Ocegik / Geometry & Mensuration
Geometry & Mensuration Formulas for CAT Preparation
Revise geometry & mensuration formulas from Ocegik’s formula library, with named identities, expressions and usage notes grouped by topic.
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Geometry & Mensuration: formulas by topic
Triangle PropertiesGeometry & Mensuration
Area — base & heightArea = 12 × b × h
Heron's FormulaArea = √(s(s − a)(s − b)(s − c))s = (a+b+c)/2
Area — included angleArea = 12ab·sin θ
Area — inradiusArea = r × s
Cosine Rulecos θ = a2 + b2 − c22ab
Law of SinesArea = abc4RR = circumradius
Angle Bisector TheoremBD/DC = AB/AC
Apollonius Theoremb2 + c2 = 2m2 + 12a2m = median to side a
Apollonius Theorem (isosceles case)m2 = b2 − a24
Acute triangle relationAC2 = AB2 + BC2 − 2 × BC × BD
Obtuse triangle relationAC2 = AB2 + BC2 + 2 × BC × BD
Right triangle, altitude to hypotenuseAD2 = BD × DC
Right triangle, leg-projection (AB)AB2 = BD × BC
Right triangle, leg-projection (AC)AC2 = CD × BC
PolygonsGeometry & Mensuration
Number of diagonalsn(n − 3)2
Exterior angle (regular polygon)360° / n
Interior angle (convex polygon)180° − 360°/n
Sum of exterior angles360°
Sum of interior angles(n − 2) × 180°
Circle TheoremsGeometry & Mensuration
Intersecting chordsPA × PB = PC × PDP = point where chords AB and CD meet, inside or outside the circle
Tangent–secant relationPA × PB = PT2AB extended to P, PT the tangent from P
Angle at centre vs. circumference∠AOB = 2 × ∠AXBO = centre, X = any point on the major arc
Exterior angle of cyclic quadrilateral∠BCE = ∠DABE is on side DC extended beyond C
Cyclic parallelograma parallelogram inscribed in a circle is always a rectangle/square
Inscribed square/rectangle diagonaldiagonal = diameter of the circle
Common tangent length (direct)√((distance between centres)2 − (r1 − r2)2)r₁, r₂ are the two circles' radii
Common tangent length (transverse)√((distance between centres)2 − (r1 + r2)2)
Alternate Segment Theoremangle between chord & tangent = angle in alternate segment
Ptolemy's TheoremAB × CD + BC × DA = AC × BD
2D MensurationGeometry & Mensuration
RectanglePerimeter = 2(a + b), Area = ab
SquarePerimeter = 4a, Area = a2
TrianglePerimeter = a+b+c, Area = 12bh
Right-angled trianglePerimeter = b+h+d, Area = 12bh, d = √(b2+h2)
Equilateral trianglePerimeter = 3a, Area = √34a2, h = √32a
Isosceles right-angled trianglePerimeter = 2a+d, Area = 12a2, d = a√2
ParallelogramPerimeter = 2(a+b), Area = ah
RhombusPerimeter = 4a, Area = 12d1 × d2d₁, d₂ are the two diagonals
QuadrilateralArea = 12(AC)(h1 + h2)AC = one diagonal; h₁, h₂ = altitudes to AC from the other two vertices
TrapeziumArea = 12h(a + b)
CircleCircumference = 2πr, Area = πr2
SemicirclePerimeter = πr + 2r, Area = 12πr2
Ring (shaded region)Perimeter = 2π(R+r), Area = π(R2 − r2)R = outer radius, r = inner radius
Sector of a circleArc length = (θ/360°) × 2πr, Area = (θ/360°) × πr2θ = central angle of the sector
Segment of a circleArea = r2[πθ360° − sin θ2]θ = angle subtended at the centre by the arc
Pathway across the middleA = w(l + b − w)l = length, b = breadth of the rectangle, w = path width
Pathway outsidePerimeter = 2[l+b+4w], A = 2w(l+b+2w)
Pathway insidePerimeter = 2[l+b−4w], A = 2w(l+b−2w)
Solids (3D Mensuration)Geometry & Mensuration
CuboidTSA = 2(lb+bh+hl), Volume = lbhTSA = total surface area; l, b, h = length, breadth, height
CubeTSA = 6a2, Volume = a3a = edge length
Right prismLSA = perimeter(base) × height, Volume = area(base) × heightLSA = lateral (side) surface area
Right circular cylinderLSA = 2πrh, TSA = 2πr(r+h), Volume = πr2hr = base radius, h = height
Right pyramidLSA = 12 × perimeter(base) × slant height, Volume = 13 × area(base) × height
Right circular coneLSA = πrl, TSA = πr(l+r), Volume = 13πr2h, l = √(r2+h2)l = slant height
SphereTSA = 4πr2, Volume = 43πr3r = radius
HemisphereTSA = 3πr2, Volume = 23πr3
Spherical shellTSA = 4π(R2+r2), Volume = 43π(R3−r3)R = outer radius, r = inner radius
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