Perimeter & Area of Composite Shapes
Match Shape to Area Formula
Draw a line to match each shape to its area formula.
Area of Composite Shapes
A shape = rectangle (10 cm × 6 cm) + triangle on top (base 10 cm, height 4 cm).
Area of the rectangle:
Area of the triangle:
Total composite area:
Perimeter of Composite Shapes
Find the perimeter. Count outer edges only.
L-shape outer edges: 8, 5, 3, 2, 5, 3 cm. Perimeter =
Rectangle 12 × 8 with 2×3 corner cut out. Perimeter =
Area Involving Semicircles
Use π ≈ 3.14.
Area of semicircle, radius 5 cm:
Rectangle (10 × 6) with semicircle (r = 3) on one end. Total area:
Add Parts or Subtract Cut-out?
Sort each composite shape problem by strategy.
Real-World Area Problems
Show all working. Include units.
A garden is a rectangle (15 m × 8 m) with a triangular bed (base 6 m, height 4 m) removed from one corner. Find the remaining area.
A running track is a rectangle (100 m × 60 m) with a semicircle on each short end. Calculate the total perimeter of the track.
Area of a Trapezium
Use A = ½(a + b)h where a and b are the parallel sides and h is the perpendicular height.
a = 5 cm, b = 9 cm, h = 4 cm
a = 7 m, b = 11 m, h = 6 m
a = 3 cm, b = 7 cm, h = 5 cm
Area of an Annulus (Ring)
An annulus is a ring between two circles. Area = π(R² − r²) where R is outer radius and r is inner radius. Use π ≈ 3.14.
R = 5 cm, r = 3 cm
R = 10 m, r = 6 m
R = 8 cm, r = 5 cm
Units of Area Conversion
Draw a line to match each conversion fact.
Finding Perimeter When Area is Given
Use the given area to find the missing dimension, then calculate the perimeter.
A rectangle has area 48 cm² and one side of length 8 cm. Find the other side and then calculate the perimeter.
A square has area 64 m². Find the side length and perimeter.
A trapezium has parallel sides 8 cm and 12 cm, and area 50 cm². Find the perpendicular height.
Correct Formula?
Sort each formula: is it correct for the named shape?
Measuring Composite Shapes at Home
Apply area and perimeter skills to spaces in your home.
- 1Sketch the floor plan of a room that is not a simple rectangle (e.g. an L-shaped lounge). Measure all dimensions and calculate the area of floor space.
- 2Find a circular object (plate, tin, manhole cover). Measure its diameter. Calculate the area and circumference. Now imagine a second smaller circle inside it — calculate the area of the ring between them.
- 3Calculate how much paint you would need to paint one wall of your home, given that paint covers 10 m² per litre. Subtract the area of any windows or doors.
Perimeter of Composite Shapes — Careful Counting
Remember: only count the outer edges. Internal boundaries are NOT part of the perimeter.
Rectangle 10 × 6 with triangle on top (two slant sides of 5 cm each). Perimeter includes:
L-shape with outer measurements: 8+5+3+2+5+3. Perimeter =
Running track: two straights of 100 m, two semicircles radius 30 m. Perimeter =
L-Shape Area and Perimeter
Show two methods for finding the area of the L-shape and verify they give the same answer.
An L-shape is 10 m tall, 8 m wide. A 4 m × 3 m rectangle is cut from the top right corner. Find the area using BOTH the addition method (two rectangles) and the subtraction method. Verify both give the same answer.
Find the perimeter of the L-shape above. List every outer edge length and add them.
Units of Area
Choose the most appropriate unit for each measurement.
Area of a postage stamp:
Area of a bedroom floor:
Area of a national park:
Area of a fingernail:
Composite Shape Problems
Sketch the shape, choose your strategy, and show full working.
A garden is a rectangle (15 m × 8 m) with a triangular bed (base 6 m, height 4 m) removed from one corner. Find the remaining area to be grassed.
A school logo is made of a rectangle (20 cm × 10 cm) with a semicircle (radius 5 cm) on each of the two short ends. Find the total area of the logo.
Find Area from Perimeter
Use the given perimeter to find the missing dimension, then calculate the area.
Rectangle: perimeter = 36 cm, length = 11 cm. Area =
Square: perimeter = 28 m. Area =
Circle: circumference = 31.4 cm. Area ≈ (use π ≈ 3.14)
Real-World Area Calculations
Show all working. Include units.
A tiler is tiling a bathroom floor that is L-shaped: 4 m × 3 m with a 1 m × 1 m section cut out for a bathtub area. Each tile is 20 cm × 20 cm. How many tiles are needed? Add 10% extra for cuts and breakages.
A farmer wants to fertilise a paddock shaped like a trapezium (parallel sides 200 m and 150 m, height 80 m). Fertiliser is spread at 50 kg per hectare and costs $2.80 per kg. Find the total cost. (1 ha = 10 000 m²)
Sector and Arc Length
Use arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². Use π ≈ 3.14.
A sector has radius 8 cm and angle 45°. Find its arc length and area.
A pizza is cut into 8 equal slices. The pizza has diameter 30 cm. Find the arc length and area of one slice.
Sort Shape Problems by Formula Needed
Sort each area problem by the formula required.
Comparing Perimeter and Area
Explore the relationship between perimeter and area.
