Practice Maths

Composite Shapes

Key Ideas

Key Terms

composite shape
a shape made by joining or cutting simpler shapes such as rectangles, triangles, and trapeziums.
decompose
to split a composite shape into simpler parts; find each part's area separately, then add or subtract.
shaded region
the part of a shape that remains after another shape is removed; shaded area = total area − unshaded area.
total area
the combined area of all parts of a composite shape; found by adding the areas of each simpler part.
outer boundary
the perimeter of a composite shape; trace only the outside edge — internal shared edges are NOT included in the perimeter.
Hot Tip For perimeter of composite shapes, trace your finger around the outside only — internal edges do not form part of the perimeter. For shaded area, always work out: Total − Unshaded.

Worked Example

Question: An L-shaped room has outer dimensions 10 m × 8 m, with a 4 m × 3 m rectangle removed from one corner. Find the area and perimeter.

Area: Total − removed = (10 × 8) − (4 × 3) = 80 − 12 = 68 m²

Perimeter: Trace outer boundary: 10 + 8 + 6 + 3 + 4 + 5 = 36 m
(The two “step” sides replace what would have been part of the full rectangle’s edge.)

What Is a Composite Shape?

A composite shape (also called a compound shape) is made up of two or more simpler shapes joined together or with pieces cut out. Real objects are almost never perfect rectangles or triangles — they're composite. A floor plan, a park, a swimming pool, or a logo might all be composite shapes.

The strategy for composite shapes: decompose (break apart) the complex shape into simpler pieces, find the area of each piece using known formulas, then add or subtract the pieces to get the total.

The Add Strategy: Joining Shapes

When a complex shape is formed by joining simpler shapes, add their areas.

Example: An L-shaped room. Extend the shape mentally to see two rectangles. Identify the dimensions of each rectangle from the given measurements, calculate each area, then add.

L-shape: total width 10 m, total height 8 m, with a 4 m × 5 m section cut from the corner.
Method A (add): split into two rectangles. Top piece: 6 m × 3 m = 18 m2. Bottom piece: 10 m × 5 m = 50 m2. Total = 68 m2.
Method B (subtract): start with full 10 × 8 = 80 m2, subtract the cut 4 × 3 = 12 m2. Total = 80 − 12 = 68 m2. Same answer.

The Subtract Strategy: Cutting Out Pieces

When one shape has another piece removed from it, subtract the removed piece's area.

Example: A square room 6 m × 6 m with a semicircular archway cut from one wall. Radius of arch = 1 m.
Square area = 36 m2. Semicircle area = ½ × π × 12 ≈ 1.57 m2.
Total floor area = 36 − 1.57 = 34.43 m2.

Example: A rectangular garden (12 m × 8 m) with a circular pond (radius 2 m) in the middle. Garden area = 96 m2. Pond = π × 4 ≈ 12.57 m2. Planting area = 96 − 12.57 = 83.43 m2.

Identifying the Simpler Shapes

The key skill is spotting which simpler shapes make up the composite. Look for:

• Right angles → rectangles or squares
• Triangles at edges or corners
• Curved parts → semicircles or quarter circles
• Trapezia where two parallel lines meet a slanted edge

Draw dividing lines on the diagram to separate the parts. Label any missing lengths by using the given dimensions — opposite sides of a rectangle are equal, so you can often work out unlabelled lengths by subtracting.

Composite Shape in Context: Floor Plans

Architects use composite shape calculations constantly. A house floor plan is a composite of rectangles, sometimes with circular windows or triangular roof sections. Tilers calculate composite areas for irregular-shaped bathrooms. Gardeners calculate irregular plot areas for turf, mulch, or fertiliser.

Example: A garden path runs around a rectangular lawn. Lawn = 10 m × 6 m. Path is 1 m wide all around. Outer dimensions of lawn + path = 12 m × 8 m. Path area = (12 × 8) − (10 × 6) = 96 − 60 = 36 m2.

Key tip: Whenever a length is not labelled in a composite shape problem, work it out from the other given lengths before calculating. Opposite sides of rectangular sections must be equal — add or subtract the labelled lengths to find any missing ones. Getting the dimensions wrong is the main cause of errors in these problems.

