Geometry and Measurement
Geometry and Measurement
This domain tests spatial reasoning and measurement in two and three dimensions: perimeter, area, surface area, volume, Pythagoras, trigonometric ratios, the sine and cosine rules, angles, bearings, and unit conversion. Questions often embed the geometry in a context such as a ramp, roof, or container, so first identify the triangle or solid, then choose the rule that matches the information given.
Core rules and concepts
- Pythagoras applies only in right-angled triangles: (a^2+b^2=c^2), where (c) is the hypotenuse.
- Right-triangle trigonometry: (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}), (\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}), (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}).
- For non-right triangles, the cosine rule is (c^2=a^2+b^2-2ab\cos C); the sine rule is (\frac{a}{\sin A}=\frac{b}{\sin B}).
- Volume formulas: prism (V=Ah); cylinder (V=\pi r^2h); sphere (V=\frac43\pi r^3). Surface area counts every exposed face.
- Bearings are measured clockwise from north and written with three digits.
Worked example
A triangular garden has sides (a=7\text{ m}), (b=8\text{ m}), and included angle (C=60^\circ). Find the third side (c).
Use the cosine rule: (c^2=7^2+8^2-2(7)(8)\cos60^\circ) (=49+64-112(0.5)=113-56=57). So (c=\sqrt{57}\approx7.55\text{ m}).
Reasoning: the angle is between the two known sides, so the cosine rule is the direct choice. Keep the exact square root until the final rounding, and state the unit.
Traps that cost marks
- Using the sine rule when the cosine rule is required, or vice versa.
- Calculator in radian mode instead of degree mode.
- Confusing slant height with perpendicular height in cones and pyramids.
- Mixing units, for example centimetres and metres, before calculating.
- Measuring a bearing anticlockwise or omitting the leading zero.