Practice Maths

L50 — Geometry, Measurement and Trigonometry Review

Key Terms

Sine rule
a/sin A = b/sin B = c/sin C — use when two angles and one side are known, or two sides and a non-included angle.
Cosine rule
c² = a² + b² − 2ab cos C — use for two sides + included angle, or all three sides known.
Bearing
A direction measured clockwise from North (0° to 360°); convert to standard angles when applying trig.
Angle of elevation / depression
The angle measured upward / downward from the horizontal to a line of sight.
Surface area and volume
Key formulas: cylinder (V = πr²h), cone (V = ⅓πr²h), sphere (V = &frac43;πr³); composite solids are split into parts.
Circle geometry
Key theorems: angle at centre = 2 × angle at circumference; opposite angles of cyclic quadrilateral sum to 180°; tangent ⊥ radius.

Trigonometry summary

SituationToolKey formula
Right triangle — find side or angleSOH-CAH-TOAsinθ=O/H, cosθ=A/H, tanθ=O/A
Two sides, included angle → areaArea formulaA = ½ab sin C
Two angles, one side → other sidesSine rulea/sin A = b/sin B
Three sides → angle, or two sides + included angle → third sideCosine rulec²=a²+b²−2ab cos C
BearingsNorth-based clockwise angleConvert to standard angles for trig
Hot Tip: Choose sine rule when you know two angles + one side, or two sides + a non-included angle. Choose cosine rule when you know two sides + the included angle, or all three sides. If in doubt, draw the triangle and mark what's given.

Worked Example — Sine and Cosine Rules Together

In triangle ABC, AB = 8 m, BC = 6 m, angle B = 40°. Find AC and angle A.

Step 1 — Cosine rule to find AC.
AC² = AB² + BC² − 2(AB)(BC)cos B = 64 + 36 − 96cos 40° ≈ 100 − 73.53 = 26.47.
AC ≈ √26.47 ≈ 5.14 m.

Step 2 — Sine rule to find angle A.
sin A / BC = sin B / AC ⇒ sin A = 6 sin 40° / 5.14 ≈ 6 × 0.643 / 5.14 ≈ 0.750.
Angle A ≈ sin−1(0.750) ≈ 48.6°.

Trigonometry (T2T1)

The three trigonometric ratios work only in right-angled triangles. For non-right triangles, use the sine rule or cosine rule. The ambiguous case of the sine rule occurs when given two sides and a non-included angle — always check for two possible triangles.

Measurement (T2T2)

Surface area and volume formulas for the main solids:

  • Prism: SA = 2 × base area + perimeter × height. V = base area × height.
  • Pyramid: SA = base area + ½ × perimeter × slant height. V = ⅓ × base area × height.
  • Cylinder: V = πr²h. Cone: V = ⅓πr²h. Sphere: V = &frac43;πr³.

For composite solids, split into known shapes, calculate each part, then add or subtract.

Circle Geometry (T2T3)

Key theorems (all require proof in exams):

  • The angle at the centre is twice the angle at the circumference subtended by the same arc.
  • Angles in the same segment (same side of chord) are equal.
  • Opposite angles of a cyclic quadrilateral are supplementary (add to 180°).
  • The tangent to a circle is perpendicular to the radius at the point of contact.
  • Tangents from an external point are equal in length.

Worked Example 2 — Composite Solid

A cylinder of radius 4 cm and height 10 cm has a hemisphere on top (radius 4 cm). Find the total volume.

Solution

Cylinder: V = π(4)²(10) = 160π cm³.
Hemisphere: V = ½ × &frac43;π(4)³ = ½ × 256π/3 = 128π/3 cm³.
Total: V = 160π + 128π/3 = (480π+128π)/3 = 608π/3 ≈ 636.7 cm³.

See Answers ➔

  1. Right-triangle trigonometry. Fluency

    In right triangle ABC with the right angle at C, AB = 13 m and BC = 5 m.

    • (a) Find AC.
    • (b) Find sin A, cos A and tan A as exact fractions.
    • (c) Find angle A (to 1 decimal place).
    • (d) Find angle B.

