Formula sheet
Every formula in Math Lab, grouped by topic. Pick one to work it out with your own numbers, a step at a time.
Numbers
- Multiply in two partsGrades 3–5
a × (b + c) = a × b + a × c
Work one out - Dividing on paperGrades 4–5
Number = divisor × answer + remainder
Work one out - Which part firstGrades 5–6
Parentheses, powers, × and ÷, then + and −
Work one out - PowersGrades 6–8
aⁿ = a × a × … × a (n of them)
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Fractions & decimals
- Equal fractionsGrades 3–4
a/b = (a × k)/(b × k)
Work one out - Adding fractionsGrades 5–6
a/b + c/d = (ad + bc)/bd
Work one out - Multiplying fractionsGrades 5–6
a/b × c/d = ac/bd
Work one out - A fraction of a numberGrades 4–5
a/b of N = (N ÷ b) × a
Work one out - Dividing by a fractionGrades 6–7
W ÷ a/d = W × d/a
Work one out - Multiplying a decimalGrades 5–6
0.1 × k = k tenths
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Ratios & percent
- Price for oneGrades 6–7
Price of one = total ÷ how many
Work one out - A percent of a numberGrades 6–7
p% of N = p ÷ 100 × N
Work one out - Sharing in a ratioGrades 6–7
Share = total ÷ (a + b) × a
Work one out - Percent up or downGrades 7–8
New = old × (1 ± p ÷ 100)
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Algebra & graphs
- Reading a straight lineGrades 8–9
y = mx + b
Work one out - Where a curve turnsGrades 9–11
y = x² − bx + c turns at x = b ÷ 2
Work one out - Solving ax + b = cGrades 7–8
ax + b = c → x = (c − b) ÷ a
Work one out - Solving a(x + b) = cGrades 7–8
a(x + b) = c → x = c ÷ a − b
Work one out - Two equations at onceGrades 8–9
x = (s + d) ÷ 2, y = (s − d) ÷ 2
Work one out - Slope from two pointsGrades 8–9
m = (y₂ − y₁) ÷ (x₂ − x₁)
Work one out - Factoring a quadraticGrades 9–10
x² − (p + q)x + pq = (x − p)(x − q)
Work one out - The quadratic formulaGrades 9–11
x = (−b ± √(b² − 4ac)) ÷ 2a
Work one out - Growing by a percentGrades 9–11
A = P(1 + r ÷ 100)ᵗ
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Algebra identities
- (a + b)²Grades 8–10
(a + b)² = a² + 2ab + b²
Work one out - (a − b)²Grades 8–10
(a − b)² = a² − 2ab + b²
Work one out - (a + b)(a − b)Grades 8–10
(a + b)(a − b) = a² − b²
Work one out - (a + b)³Grades 9–11
(a + b)³ = a³ + 3a²b + 3ab² + b³
Work one out - (a − b)³Grades 9–11
(a − b)³ = a³ − 3a²b + 3ab² − b³
Work one out - a³ + b³Grades 9–11
a³ + b³ = (a + b)(a² − ab + b²)
Work one out - a³ − b³Grades 9–11
a³ − b³ = (a − b)(a² + ab + b²)
Work one out - (a + b + c)²Grades 9–11
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Work one out - a³ + b³ + c³ − 3abcGrades 10–12
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
Work one out - a⁴ − b⁴Grades 9–11
a⁴ − b⁴ = (a² − b²)(a² + b²)
Work one out - (a + b)² and (a − b)² togetherGrades 9–11
(a + b)² + (a − b)² = 2(a² + b²), (a + b)² − (a − b)² = 4ab
Work one out - x + 1/x = mGrades 10–12
If x ± 1/x = m, then x² + 1/x² = m² ∓ 2
Work one out - Sum and product of rootsGrades 9–11
α + β = −b ÷ a, αβ = c ÷ a
Work one out - (a + b + c)(a + b − c)Grades 9–11
(a + b + c)(a + b − c) = (a + b)² − c²
Work one out
Exponents & logs
- Laws of powersGrades 8–10
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
Work one out - LogarithmsGrades 10–12
log(x × y) = log x + log y (same base)
