1. Units of measure
The conversions to know
- Metric conversions all work by multiplying or dividing by a power of ten
| Measure | Conversions |
|---|---|
| Length | 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m |
| Mass | 1 g = 1000 mg, 1 kg = 1000 g, 1 tonne = 1000 kg |
| Capacity | 1 litre = 1000 ml, 1 cl = 10 ml |
| Volume to capacity | 1 ml = 1 cm³, 1 litre = 1000 cm³, 1 m³ = 1000 litres |
- Decide between multiplying and dividing by asking whether the count of units should go up or down
- Changing to a smaller unit gives a bigger number, so multiply; changing to a larger unit gives a smaller number, so divide
- Awkward conversions are safer in stages, so centimetres to kilometres goes through metres first
Squared and cubed units
- Area units are squared, so the conversion factor is squared too
- Volume units are cubed, so the conversion factor is cubed
- 1 cm² = 10² mm² = 100 mm²
- 1 m² = 100² cm² = 10 000 cm²
- 1 km² = 1000² m² = 1 000 000 m²
- 1 cm³ = 10³ mm³ = 1000 mm³
- 1 m³ = 100³ cm³ = 1 000 000 cm³
- Picture a square metre as a square measuring 100 cm by 100 cm, which makes the 10 000 obvious rather than something to memorise
- The syllabus names conversion between area units and between volume and capacity explicitly, so both are examinable
Worked example
Converting an area
Convert 2.5 m² into cm², and 45 000 cm² into m².
Solution:
- One metre is 100 cm, so one square metre is 100 × 100 = 10 000 cm²
- Changing to the smaller unit means multiplying
- 2.5 × 10 000 = 25 000 cm²
- Going the other way means dividing by the same factor
- 45 000 ÷ 10 000 = 4.5 m²
- Check: 4.5 × 10 000 = 45 000
2. Perimeter
Adding up the outside
- The perimeter is the total distance round the edge of a shape, so it is a length and its units are mm, cm, m or km
- Find it by adding every side, working round the shape once so nothing is counted twice or missed
- For a regular shape it is the number of sides times the length of one, so a square of side x has perimeter 4x
- A circle's perimeter has its own name, the circumference, and belongs to the circles topic
Missing lengths
- Compound shapes usually leave some sides unlabelled, and they have to be worked out before adding
- On an L-shape the two shorter horizontal sides together equal the long horizontal side opposite, and the same holds vertically
- Dashes on a diagram mark sides that are equal, and the question may state equalities in words instead
- Pythagoras supplies a sloping side where a right angle is available
Worked example
Perimeter of an L-shaped room
The floor plan below shows a room. All the corners are right angles. Work out the perimeter of the room.
Solution:
- Two sides are unlabelled, so find those first
- The two horizontal sides at the top must add to the 6 m base, so the top edge is 6 − 2.4 = 3.6 m
- The two vertical sides on the right must add to the 3.2 m left side, so the step down is 3.2 − 1.8 = 1.4 m
- Now add every side once, going round the shape
- 3.6 + 1.4 + 2.4 + 1.8 + 6 + 3.2 = 18.4 m
- Perimeter is a length, so the answer is in metres and not square metres
3. Area formulas
The four shapes
- Area is the space inside a shape, measured in squared units such as mm², cm² or m²
| Shape | Area |
|---|---|
| Rectangle | length × width |
| Triangle | ½ × base × perpendicular height |
| Parallelogram | base × perpendicular height |
| Trapezium | ½ × (sum of the parallel sides) × distance between them |
- Only the triangle formula is printed in the paper, so the other three have to be known
- Every one of them uses the perpendicular height, measured at right angles to the base (up to the opposite corner of a triangle, or across to the opposite parallel side) rather than along a sloping side
- The trapezium formula needs the two parallel sides identified first, since the other two play no part
- If a formula escapes you, cut the shape into a rectangle plus one or two triangles instead and add those areas: the total is the same
Exam tip
Find the parallel sides before the height
Mark the two parallel sides before you look for the height. A trapezium may be drawn on its side, with the distance between them along the bottom, at right angles to both; the height may also lie outside the shape, or be left for you to work out. The method mark in the schemes uses that perpendicular distance wherever it sits, and a sloping side printed on the diagram is never the height.
Worked example
Three areas from the formulas
Find the area of each shape.
Solution:
- The triangle is right-angled, so the two sides meeting at the right angle are the base and the perpendicular height
- Triangle: ½ × 8 × 7 = 28 cm²
- On the parallelogram the 12 cm is the perpendicular height, not the sloping side
- Parallelogram: 15 × 12 = 180 cm²
- On the trapezium the parallel sides are 15 cm and 30 cm, and they are 20 cm apart
- Trapezium: ½ × (15 + 30) × 20 = ½ × 45 × 20 = 450 cm²
- Every answer is in cm² because area is measured in squared units
Areas on a grid
- Count the whole squares inside the shape first, marking them off as you go
- Then pair up the part-squares round the edge to make whole ones
- Check what one square stands for: on a 1 cm² grid the count is the area in cm², and if a square stands for any other area, multiply the count by that area
4. Compound shapes and problem solving
Adding or subtracting
- A compound shape is built from simpler ones, and there are two ways to handle it
- Split it into rectangles and triangles, find each area, and add them
- Subtract, by taking the whole enclosing rectangle and removing the missing piece
- Subtracting is usually quicker where a corner or a hole has been cut out
- Sketch the split on the diagram and label each piece, so the working shows where each number came from
- Check the split covers the shape exactly once, with no overlap and no gap
Worked example
The same area two ways
The floor plan of the same L-shaped room is split into two rectangles by the dashed line. Work out the floor area.
Solution:
- Area A is a rectangle 3.6 m by 3.2 m, so its area is 3.6 × 3.2 = 11.52 m²
- Area B is a rectangle 2.4 m by 1.8 m, so its area is 2.4 × 1.8 = 4.32 m²
- Adding: 11.52 + 4.32 = 15.84 m²
- Subtracting gives the same answer, which is a quick check
- The whole enclosing rectangle is 6 × 3.2 = 19.2 m²
- The piece missing from the top right is 2.4 m by 3.2 − 1.8 = 1.4 m, so 2.4 × 1.4 = 3.36 m²
- 19.2 − 3.36 = 15.84 m², as before
Working backwards
- A question may give the area and ask for a missing length, which means substituting into the formula and solving
- The same applies to perimeter, and a shape with an algebraic side turns into an equation
- Check the units the answer needs, converting only at the end
Worked example
An area given, a length wanted
A trapezium has parallel sides of 9 cm and 13 cm and an area of 77 cm². Find the distance between the parallel sides.
Solution:
- Write the formula first, then substitute what is known
- Area = ½ × (sum of the parallel sides) × distance between them
- 77 = ½ × (9 + 13) × h
- 77 = ½ × 22 × h
- 77 = 11h
- h = 77 ÷ 11 = 7 cm
- Check by substituting back: ½ × 22 × 7 = 77, which matches the area given
- The answer is a length, so it is in cm and not cm²