  2. Sine and cosine rules. Fluency

    • (a) In triangle PQR, angle P = 50°, angle Q = 70° and PQ = 12 cm. Find QR using the sine rule.
    • (b) In triangle XYZ, XY = 7 cm, YZ = 9 cm and angle Y = 55°. Find XZ using the cosine rule.
    • (c) Find the area of triangle XYZ from (b).
    • (d) In triangle DEF, DE = 5, EF = 8, DF = 10. Find angle E.

  3. Surface area and volume. Fluency

    • (a) Find the total surface area of a cylinder with radius 3 cm and height 8 cm. (Leave answer in terms of π.)
    • (b) Find the volume of a cone with radius 6 cm and height 10 cm. (Leave in terms of π.)
    • (c) Find the surface area of a sphere with diameter 14 cm. (Round to 1 decimal place.)
    • (d) A square pyramid has base side 10 m and slant height 13 m. Find the total surface area.

  4. Circle geometry theorems. Fluency

    • (a) O is the centre of a circle. Angle AOB = 140° where A and B are on the circle. Find angle ACB where C is any other point on the major arc AB.
    • (b) PQRS is a cyclic quadrilateral. Angle P = 105° and angle Q = 80°. Find angles R and S.
    • (c) A tangent from point T touches a circle at A. The radius OA = 5 cm and OT = 13 cm. Find the length of TA.
    • (d) Two tangents from external point P touch a circle at A and B. PA = 8 cm. Find PB.

  5. Area of a triangle and applications. Understanding

    A triangular field has sides of 80 m and 60 m meeting at an angle of 72°.

    • (a) Find the area of the field.
    • (b) Find the third side of the field.
    • (c) A fence is to be built along the third side. Fencing costs $45 per metre. Find the cost.
    • (d) Find the angle between the side of 80 m and the third side.

  6. Composite solids. Understanding

    A storage tank consists of a cylinder of radius 2 m and height 5 m with a hemispherical cap on top (radius 2 m).

    • (a) Find the total volume of the tank (leave in terms of π).
    • (b) Find the total external surface area (the base is flat — include only the base circle, the cylinder sides, and the curved hemisphere).
    • (c) The tank is filled with water at a rate of 0.5 m³ per minute. How long to fill it? (Use π ≈ 3.14.)
    • (d) When full, the water level drops 0.2 m per hour. How many hours until only the cylindrical part is full?

  7. Angles of elevation and bearings. Understanding

    • (a) From a point 80 m from the base of a building, the angle of elevation to the top is 32°. Find the height of the building.
    • (b) A ship sails from port A on a bearing of 060° for 20 km to reach B. It then sails on a bearing of 150° for 15 km to reach C. Find AC.
    • (c) Find the bearing from A to C (to the nearest degree).
    • (d) A plane at 3 000 m altitude is directly above a point. An observer on the ground 5 km (horizontal) away sees it. Find the angle of elevation.

  8. Similarity and scale factors. Understanding

    Two similar cylinders have radii 4 cm and 10 cm.

    • (a) What is the scale factor of the linear dimensions?
    • (b) What is the ratio of their surface areas?
    • (c) What is the ratio of their volumes?
    • (d) The smaller cylinder has volume 64π cm³. Find the volume of the larger cylinder.

  9. Navigation problem. Problem Solving

    Two ships leave a port P at the same time. Ship A travels on bearing 040° at 30 km/h. Ship B travels on bearing 310° at 25 km/h. After 2 hours:

    • (a) Find how far each ship has travelled.
    • (b) Find the angle between the two paths.
    • (c) Use the cosine rule to find the distance between the two ships.
    • (d) Find the bearing of Ship B from Ship A.

  10. Extended: geometry and measurement. Problem Solving

    A cone has base radius r cm and slant height l cm. The vertical height h = 12 cm and the slant height l = 13 cm.

    • (a) Find the radius r.
    • (b) Find the exact volume of the cone.
    • (c) The cone is melted and recast as a sphere. Find the radius of the sphere (leave in surd/exact form).
    • (d) Compare the surface areas of the original cone and the sphere. Which is greater? (Use π ≈ 3.14.)