Exam-Ready Maths with MyCareerVerse
We help students turn formulas into marks: timed practice, error-checking, and AI-supported feedback using our Interactive Mathematics Formula Booklet. Built for the Irish Leaving Certificate and aligned with UNESCO’s AI education guidance.
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<text x='420' y='346'>A = πr²</text>
<text x='480' y='406'>c² = a² + b²</text>
<text x='450' y='466'>m = (y₂ − y₁)/(x₂ − x₁)</text>
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Parallelogram
where A = area, b = base, h = height mar a bhfuil A = achar, b = bonn, h = airde
Worked Example:
Find the area of a parallelogram with base 8cm and height 5cm.
A = bh = 8 × 5 = 40 cm²
Interactive Calculator
Trapezium
where a, b = parallel sides, h = height mar a bhfuil a, b = taobhanna comhthreomhara, h = airde
Worked Example:
Find the area of a trapezium with parallel sides 6cm and 10cm, height 4cm.
A = ½h(a + b) = ½ × 4 × (6 + 10) = ½ × 4 × 16 = 32 cm²
Interactive Calculator
Circle / Disc
Area: A = πr²
Worked Example:
Find the circumference and area of a circle with radius 3cm.
l = 2πr = 2π × 3 = 6π ≈ 18.85 cm
A = πr² = π × 3² = 9π ≈ 28.27 cm²
Interactive Calculator
Arc / Sector
l = rθ (arc length)
A = ½r²θ (sector area)
Note: tan A and sec A are not defined when cos A = 0, sin A = 0 Nóta: níl tan A agus sec A sainithe nuair a bhíonn cos A = 0, sin A = 0
Worked Example:
Find arc length and sector area for r = 5cm, θ = 1.2 radians.
l = rθ = 5 × 1.2 = 6 cm
A = ½r²θ = ½ × 5² × 1.2 = 15 cm²
Interactive Calculator
Right-angled Triangle
cos A = b/c = adjacent/hypotenuse
tan A = a/b = opposite/adjacent
a² = b² + c² - 2bc cos A (cosine rule)(riail chosamháile)
Worked Example:
In a right triangle, if opposite = 3 and adjacent = 4:
tan A = 3/4 = 0.75
A = arctan(0.75) ≈ 36.87°
hypotenuse = √(3² + 4²) = √25 = 5
Trigonometric Calculator
Triangle Area
where a, b are two sides and C is the included angle mar a bhfuil a, b dhá thaobh agus C an uillinn san áireamh
Triangle Area Calculator
Compound & Double Angle Formulas
Compound Angle
cos(A - B) = cos A cos B + sin A sin Bsin(A + B) = sin A cos B + cos A sin B
tan(A - B) = (tan A - tan B)/(1 + tan A tan B)
Double Angle
tan 2A = 2tan A/(1 - tan²A)cos 2A = 1 - tan²A/(1 + tan²A)
sin 2A = 2tan A/(1 + tan²A)
Double Angle Calculator
Pythagorean Theorem
For right-angled triangles: c = hypotenuse, a & b = other sides Do thriantáin chearta: c = hipeiteiniús, a & b = na taobhanna eile
Pythagorean Calculator
Line Equations
m = (y₂ - y₁)/(x₂ - x₁)
Equation of line (given center (h,k) and radius r): Cothromóid na líne (tugtha lárphointe (h,k) agus ga r):
(x - h)² + (y - k)² = r²
Distance from (x₁, y₁) to line ax + by + c = 0: Fad ó (x₁, y₁) go líne ax + by + c = 0:
d = |ax₁ + by₁ + c|/√(a² + b²)
Worked Example - Slope:
Find the slope of line through points (2, 3) and (6, 11).
m = (11 - 3)/(6 - 2) = 8/4 = 2
Slope Calculator
Distance Formula
Distance between two points P(x₁, y₁) and Q(x₂, y₂) Fad idir dhá phointe P(x₁, y₁) agus Q(x₂, y₂)
Worked Example:
Find distance between points (1, 2) and (4, 6).