Mastery Practice

  1. Find the total area of each composite shape made from two rectangles joined together. Fluency

    1. Rectangle A: 8 cm × 5 cm joined to Rectangle B: 4 cm × 3 cm (end-to-end)
    2. Rectangle A: 12 m × 6 m joined to Rectangle B: 5 m × 4 m (side-by-side)
    3. Rectangle A: 9 cm × 7 cm joined to Rectangle B: 6 cm × 4 cm
    4. Rectangle A: 15 m × 8 m joined to Rectangle B: 7 m × 5 m
    5. Rectangle A: 10 cm × 6 cm joined to Rectangle B: 10 cm × 3 cm (stacked)
    6. Rectangle A: 20 mm × 12 mm joined to Rectangle B: 8 mm × 10 mm
    7. Rectangle A: 6.5 m × 4 m joined to Rectangle B: 3 m × 2 m
    8. Rectangle A: 14 cm × 9 cm joined to Rectangle B: 6 cm × 5 cm

  2. Find the total area of each composite shape made from a rectangle and a triangle. Fluency

    1. Rectangle 10 cm × 6 cm with a triangle on top: base 10 cm, height 4 cm
    2. Rectangle 8 m × 5 m with a right-angled triangle attached: base 4 m, height 5 m
    3. Rectangle 12 cm × 7 cm with a triangle on one end: base 7 cm, height 5 cm
    4. Rectangle 15 m × 9 m with a triangle on top: base 15 m, height 6 m
    5. A house-shaped figure: rectangle 8 cm × 6 cm with a triangular roof, base 8 cm, height 3 cm
    6. Rectangle 20 mm × 10 mm with a triangle on the right: base 10 mm, height 8 mm
    7. Rectangle 7 m × 4 m with a right-triangle removed from one corner: triangle base 2 m, height 4 m (find area remaining)
    8. Rectangle 11 cm × 8 cm with a triangle on top: base 11 cm, height 5 cm

  3. Find the shaded area in each diagram (large rectangle minus small rectangle inside). Fluency

    1. Outer rectangle 10 cm × 8 cm, inner rectangle 4 cm × 3 cm
    2. Outer rectangle 14 m × 10 m, inner rectangle 6 m × 5 m
    3. Outer square 12 cm × 12 cm, inner square 5 cm × 5 cm
    4. Outer rectangle 20 mm × 15 mm, inner rectangle 8 mm × 6 mm
    5. Outer rectangle 8 m × 6 m, inner rectangle 3 m × 2 m (border/frame)
    6. Outer square 16 cm × 16 cm, inner square 10 cm × 10 cm (picture frame)
    7. Outer rectangle 25 cm × 18 cm, inner rectangle 15 cm × 10 cm
    8. Outer rectangle 30 m × 20 m, inner rectangle 22 m × 14 m (garden path around lawn)

  4. Find the perimeter of each composite shape. Be careful to trace only the outer boundary. Understanding

    1. L-shape: outer 12 cm × 8 cm, notch removed is 5 cm × 3 cm (from one corner)
    2. T-shape: horizontal top bar 14 m × 4 m, vertical stem 4 m × 8 m from centre of base
    3. A staircase shape with 3 steps: each step is 4 cm wide and 3 cm tall
    4. Plus-sign: two rectangles 12 cm × 3 cm crossing at centre (vertical and horizontal)
    5. L-shape: outer dimensions 18 m × 12 m, removed corner is 8 m × 5 m
    6. House shape: rectangle 10 m × 6 m with isoceles triangle on top, base 10 m, slant sides each 6.5 m
    7. U-shape: outer 20 cm × 12 cm, inner cut-out 10 cm × 8 cm from top centre
    8. F-shape: outer rectangle 12 m × 10 m, two rectangular notches removed from the right side, each 4 m × 3 m

  5. Find the area of each composite shape that includes a trapezium. Understanding

    1. Rectangle 8 m × 5 m attached to a trapezium: parallel sides 8 m and 5 m, height 4 m
    2. A swimming pool shape: rectangle 12 m × 6 m with a trapezium at one end, parallel sides 6 m and 4 m, height 3 m
    3. A pentagon-like shape: square 6 cm × 6 cm with a trapezium on top, parallel sides 6 cm and 3 cm, height 3 cm
    4. Two trapeziums joined: Trapezium A has parallel sides 10 cm and 6 cm, height 4 cm. Trapezium B has parallel sides 6 cm and 3 cm, height 3 cm.
    5. A cross-section of a road embankment: trapezium with parallel sides 12 m and 8 m, height 3 m, on top of rectangle 12 m × 1 m
    6. An irregular hexagon formed by a rectangle 14 cm × 6 cm with two identical trapeziums attached to each short end, each with outer parallel side 4 cm and height 4 cm.

