angle 135°.

  3. Q3 — Circumference and area of circles

    Fluency

    (a) Circle with r = 4.5 cm: find circumference and area.

    (b) Circle with d = 14 m: find circumference and area.

    (c) Circle with r = 9.2 mm: find circumference and area.

  4. Q4 — Composite shape: rectangle with semicircle

    Understanding

    A shape consists of a rectangle 10 m × 6 m with a semicircle attached to one of the long ends (the diameter of the semicircle equals the short side = 6 m, so radius = 3 m).

    Wait — re-read: the semicircle sits on one short end (width = 10 m... no). Actually: the rectangle is 10 m long, 6 m wide. The semicircle is on one short end. Short end has length 6 m, so the semicircle has diameter = 6 m and radius = 3 m.

    Actually the question states: semicircle on one end, diameter = 10. So the rectangle is 10 × 6, and the semicircle diameter = 10 (sits on a long end), radius = 5 m.

    (a) Find the total perimeter of the composite shape.

    (b) Find the total area.

  5. Q5 — L-shaped room

    Understanding

    An L-shaped room has outer dimensions 8 m × 6 m, with a 3 m × 2 m rectangular section cut from one corner.

    (a) Find the perimeter of the L-shape.

    (b) Find the area of the L-shape.

  6. Q6 — Sector problems

    Understanding

    (a) Find the arc length of a sector with radius 15 cm and angle 72°.

    (b) Find the area of a sector with radius 10 cm and angle 240°.

    (c) A pizza slice has an arc length of 18.85 cm and a radius of 15 cm. Find the central angle.

  7. Q7 — Fencing a rectangular paddock

    Understanding

    A rectangular paddock measures 240 m × 185 m.

    (a) Find the total fencing required.

    (b) Fencing costs $12.50 per metre. Find the total cost.

    (c) If one long side borders a river and no fence is needed there, recalculate the fencing needed and its cost.

  8. Q8 — Tiling a floor

    Understanding

    A floor is 7.2 m × 4.8 m. Square tiles measure 30 cm × 30 cm.

    (a) Find the floor area in m².

    (b) How many tiles are needed?

    (c) Tiles cost $4.80 each. Add 10% extra for waste. Find the total cost.

  9. Q9 — Running track

    Problem Solving

    A standard running track has two semicircular ends (each with radius 36 m) and two straight sections. The total track length is 400 m.

    (a) Find the length of each straight section.

    (b) Find the total area enclosed by the track.

  10. Q10 — Stained glass window

    Problem Solving

    A stained glass window consists of a rectangle 60 cm wide × 80 cm tall, with an equilateral triangle (side length 60 cm) on top.

    (a) Find the total perimeter of the window.

    (b) Find the total area of the window in cm².

    (c) Find the cost to frame the window at $8.50 per metre (convert the perimeter to metres).