d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5
Distance Calculator
Midpoint Formula
Midpoint between two points P(x₁, y₁) and Q(x₂, y₂) Lárphointe idir dhá phointe P(x₁, y₁) agus Q(x₂, y₂)
Midpoint Calculator
Circle Equation
Circle with center (h, k) and radius r Ciorcal le lárphointe (h, k) agus ga r
General form: Foirm ghinearálta:
x² + y² + 2gx + 2fy + c = 0
Center: (-g, -f), Radius: √(g² + f² - c) Lárphointe: (-g, -f), Ga: √(g² + f² - c)
Circle Equation Calculator
Point to Line Distance
Distance from point (x₁, y₁) to line ax + by + c = 0 Fad ó phointe (x₁, y₁) go líne ax + by + c = 0
Point to Line Distance Calculator
Point coordinates:
Line equation (ax + by + c = 0):
Quadratic Equations
ax² + bx + c = 0
Quadratic formula: Foirmle chearnógach:
x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Idirdhealaitheoir:
Δ = b² - 4ac
• Δ > 0: two real roots
• Δ = 0: one repeated root
• Δ < 0: no real roots • Δ > 0: dhá fhréamh fhíor
• Δ = 0: fréamh amháin arís
• Δ < 0: gan fréamhacha fíora
Worked Example:
Solve: 2x² + 5x - 3 = 0
a = 2, b = 5, c = -3
Δ = 5² - 4(2)(-3) = 25 + 24 = 49
x = (-5 ± √49)/4 = (-5 ± 7)/4
x = 0.5 or x = -3
Quadratic Equation Solver
Indices and Logarithms
Index Laws
a^m × a^n = a^(m+n)a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(mn)
a^(-n) = 1/a^n
a^0 = 1
a^(1/n) = ⁿ√a
Logarithm Laws
log_a(xy) = log_a(x) + log_a(y)log_a(x/y) = log_a(x) - log_a(y)
log_a(x^n) = n log_a(x)
log_a(a) = 1
log_a(1) = 0
a^(log_a(x)) = x
Logarithm Calculator
Sequences and Series
Arithmetic Sequence
T_n = a + (n-1)dS_n = n/2[2a + (n-1)d]
where a = first term, d = common difference mar a bhfuil a = an chéad téarma, d = comhdhifríocht
Geometric Sequence
T_n = ar^(n-1)S_n = a(r^n - 1)/(r - 1)
where a = first term, r = common ratio mar a bhfuil a = an chéad téarma, r = comhchóimheas
Sequence Calculator
Binomial Theorem
where (n choose r) = n!/(r!(n-r)!) mar a bhfuil (n choose r) = n!/(r!(n-r)!)
Binomial coefficient: Comhéifeacht déthéarmach:
C(n,r) = n!/(r!(n-r)!)
Binomial Coefficient Calculator
Algebraic Identities
Basic Identities
(a + b)² = a² + 2ab + b²(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
Cubic Identities
a³ + b³ = (a + b)(a² - ab + b²)a³ - b³ = (a - b)(a² + ab + b²)
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
Identity Expansion Calculator
Derivatives
Product Rule
d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)Derivative of product Díorthach an iolraithe
Quotient Rule
d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²Derivative of quotient Díorthach an chánóis
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)Derivative of composite function Díorthach feidhme comhdháite
Worked Example - Product Rule:
Find d/dx[x²sin(x)]
f(x) = x², f'(x) = 2x
g(x) = sin(x), g'(x) = cos(x)
d/dx[x²sin(x)] = 2x·sin(x) + x²·cos(x)
Derivative Practice Calculator
Integration
Integration by Parts
∫u dv = uv - ∫v duwhere u and dv are chosen parts of the integrand mar a roghnaítear u agus dv mar chodanna den imeascthach
Basic Integrals
∫x^n dx = x^(n+1)/(n+1) + C∫e^x dx = e^x + C
∫1/x dx = ln|x| + C
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
Substitution
∫f(g(x))g'(x) dx = ∫f(u) duwhere u = g(x), du = g'(x)dx mar a bhfuil u = g(x), du = g'(x)dx
Definite Integration Calculator
Limits
f'(x) = lim[h→0] [f(x+h) - f(x)]/h
L'Hôpital's Rule: Riail L'Hôpital:
If lim f(x)/g(x) gives 0/0 or ∞/∞, then: Má thugann lim f(x)/g(x) 0/0 nó ∞/∞, ansin:
lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)
Simple Limits Calculator
Applications of Calculus
Area Under Curve
A = ∫[a to b] f(x) dxArea between curve and x-axis Achar idir cuar agus x-ais
Rate of Change
Rate = dy/dxInstantaneous rate of change Ráta athraithe láithreach
Area Under Curve Calculator
Cylinder
V = πr²h (volume)
Interactive Calculator
Sphere
V = 4/3 πr³ (volume)
Interactive Calculator
Explore All Geometry Topics
Formulas, worked examples, and interactive calculators in one place.
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