DSE Maths formulas to learn, topic by topic

Q: Does DSE Maths give a formula sheet?

No formula sheet is listed for the Compulsory Part. The M2 paper provides trigonometric formulas, and the M1 paper prints statistical tables where appropriate.

Q: Which standard deviation formula should I use in the Compulsory Part?

The population formula, which divides by n. The sample formula, which divides by n − 1, is used in M1.

Q: Do I need Heron's formula?

It helps whenever you know all three sides of a triangle and need its area. You can also find one angle with the cosine formula and then use ½ab sin C.

Q: Can I store formulas in my calculator?

Check HKEAA's current rules on approved calculators before relying on stored programs or notes, and learn the formulas anyway.

Sprout Classroom・19 October 2026

The formulas you need for HKDSE Mathematics Compulsory Part, grouped by topic: quadratics, straight lines, sequences, mensuration, trigonometry and statistics.

There is no DSE Maths formula sheet for the Compulsory Part papers, so you need to know the formulas yourself. This page groups them by topic, from the quadratic formula to standard deviation, and shows which ones you can rebuild quickly instead of memorising.

Key points

The Compulsory Part papers give no formula sheet.
M1 prints statistical tables where needed; M2 gives trigonometric formulas.
Many formulas can be rebuilt from a simpler idea, such as a sector being a fraction of a circle.

Is there a formula sheet?

Not for the Compulsory Part. HKEAA's assessment framework allows calculators and drawing instruments in Paper 1 and Paper 2 but lists no formula sheet. The M2 paper gives trigonometric formulas, and the M1 paper prints statistical tables where needed.

Quadratics, indices and logarithms

For ax² + bx + c = 0 with a ≠ 0, the quadratic formula gives x = (−b ± √(b² − 4ac)) / 2a. The discriminant Δ = b² − 4ac tells you the number of real roots: two if Δ > 0, one (a double root) if Δ = 0, and none if Δ < 0.

If the roots are α and β, then α + β = −b/a and αβ = c/a. Questions often ask for α² + β², which is (α + β)² − 2αβ.

aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
log(MN) = log M + log N and log(M/N) = log M − log N.
log Mᵏ = k log M, and change of base: logₐ M = log M / log a.

Straight lines and circles

For points A(x₁, y₁) and B(x₂, y₂), the slope formula, distance formula and mid-point formula below cover most coordinate questions. Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

A circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². In the general form x² + y² + Dx + Ey + F = 0, the centre is (−D/2, −E/2) and the radius is √((D/2)² + (E/2)² − F).

Coordinate formulas

Formula	Expression
Slope of AB	(y₂ − y₁) / (x₂ − x₁)
Distance AB	√((x₂ − x₁)² + (y₂ − y₁)²)
Mid-point of AB	((x₁ + x₂)/2, (y₁ + y₂)/2)
Point dividing AB in the ratio r : s	((s x₁ + r x₂)/(r + s), (s y₁ + r y₂)/(r + s))
Point-slope form	y − y₁ = m(x − x₁)

Sequences and compound interest

Use a for the first term, d for the common difference and r for the common ratio. The sum to infinity of a geometric sequence exists only when −1 < r < 1.

Compound interest is a geometric sequence in disguise: after n periods at r% per period, the amount is P(1 + r%)ⁿ. For depreciation, replace the plus sign with a minus sign.

Sequence formulas

Item	Arithmetic	Geometric
nth term	a + (n − 1)d	arⁿ⁻¹
Sum of first n terms	n/2 × [2a + (n − 1)d]	a(rⁿ − 1)/(r − 1), r ≠ 1
Sum to infinity	Does not exist (unless every term is 0)	a/(1 − r)

Arcs, sectors and solids

An arc or sector with angle θ at the centre is θ/360 of the whole circle, so you can rebuild both formulas from the circumference 2πr and the area πr². For similar figures, areas scale with the square of the length ratio and volumes with the cube.

Mensuration formulas

Shape	Volume	Surface area
Cylinder	πr²h	Curved: 2πrh
Cone (slant height l)	⅓πr²h	Curved: πrl
Sphere	⁴⁄₃πr³	4πr²
Pyramid	⅓ × base area × height	Sum of the faces
Prism	Base area × height	Sum of the faces

Trigonometry

In any triangle ABC with sides a, b and c opposite the angles A, B and C, these formulas let you find unknown sides, angles and areas. Use the cosine formula when you know two sides and the angle between them, or all three sides.

sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.
Sine formula: a/sin A = b/sin B = c/sin C.
Cosine formula: c² = a² + b² − 2ab cos C.
Area = ½ab sin C.
Heron's formula: area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2.

Statistics, counting and probability

The Compulsory Part uses the population standard deviation, σ, which divides by n. The sample version, s, which divides by n − 1, belongs to M1.

Mean: x̄ = Σx / n.
Standard deviation: σ = √(Σ(x − x̄)² / n).
Permutations: ⁿPᵣ = n! / (n − r)!.
Combinations: ⁿCᵣ = n! / (r!(n − r)!).
P(A or B) = P(A) + P(B) − P(A and B).
If A and B are independent, P(A and B) = P(A) × P(B).

How to remember them

Learn the idea behind a formula, not just its letters. The arithmetic sum comes from pairing the first and last terms; the sector area is a fraction of the circle; the cosine formula becomes Pythagoras' theorem when C is a right angle.

Write each formula from memory once a week, then check it against this page. A formula you can rebuild in half a minute is one you will not lose under exam pressure.

Try one

A geometric sequence has first term 8 and common ratio ½. Find the sum of its first 5 terms and its sum to infinity.

Hint: Use the geometric sequence formula in the form a(1 − rⁿ)/(1 − r), which suits r below 1.

Answer: S₅ = 8(1 − (½)⁵)/(1 − ½) = 16 × 31/32 = 15.5. The sum to infinity is 8/(1 − ½) = 16.

Try the quadratic equation calculator

Frequently asked questions

Does DSE Maths give a formula sheet?