Work one out - Very big and very small numbersGrades 8–9
number = a × 10ⁿ, 1 ≤ a < 10
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Triangles & shapes
- Area and perimeterGrades 3–5
Area = l × w, Perimeter = 2(l + w)
Work one out - The long side of a right triangleGrades 8–10
a² + b² = c²
Work one out - Area of a triangleGrades 6–7
A = ½ × base × height
Work one out - The missing angleGrades 7–8
A + B + C = 180°
Work one out - Scaled-up trianglesGrades 8–10
A ÷ a = B ÷ b = scale factor
Work one out - Area of a parallelogramGrades 6–7
Area = base × height
Work one out - Area of a trapezoidGrades 6–7
A = (a + b) ÷ 2 × h
Work one out - Area of a rhombusGrades 7–8
A = (d₁ × d₂) ÷ 2
Work one out - Angles of a polygonGrades 7–9
Sum = (n − 2) × 180°; each = (n − 2) × 180° ÷ n
Work one out - The outside angle of a triangleGrades 7–8
outside angle = A + B
Work one out - Isosceles triangle anglesGrades 7–8
base angle = (180° − top angle) ÷ 2
Work one out - Area from three sidesGrades 9–11
s = (a + b + c) ÷ 2, A = √(s(s − a)(s − b)(s − c))
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Circles
- Around and across a circleGrades 7–8
d = 2r, C = πd, A = πr²
Work one out - Arc lengthGrades 9–11
arc length = (θ ÷ 360) × 2πr
Work one out - Area of a sliceGrades 9–11
Sector area = θ/360 × πr²
Work one out - A chord's lengthGrades 9–11
chord = 2√(r² − d²)
Work one out - Angles in a circleGrades 9–11
angle at center = 2 × angle at edge
Work one out - A tangent from a pointGrades 9–11
t = √(D² − r²)
Work one out - A circle on a gridGrades 10–12
(x − h)² + (y − k)² = r²
Work one out
Solids
- Volume of a canGrades 8–9
V = πr²h
Work one out - A cubeGrades 6–8
Surface = 6s² Volume = s³
Work one out - Surface of a boxGrades 7–8
Surface = 2(lw + lh + wh)
Work one out - Volume of a coneGrades 8–10
V = ⅓πr²h
Work one out - A ballGrades 8–10
Surface = 4πr², volume = ⁴⁄₃πr³
Work one out - Surface of a canGrades 8–10
Surface = 2πr² + 2πrh
Work one out - Volume of a pyramidGrades 8–10
V = ⅓ × base area × h
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Coordinate geometry
- Distance between two pointsGrades 8–10
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Work one out - The middle of two pointsGrades 8–9
M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Work one out - The line through two pointsGrades 8–10
y = mx + c
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Statistics
- The meanGrades 6–7
Mean = sum ÷ how many
Work one out - Chances from a bagGrades 7–8
P(event) = ways it happens ÷ all outcomes
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Trig & calculus
- A side from an angleGrades 10–12
sin θ = opposite ÷ hypotenuse
Work one out - Slope of a curve at a pointGrades 11–12
d/dx (cxⁿ) = ncxⁿ⁻¹
Work one out - Area under a lineGrades 11–12
Area = (start height + end height) ÷ 2 × width
Work one out - sin² + cos² = 1Grades 10–12
sin²θ + cos²θ = 1
Work one out - Exact trig valuesGrades 10–12
Sides 1 : √3 : 2 (30-60-90) and 1 : 1 : √2 (45-45-90)
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Vectors & motion
- Adding two arrowsGrades 11–12
(a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂)
Work one out - A thrown ballGrades 11–12
Range = v cos θ × (2v sin θ ÷ g)
Work one out