  6. Solve each real-world problem. Show all working including units. Problem Solving

    1. A floor plan shows an L-shaped room. The full rectangle would be 9 m × 6 m, but a rectangular alcove 3 m × 2 m is cut from one corner.
      1. Find the floor area.
      2. Carpet costs $55 per m². Find the total carpet cost.
      3. Skirting board is needed around the entire perimeter. Find the length of skirting board required (assume no doorways).
    2. A garden path runs around the outside of a rectangular lawn 18 m × 12 m. The path is 1.5 m wide on all sides.
      1. Find the outer dimensions of the path and lawn combined.
      2. Find the area of just the path (outer area minus lawn area).
      3. Paving stones cost $38 per m². Find the total cost.
    3. A block of land is shaped like a rectangle 40 m × 30 m with a triangular section attached to one 30 m end. The triangle has a base of 30 m and a height of 15 m.
      1. Find the total area of the block.
      2. Express the area in hectares.
      3. The owner fences the outer boundary. The triangular section has two slant sides each 17 m long. Find the total fencing needed.

  7. Find the area and perimeter of the composite shape shown. Split it into simpler parts first. Understanding

    1. Find the area of this L-shape. Show how you split it. 6 cm ←5 cm→ 10 cm 12 cm 4 cm
    2. A house-shaped figure: find the total area. 14 cm 9 cm h = 5 cm

  8. For each composite shape, describe how you would split it into simpler parts, then find the area. Understanding

    1. A shape made of a rectangle 16 m × 8 m with a semicircle of diameter 8 m added to one short end. (Use π ≈ 3.14.)
    2. A shape made of a triangle with base 10 cm and height 6 cm sitting on top of a trapezium with parallel sides 10 cm and 16 cm, height 5 cm.
    3. A square 10 m × 10 m with a right-angled triangle of base 4 m and height 4 m removed from one corner.
    4. A rectangle 20 mm × 8 mm with two small squares of side 3 mm removed from each corner of one short end.

  9. Each student made an error calculating the perimeter or area of a composite shape. Identify and correct it. Understanding

    1. For an L-shape with outer dimensions 10 m × 8 m and a 4 m × 3 m piece removed, Sam calculated the perimeter as P = 2(10 + 8) = 36 m. What is wrong?
    2. For the same L-shape, Jordan calculated the area as A = 10 × 8 = 80 m². What did Jordan forget?
    3. For a shape made of two rectangles (6 cm × 4 cm and 3 cm × 2 cm), Alex added them: A = (6 × 4) + (3 × 2) = 30 cm². This is correct. But Alex then said the perimeter was P = 2(6 + 4) + 2(3 + 2) = 30 cm. Explain why the perimeter is not simply the sum of both rectangles’ perimeters.
    4. A picture frame has outer dimensions 30 cm × 20 cm and a rectangular hole 24 cm × 14 cm. Morgan calculated the frame area as 30 + 20 + 24 + 14 = 88 cm². What should the area be?

  10. A builder is designing an outdoor entertaining area. Use the diagram description to answer all parts. Show full working. Problem Solving

    10 m 7 m 3 m Patio Deck

    The patio is a rectangle 10 m × 7 m. A triangular deck is attached to one 7 m side with base 7 m and perpendicular height 3 m.

    1. Find the area of the patio.
    2. Find the area of the triangular deck.
    3. Find the total area of the entertaining area.
    4. Decking timber costs $65 per m² (for the triangular deck) and paving costs $45 per m² (for the patio). Find the total cost to complete the area.
    5. The owner wants to put a fence around the outer boundary only (not between the patio and deck). If the two slant sides of the triangle are each 5 m, find the total fencing length needed.
See Answers ➔