8.2 Area & Composite Shapes
Key Takeaways
- Area measures the surface inside a shape in square units: 1 cm^2 is a square 1 cm by 1 cm, and 1 m^2 = 10,000 cm^2
- Area formulas by paper level: rectangle A = l x w, triangle A = (b x h) / 2, parallelogram A = b x h, trapezium A = ((a + b) / 2) x h
- Split composite shapes into rectangles and add the parts, or complete the bounding rectangle and subtract the missing piece
- Shaded-region problems ask for outer area minus inner (unshaded) area; find each area separately first
- Area uses square units and perimeter uses plain units — a rectangle 5 cm by 3 cm has area 15 cm^2 but perimeter 16 cm, and ICAS tests that you can tell them apart
Area is where the Measures & Units strand gets interesting. Papers B-C (Years 4-5) explicitly list "areas and perimeters" as core skills, and by Papers G-H you will meet composite shapes, shaded regions and conversions between square units, often inside word problems about carpets, tiles and paint. The skills layer on top of each other, so a shaky rectangle formula will hurt you all the way to Year 10.
What Area Means
Area is the amount of flat surface inside a shape, measured in square units. One square centimetre (1 cm^2) is a square with sides of 1 cm. Younger papers show a shape drawn on a centimetre grid and ask you to count the squares — count full squares first, then pair up half squares.
The Area Formulas, in ICAS Order
| Shape | Formula | First appears | Example |
|---|---|---|---|
| Rectangle / square | A = l x w | Papers B-C | 6 cm x 4 cm = 24 cm^2 |
| Triangle | A = (b x h) / 2 | Papers D-E | b = 10 cm, h = 6 cm: A = 30 cm^2 |
| Parallelogram | A = b x h | Papers E-F | b = 8 cm, h = 5 cm: A = 40 cm^2 |
| Trapezium | A = ((a + b) / 2) x h | Papers G-H | a = 6, b = 10, h = 4: A = 32 cm^2 |
Two warnings ICAS exploits every year:
- The height of a triangle or parallelogram is the perpendicular distance to the base, not the slanted side. Diagrams deliberately label a slant side to tempt you.
- For a triangle, "half of base times height" means you can halve either number first. Halve the even one: (14 x 5) / 2 = 7 x 5 = 35. This keeps the arithmetic mental.
Worked Example 1 (Paper D style)
A triangle has base 12 cm and perpendicular height 7 cm. Find its area.
A = (12 x 7) / 2 = 84 / 2 = 42 cm^2. Check: a 12 by 7 rectangle would be 84 cm^2, and the triangle is half of it, so 42 cm^2 is sensible.
Areas and Perimeters Together (Papers B-C)
ICAS loves asking for both in one question, or asking which shape has a given area but a different perimeter. Keep them straight:
- Area counts the squares inside — multiply.
- Perimeter measures the fence around the outside — add the sides.
A 5 cm by 3 cm rectangle has area 15 cm^2 and perimeter 16 cm. Note that shapes can share an area but have different perimeters: a 6 by 2 rectangle also has area 12 cm^2 like a 4 by 3 rectangle, but its perimeter is 16 cm rather than 14 cm. This "same area, different perimeter" idea is a favourite multiple-choice set-up.
Composite Shapes: Split or Subtract
A composite shape is built from simple shapes glued together. Two strategies cover every ICAS question:
- Split: cut the shape into rectangles (and triangles), find each area, and add.
- Subtract: complete the shape into one big rectangle, then subtract the missing piece.
Worked Example 2
An L-shaped room is made from a 6 m by 4 m rectangle with a 2 m by 3 m corner removed. Find its area.
Subtract method: bounding rectangle = 6 x 4 = 24 m^2; removed corner = 2 x 3 = 6 m^2; area = 24 - 6 = 18 m^2.
Split method check: a 6 x 2 strip (12 m^2) plus a 3 x 2 strip (6 m^2) gives 18 m^2. Both agree, which is a good habit when you have time.
Shaded Regions
Older papers show a large shape with a smaller shape cut out (or overlapping) and ask for the shaded area. The rule is always:
Shaded area = outer area - unshaded area
Worked Example 3
A rectangular photo 10 cm by 8 cm sits in a frame 2 cm wide on every side. What is the area of the frame?
Outer rectangle: (10 + 4) x (8 + 4) = 14 x 12 = 168 cm^2 (add 2 cm to both ends of each dimension — a common trap is adding only 2 cm total). Frame area = 168 - 80 = 88 cm^2.
Converting Square Units
This is the single most-missed area skill. Because 1 m = 100 cm, one square metre is 100 x 100 = 10,000 cm^2, not 100 cm^2.
- m^2 to cm^2: multiply by 10,000. 0.5 m^2 = 5,000 cm^2.
- cm^2 to m^2: divide by 10,000. 35,000 cm^2 = 3.5 m^2.
Also know that 1 hectare = 10,000 m^2 for land area in the older papers.
Worked Example 4 (Papers G-H style)
A floor is 4 m by 3 m. Tiles are 20 cm by 20 cm. How many tiles are needed?
Method by area: floor = 12 m^2 = 120,000 cm^2. Each tile = 400 cm^2. Tiles needed = 120,000 / 400 = 300.
Method by rows (a good check): 4 m = 400 cm gives 20 tiles along the length; 3 m = 300 cm gives 15 along the width; 20 x 15 = 300 tiles.
Common Traps Summary
- Confusing area and perimeter (read the units in the question: cm vs cm^2).
- Using the slant side instead of the perpendicular height.
- Forgetting the half in the triangle formula.
- Converting square units by 100 instead of 10,000.
- Adding a border to only one side of a dimension in frame problems.
A parallelogram has a base of 9 cm and a perpendicular height of 6 cm. Its slanted side is 7 cm. What is its area?
A rectangular courtyard is 5 m by 4 m. A square garden bed of side 2 m sits in the middle. What is the paved (unplanted) area around the garden bed?