Three rectangles all have a perimeter of 24 cm: 10×2, 8×4, and 6×6. Calculate the area of each. Which has the largest area? What shape would maximise area for a fixed perimeter?
A farmer has 120 m of fencing to enclose a rectangular paddock. What dimensions maximise the area? (Try at least 4 different rectangles and record their areas.)
Maximising Area — Optimisation
Investigate and draw conclusions.
Show that among all rectangles with a fixed perimeter of P metres, the square maximises the area. (Try specific values: P = 20. Compare squares vs. non-squares.)
For a circle and a square each with perimeter 20 cm, which encloses a larger area? Calculate both and compare.
Area Calculations — Set A
Calculate the area of each shape. Show all working.
Triangle with base 12 cm and perpendicular height 7 cm.
Parallelogram with base 15 m and height 8 m.
Trapezium with parallel sides 6 cm and 10 cm, height 4 cm.
Rhombus with diagonals 14 cm and 10 cm. (Area = d₁ × d₂ / 2)
Perimeter Calculations
Calculate the perimeter of each shape. Show all working.
A shape is a rectangle 8 m × 5 m with a 2 m × 2 m square removed from one corner. Find the perimeter.
A semicircle of diameter 10 cm sits on top of a rectangle 10 cm × 6 cm. Find the perimeter of the composite shape (include both straight sides of rectangle and the semicircle arc).
Composite Area — Set A
Find the area of each composite shape. Sketch and label first.
An L-shaped floor plan: 10 m × 8 m with a 3 m × 4 m rectangle cut from one corner.
A shape made from a square 6 m × 6 m with a right-angled triangle of base 6 m and height 4 m attached to one side.
Composite Area — Set B
Find the area of each composite shape involving circles.
A rectangle 12 m × 6 m with a semicircle of diameter 6 m removed from one end. Find the remaining area.
A square 10 cm × 10 cm with a circular hole of radius 3 cm. Find the area of the square with the hole.
Selecting the Right Formula
Choose the correct formula for each situation.
Area of a triangle:
Area of a trapezium:
Perimeter of a circle:
Area of a parallelogram:
Area in Context — Paint
Calculate the area then answer the practical question.
A room has 4 walls: two walls are 5 m × 2.7 m and two walls are 4 m × 2.7 m. Each wall has one window (1.2 m × 1.5 m). One tin of paint covers 12 m². How many tins are needed to paint all 4 walls (excluding windows) with two coats?
Area in Context — Flooring
Calculate the area then find the cost.
An L-shaped room has dimensions: main part 8 m × 5 m, extension 3 m × 2 m. Carpet costs $45/m². How much will it cost to carpet the entire room?
If tiles are 40 cm × 40 cm and cost $4.50 each, how many tiles are needed for the room and what is the cost? (Add 10% for wastage.)
Add or Subtract?
For each composite shape description, decide whether you add or subtract areas.
Mixed Perimeter and Area
Find both perimeter and area for each shape.
A running track: two straight sections each 100 m, and two semicircles each with diameter 60 m. Find total perimeter (length of one lap) and total area enclosed.
Converting Units of Area
Convert each area to the specified unit. Show working.
Convert 2.5 m² to cm².
Convert 45 000 cm² to m².
A farm is 3.2 km². Express this in hectares. (1 km² = 100 ha)
Error Analysis — Area and Perimeter
Find and fix the errors.
Student: 'An L-shape has two rectangles: 8×4 and 3×2. Perimeter = 2(8+4) + 2(3+2) = 34.' What is wrong? Find the correct perimeter.
Student: 'Area of triangle = base × height = 6 × 8 = 48 cm².' What is wrong? Find the correct area.
Reflection
Summarise your learning about perimeter and area.
What is the key difference between perimeter and area? Give a real-world example where each matters.
Describe your strategy for finding the area of a composite shape. Use an example.
Design Challenge: Maximum Area
Solve this design problem with full working.
A garden bed uses 24 m of edging (perimeter = 24 m). List three different rectangle dimensions. Calculate the area of each. Which gives the largest area?
If the garden does not need to be a rectangle, what shape would give the largest area? Explain why.
Real-World Context: Painting a Wall
Solve the problem below. Show all steps.
A wall is 8 m wide and 3 m tall. It has two windows (each 1.2 m × 1.5 m) and one door (0.9 m × 2.1 m). Paint covers 5 m² per litre. How many litres of paint are needed to paint the wall (not the windows or door)? Round up to the nearest litre.
Classify the Measurement
Drag each item to 'Perimeter' or 'Area'.
Semester Review: Extended Problem
Complete this multi-step problem. Show all working.
A sports complex has a rectangular field (80 m × 50 m) with a semicircular goal area at each end (diameter = 20 m). Find: (a) total area of the playing surface; (b) total perimeter of the playing surface.
The complex manager wants to re-turf the field. Turf costs $12 per m². What is the total cost? If budget is $60 000, is there enough money?
Self-Assessment
Rate your confidence and identify areas for further study.
Rate your confidence (1–5) for each skill: finding perimeter of composite shapes, finding area of rectangles/triangles, finding area of circles, finding area of composite shapes. Explain your ratings.
Which problem in this worksheet did you find most challenging? What would you do differently next time?