Measuring, bearings and scale drawings
A bearing has three rules, and all three must be followed or the answer is wrong: measure from north, measure clockwise, and always write it with three figures. So an angle of 45° east of north is written 045°, and due west is 270°.
Back bearings — the bearing from B to A when you know the bearing from A to B — differ by 180°. Add 180 if the original is less than 180, subtract 180 if it is more.
Scale drawings use the ratio methods from Ratio, proportion and rates of change: measure the drawing, then multiply by the scale to get the real distance, converting units at the end.
- Bearing
- a direction measured from north, clockwise, and always written with three figures — so 45° east of north is written 045°
Area, volume and surface area
Composite shapes are split into pieces whose formulae you know, worked out separately, then added — or subtracted, when a piece has been cut out. Sketch the split and label each piece before calculating.
A composite solid works the same way. A cone on top of a cylinder has volume equal to the sum of the two, but its surface area does not include the circle where they join, because that surface is no longer on the outside. Deciding which surfaces are actually exposed is where the marks are.
Keep π in the calculation until the end and round once, or leave the answer in terms of π when the question asks for an exact value.
Worked example
A sector has radius 9 cm and angle 40°. Find its perimeter, to 3 significant figures.
- The perimeter is the arc plus the two straight radii — not just the arc.
- Arc length = (40 ÷ 360) × 2 × π × 9
- 40 ÷ 360 = 1⁄9, so the arc is (1⁄9) × 18π = 2π cm
- 2π = 6.2832 cm
- Add the two radii: 6.2832 + 9 + 9 = 24.2832
Answer: 24.3 cm to 3 significant figures. Forgetting the two radii is the standard error in sector perimeter questions — the exact arc length is 2π cm.
Pythagoras and trigonometry
The exact values are a specification requirement in their own right, so they are examined directly as well as being needed on the non-calculator paper. They can be reconstructed from two triangles if you would rather not memorise them: half an equilateral triangle of side 2 gives the 30° and 60° values, and a right-angled isosceles triangle with two sides of 1 gives the 45° values.
In three dimensions the method is to find a right-angled triangle inside the solid and work in two dimensions within it. Usually this takes two steps: use Pythagoras on the base to find a diagonal, then use that diagonal as one side of a second triangle standing up through the solid. Sketching the triangle separately from the 3D diagram makes it far easier to see.
Worked example
A cuboid measures 6 cm by 8 cm by 5 cm. Find the length of the diagonal from one corner to the opposite corner, to 3 significant figures.
- First find the diagonal across the base, using the 6 cm and 8 cm sides.
- base diagonal² = 6² + 8² = 36 + 64 = 100, so the base diagonal is 10 cm.
- Now form a right-angled triangle standing up: the base diagonal of 10 cm, the height of 5 cm, and the space diagonal as the hypotenuse.
- space diagonal² = 10² + 5² = 100 + 25 = 125
- space diagonal = √125 = 11.180…
Answer: 11.2 cm to 3 significant figures. Every 3D Pythagoras question follows this two-triangle pattern — find a diagonal in the base, then stand a triangle on it.
- Pythagoras' theorem
- in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides
- Hypotenuse
- the longest side of a right-angled triangle, opposite the right angle
Triangles without a right angleHigher tier
All three formulae are on the formula sheet, so the marks are for labelling and choosing rather than recalling.
The sine rule needs a matching pair: a side and the angle opposite it. If you have that pair plus one more fact, use it. Write it with the unknown on top when finding a side, and inverted when finding an angle, to avoid rearranging under pressure.
The cosine rule is for the two cases the sine rule cannot handle: three sides with an angle wanted, or two sides and the angle between them with the third side wanted.
The area formula ½ ab sin C needs the angle between the two sides. Any other angle gives the wrong area.
Keep full accuracy through the working and round only at the end — rounding an intermediate value and then using it is a common source of a final answer that is close but wrong.
Common mistake
The sine rule can give an ambiguous answer, because sin θ and sin(180° − θ) are equal. When the question describes an obtuse triangle, or the diagram shows one, check whether the obtuse solution is the intended one before discarding it.
- Sine rule
- a rule for non-right-angled triangles, used when you have a side and the angle opposite it plus one more fact
- Cosine rule
- a rule for non-right-angled triangles, used when there is no matching pair of a side and its opposite angle
Before you practise
- Bearings: from north, clockwise, three figures. A back bearing differs by 180°.
- Triangle area uses the perpendicular height, not a slanted side.
- Circumference = 2πr = πd; area of a circle = πr².
- Volume of any prism = cross-sectional area × length.
- Sphere: 4⁄3 πr³ and 4πr². Cone: ⅓ πr²h and curved surface πrl, with l the slant height.
- Sector arc and area are the fraction θ⁄360 of the whole circle.
- A sector's perimeter includes the two radii as well as the arc.
- For a composite solid, decide which surfaces are actually exposed before adding areas.
- Two sides and you want the third: Pythagoras. An angle involved: SOH CAH TOA.
- Know the exact values of sin, cos and tan at 0°, 30°, 45°, 60° and 90°.
- In 3D, find a right-angled triangle inside the solid — usually a base diagonal first, then stand a triangle on it.
- Sine rule needs a matching side-and-angle pair; otherwise use the cosine rule.
- Area = ½ ab sin C needs the angle between the two sides.
- Keep full accuracy until the final rounding.