Free reference guide · HSC
Worked examples: HSC Maths Standard 2 Formula Sheet
Read the NESA Standard 1 and 2 formulas with units, original examples and the official reference PDF.
Board sources checked
Official PDFs
Check the syllabus and examination materials for your cohort. A formula does not decide which values belong in it: identify the quantity requested, convert units, and state the meaning of the result. This is a selection of revision examples, not a complete syllabus checklist.
Download the worked examples (PDF) →
Formula index with worked examples
Measurement · error and bounds
Formula or reference: e = precision ÷ 2. Lower bound = measurement − e. Upper bound = measurement + e.
Units and conditions: Use the same unit for the reading and precision.
Worked example: A length of 16 cm to the nearest centimetre has e = 0.5 cm. The rounding bounds are 15.5 cm and 16.5 cm.
Measurement · arc length
Formula or reference: ℓ = (θ ÷ 360°) × 2πr
Units and conditions: θ in degrees. ℓ and r in the same length unit.
Worked example: A 60° arc of radius 3 cm has length π cm.
Measurement · sector area
Formula or reference: A = (θ ÷ 360°) × πr²
Units and conditions: θ in degrees. Area in square units.
Worked example: A 90° sector of radius 4 cm has area 4π cm².
Measurement · trapezium
Formula or reference: A = h(a + b) ÷ 2
Units and conditions: a and b are parallel lengths. h is perpendicular to them.
Worked example: Parallel sides of 4 m and 6 m and height 3 m give 15 m².
Measurement · trapezoidal approximation
Formula or reference: A ≈ h(d₁ + d₂) ÷ 2
Units and conditions: h is the spacing of adjacent offsets. The offsets and spacing use the same length unit.
Worked example: Offsets of 5 m and 7 m separated by 4 m approximate one strip as 24 m².
Measurement · cylinder surface
Formula or reference: A = 2πr² + 2πrh
Units and conditions: Square units. Includes both circular ends.
Worked example: A closed cylinder of radius 2 cm and height 5 cm has area 28π cm².
Measurement · sphere surface
Formula or reference: A = 4πr²
Units and conditions: Square units.
Worked example: A sphere of radius 3 cm has surface area 36π cm².
Measurement · pyramid or cone volume
Formula or reference: V = Ah ÷ 3
Units and conditions: A is base area. h is perpendicular height. Volume uses cubic units.
Worked example: Base area 18 cm² and height 5 cm give volume 30 cm³.
Measurement · sphere volume
Formula or reference: V = 4πr³ ÷ 3
Units and conditions: Cubic units.
Worked example: A sphere of radius 3 cm has volume 36π cm³.
Trigonometry · right-triangle ratios
Formula or reference: sin A = opposite ÷ hypotenuse. cos A = adjacent ÷ hypotenuse. tan A = opposite ÷ adjacent.
Units and conditions: Ratios are unitless. Choose degree mode when angles are given in degrees.
Worked example: For sides 3, 4 and 5 and A opposite the side of length 3, sin A = 3/5, cos A = 4/5 and tan A = 3/4.
Trigonometry · triangle area
Formula or reference: A = ab sin C ÷ 2
Units and conditions: C is the included angle. Use square units for area.
Worked example: Sides 6 cm and 4 cm enclosing 30° give 6 cm².
Trigonometry · sine rule
Formula or reference: a/sin A = b/sin B = c/sin C
Units and conditions: Sides in the same unit. Each side is opposite its corresponding angle.
Worked example: If a = 10 cm opposite 30° and B = 45°, then b = 10√2 cm.
Trigonometry · cosine rule
Formula or reference: c² = a² + b² − 2ab cos C. cos C = (a² + b² − c²) ÷ (2ab).
Units and conditions: Sides in the same unit. C is opposite c.
Worked example: With a = 3 cm, b = 4 cm and C = 90°, c = 5 cm. Reversing the formula gives cos C = 0.
Finance · compound growth
Formula or reference: FV = PV(1 + r)ⁿ
Units and conditions: PV and FV are monetary amounts. r is a decimal rate per period and n counts those periods.
Worked example: $800 at 5% per year for two years becomes $882.
Finance · straight-line depreciation
Formula or reference: S = V₀ − Dn
Units and conditions: D is a monetary decrease per period. n counts those periods.
Worked example: A $1,000 item losing $120 each year is worth $640 after three years under this model.
Finance · declining-balance depreciation
Formula or reference: S = V₀(1 − r)ⁿ
Units and conditions: r is the decimal rate per period. S and V₀ use the same monetary unit.
Worked example: A $1,000 item losing 20% of its value each year is worth $640 after two years.
Statistics · outlier fences
Formula or reference: Lower = Q₁ − 1.5 × IQR. Upper = Q₃ + 1.5 × IQR.
Units and conditions: Fences have the data’s units.
Worked example: With Q₁ = 10, Q₃ = 14 and IQR = 4, the fences are 4 and 20. A value of 21 is beyond the upper fence.
Statistics · standard score
Formula or reference: z = (x − μ) ÷ σ
Units and conditions: z is unitless. x, μ and σ use the same units.
Worked example: A score of 72 from a distribution with mean 60 and standard deviation 8 gives z = 1.5.
Statistics · normal distribution bands
Formula or reference: About 68%, 95% and 99.7% lie within 1, 2 and 3 standard deviations of the mean, respectively.
Units and conditions: Percentages are approximate and assume a normal distribution.
Worked example: With mean 50 and standard deviation 4, about 95% of normal observations are between 42 and 58.
More worked examples
Use the Mathematics Standard 1 and 2 reference sheet for Standard 2. NESA links it from the Mathematics Standard syllabus page. These original examples explain how to select and use relationships; they are not a replacement for the official sheet.
Compound growth
Relationship: A = P(1 + r)^n, with r as a decimal rate per compounding period.
An original practice scenario: $2,000 grows at 4% annually for three years without deposits or fees. Find its value.
Show worked response and error check
P = 2000, r = 0.04 and n = 3. A = 2000 × 1.04^3 = 2249.728, so the final balance is $2,249.73. Interest earned is $249.73, which is different from the balance.
Check: Do not use r = 4 or mix a monthly rate with a count of years. This simplified exercise is not an investment forecast.
A non-right-angled triangle
Relationship: Area = ½ab sin C when C is the angle between sides a and b.
Two sides are 8 cm and 5 cm with an included angle of 30°. Find the triangle's area.
Show worked response and error check
Area = ½ × 8 × 5 × sin 30° = 10 cm². The two side lengths and the included angle are enough; no third side is required.
Check: Check degree mode and use the angle between the stated sides. Area needs square units.
Standardised scores
Relationship: z = (x − mean) / standard deviation.
A value of 74 comes from a distribution with mean 62 and standard deviation 8. Find and interpret its z-score.
Show worked response and error check
z = (74 − 62) / 8 = 1.5. The value is 1.5 standard deviations above the mean. No normality assumption is required to calculate this standardised score.
Check: A z-score has no units. It is not a percentage or an ATAR; normal probabilities need an appropriate distribution assumption.
What to work out beyond the sheet
The sheet gives the outlier fences. Work out the interquartile range first using IQR = Q₃ − Q₁.
Subject resources · Official past papers · ATAR calculator guide