Learning Objectives
  • Calculate perimeters and areas of common plane figures.
  • Use formulas for triangles, parallelograms and trapezia.
  • Solve compound-shape and missing-length problems.
  • Convert correctly between units of length and area.
Key Terms
Perimeter
The total distance around the boundary of a plane figure.
Area
The amount of surface enclosed by a plane figure.
Perpendicular height
The shortest distance measured at right angles to a base.
Compound shape
A shape formed from two or more simpler shapes.
Square unit
A unit of area, such as cm² or m².
Perimeter And Units

Perimeter is found by adding boundary lengths only. Internal dividing lines are not part of the perimeter. All lengths must be in the same unit before addition.

Worked Example: Perimeter With Units

Question: A rectangle is 1.8 m by 75 cm. Find its perimeter in metres.

  1. Convert 75 cm to 0.75 m.
  2. Use P=2(1.8)+2(0.75).

Answer: 5.1 m.

Area Formulas
Shape Area
Rectangle A=lw
Square A=s^2
Triangle A=\frac12 bh
Parallelogram A=bh
Trapezium A=\frac12(a+b)h

For triangles, parallelograms and trapezia, h is the perpendicular height. A sloping side cannot be used as height unless it is perpendicular to the selected base.

Worked Example: Trapezium

Question: A trapezium has parallel sides 8 cm and 15 cm and perpendicular height 6 cm. Find its area.

  1. Use A=\frac12(a+b)h.
  2. Substitute: A=\frac12(8+15)(6).
  3. Calculate \frac12\times23\times6=69.

Answer: 69 cm².

Finding A Missing Dimension

Rearrange an area formula or reason using inverse operations. Keep units consistent and distinguish length units from square units.

Worked Example: Missing Height

Question: A triangle has area 54 cm² and base 12 cm. Find its perpendicular height.

  1. Use 54=\frac12(12)h.
  2. So 54=6h.
  3. Divide by 6.

Answer: 9 cm.

Compound Shapes

Split a compound shape into familiar pieces or subtract a missing part from a larger surrounding shape. Mark every derived length before calculating. Different decompositions should give the same result.

Worked Example: L-Shaped Region

Question: A 10 cm by 8 cm rectangle has a 4 cm by 3 cm rectangle removed from one corner. Find the remaining area.

  1. Large rectangle area =10\times8=80 cm².
  2. Removed area =4\times3=12 cm².
  3. Subtract.

Answer: 68 cm².

Area Unit Conversion

Length conversion factors must be squared for area. Since 1\text{ m}=100\text{ cm}, it follows that 1\text{ m}^2=100^2=10000\text{ cm}^2. Similarly, 1\text{ cm}^2=100\text{ mm}^2.

Worked Example: Area Conversion

Question: Convert 3.6\text{ m}^2 to cm².

  1. Use 1\text{ m}^2=10000\text{ cm}^2.
  2. Calculate 3.6\times10000.

Answer: 36 000 cm².

Problem Solving With Cost

Area calculations often lead to cost questions. First find total area, then account for wastage or coverage, then multiply or divide by the stated rate. Round up when whole tiles, tins or rolls are required.

Worked Example: Tiling

Question: A floor is 5.4 m by 3.8 m. Square tiles have side 30 cm. How many tiles are needed, ignoring wastage?

  1. Floor area =5.4\times3.8=20.52 m².
  2. Tile side =0.30 m, so tile area =0.30^2=0.09 m².
  3. Number =20.52\div0.09=228.

Answer: 228 tiles.

8th Edition Chapter Map
  • Conversion of area units
  • Perimeter and area of rectangles and triangles
  • Parallelograms and trapeziums
  • Circumference and area of circles
  • Composite figures
Area Unit Conversion

Area scale factors are squared. Since 1\text{ m}=100\text{ cm}, 1\text{ m}^2=100^2=10000\text{ cm}^2. This is why multiplying by only 100 is wrong. Similarly 1\text{ cm}^2=100\text{ mm}^2.

Worked Example: Area Conversion

3.72\text{ m}^2=3.72\times10000=37200\text{ cm}^2. Conversely, 84500\text{ cm}^2=8.45\text{ m}^2.

Deriving And Using Area Formulae

The area of a parallelogram is base times perpendicular height, not base times sloping side. A triangle with the same base and height has half the area. A trapezium has area equal to half the sum of its parallel sides multiplied by the perpendicular distance between them.

A_{\triangle}=\frac12bh,\qquad A_{\text{parallelogram}}=bh,\qquad A_{\text{trapezium}}=\frac12(a+b)h
Worked Example: Missing Height

A trapezium has area 154 cm^2 and parallel sides 9 cm and 13 cm. 154=\frac12(9+13)h=11h, so h=14 cm.

Circles

The circumference is a length and uses linear units; area uses square units. Use the calculator value of \pi or leave an exact answer in terms of \pi when requested.

C=2\pi r=\pi d,\qquad A=\pi r^2
Worked Example: Radius From Circumference

A circle has circumference 31.4 cm. r=31.4/(2\pi)\approx4.997, so the radius is approximately 5.00 cm.

Composite And Shaded Figures

Break a compound shape into standard pieces or subtract cut-outs from a larger shape. Mark every required length, including those found by subtraction. For perimeter, count only the outside boundary; an internal dividing line is not part of the perimeter.

Worked Example: Rectangle With Semicircle

A rectangle 12 cm by 8 cm has a semicircle of diameter 8 cm attached to one short side. Area =12\times8+\frac12\pi(4)^2=96+8\pi cm^2. The perimeter is 12+12+8+\pi(4)=32+4\pi cm; the shared diameter is internal and is not counted.

Multi-Step Cost Problems

After finding an area or perimeter, read the pricing unit carefully. Flooring may be sold per square metre, fencing per metre and tiles in whole packs. Round up when a whole number of packs is required; ordinary rounding could leave insufficient material.

Worked Example: Tiling

A floor is 6.4 m by 4.5 m. Tiles cover 0.36 m^2 per box. Floor area =28.8 m^2. Boxes =28.8/0.36=80. With 8% extra for breakage, 80\times1.08=86.4, so 87 boxes are needed.

Extended Practice

A. A parallelogram has area 126 cm^2 and height 9 cm. Find its base.

b=126/9=14 cm.

B. Find the area of an annulus with outer radius 8 cm and inner radius 5 cm.

\pi(8^2-5^2)=39\pi\approx122.5 cm^2.

C. Convert 0.084\text{ m}^2 to cm^2.

0.084\times10000=840\text{ cm}^2.

Examination Guidance
  • Write the formula before substitution.
  • Use the perpendicular height, not a sloping side.
  • Show missing lengths on compound shapes.
  • When converting area units, square the length conversion factor.
  • Round up only when the context requires complete objects or packages.
Common Mistakes
  • Including internal edges in perimeter.
  • Using A=bh for a triangle and forgetting the factor \frac12.
  • Converting m² to cm² by multiplying by 100 instead of 10 000.
  • Rounding a number of tiles down because the decimal part seems small.
Knowledge Check

1. Find the area of a parallelogram with base 13 cm and perpendicular height 7 cm.

Answer

91 cm².

2. A trapezium has area 84 cm², parallel sides 9 cm and 15 cm. Find its height.

Answer

7 cm.

3. Convert 4500 cm² to m².

Answer

0.45 m².

4. A square has perimeter 52 cm. Find its area.

Answer

169 cm².

5. A 12 m by 9 m garden contains a 4 m by 3 m pond. Find the remaining area.

Answer

96 m².

Mixed Review With Full Solutions

1. A triangle has area 96 cm^2 and base 15 cm. Find its perpendicular height.

96=\frac12(15)h, so h=192/15=12.8 cm.

2. A circular garden has diameter 18 m. Find the cost of fencing it at 1 250 per metre.

Circumference =18\pi\approx56.55 m. Cost \approx56.55\times1250=70685.8, approximately 70 686.

3. A square and a circle have equal perimeter 40 cm. Which has greater area?

Square side 10 cm, area 100 cm^2. Circle radius =40/(2\pi)=20/\pi, area =400/\pi\approx127.3 cm^2. The circle has greater area.

4. A compound figure is a 14 cm by 10 cm rectangle with a 4 cm by 3 cm corner removed. Find its area.

14\times10-4\times3=140-12=128 cm^2.

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