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Progressions (AP, GP, HP)
Every CAT Progressions formula on one page — AP, GP, HP, AM-GM-HM, special series, AGP and telescoping sums — each with a worked example.
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Geometry & Mensuration
CAT'26 QUANT CHEATSHEET
Every geometry & mensuration formula you need for CAT 2026 — on one page.
Geometry and Mensuration on CAT rewards recognition over raw computation — the same handful of relationships (Pythagoras, similar-figure ratios, the tangent-radius right angle) reappear across triangles, circles, and coordinate-geometry questions in different disguises, and the 3D solids all reduce to a small set of volume and surface-area formulas once you know the slant height or the right radius to use. This sheet lays out every formula you need for the CAT quant section: triangle and circle basics, similar figures, quadrilaterals and polygons, coordinate geometry, and the volume and surface area of cubes, cylinders, cones and spheres, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Geometry & Mensuration: every formula you need
1Triangle Basics
The three interior angles of any triangle always sum to 180°.
sum of angles = 180° | exterior angle = sum of the two remote interior angles
Example: angles 50° and 60° given → third angle = 180−110 = 70°.
CAT Hack: Triangle inequality: the sum of any two sides must exceed the third — a quick check for valid triangles.
2Pythagorean Theorem
Relates the two legs and hypotenuse of a right triangle.
a2 + b2 = c2
Example: legs 6 and 8 → hypotenuse = √(36+64) = √100 = 10.
CAT Favourite: Memorise the common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 — and their multiples.
3Area of a Triangle
Three interchangeable ways to compute the same area.
½×base×height | Heron's: √[s(s−a)(s−b)(s−c)], s=(a+b+c)/2
Example: sides 3, 4, 5 → s=6 → √(6·3·2·1) = √36 = 6.
CAT Hack: Reach for Heron's formula only when no height is given directly — base×height is faster whenever you have it.
4Similar Triangles
Matching angles and proportional sides scale the area differently.
area ratio = (side ratio)²
Example: similar triangles with sides in ratio 2:3 → area ratio = 4:9, not 2:3.
Common Mistake: The area ratio is the square of the side ratio, not the side ratio itself.
5Circle: Chords & Tangents
Key relationships between a circle, its chords and tangent lines.
perpendicular from centre bisects a chord | tangent length = √(d2−r2)
Example: radius 5, external point at distance 13 from centre → tangent = √(169−25) = 12.
CAT Hack: A tangent is always perpendicular to the radius at the point of contact — this creates a right triangle to work with.
6Circle: Area & Circumference
The two basic circle measurements, both driven by the radius.
Area = πr2 | Circumference = 2πr
Example: r=7 → area = 22/7·49 = 154; circumference = 2·22/7·7 = 44.
7Sector & Segment
A slice of a circle, and the region cut off by a chord.
Sector area = (θ/360)·πr2 | Arc length = (θ/360)·2πr
Example: r=6, θ=60° → sector area = (1/6)·π·36 = 6π ≈ 18.85.
CAT Insight: A circular segment's area is the sector area minus the triangle formed by the two radii and the chord.
8Quadrilaterals
Area formulas for the two most common four-sided shapes.
Parallelogram: base×height | Trapezium: ½(a+b)×height
Example: trapezium with parallel sides 8 and 12, height 5 → ½·20·5 = 50.
9Polygon Angles
How the interior and exterior angles scale with the number of sides.
sum of interior angles = (n−2)×180° | each exterior angle (regular) = 360°/n
Example: a regular hexagon (n=6) → interior sum = 4·180 = 720° → each angle = 120°.
Common Mistake: The exterior angles of any convex polygon always sum to 360°, regardless of how many sides it has.
10Coordinate Geometry: Distance & Midpoint
The straight-line distance and the midpoint between two points.
Distance = √[(x2−x1)2+(y2−y1)2]
Example: (1,2) and (4,6) → √(9+16) = √25 = 5.
11Coordinate Geometry: Slope
How steep a line is, from any two points on it.
slope = (y2−y1) / (x2−x1)
Example: line through (1,2) and (4,8) → (8−2)/(4−1) = 6/3 = 2.
CAT Hack: Parallel lines share the same slope; perpendicular lines have slopes whose product is −1.
12Cube & Cuboid
Volume and surface area of the two simplest rectangular solids.
Cube: V=a3, SA=6a2 | Cuboid: V=lbh, SA=2(lb+bh+hl)
Example: cube of side 4 → volume = 64, surface area = 96.
13Cylinder
Volume and surface area of a circular prism.
V = πr2h | CSA = 2πrh | TSA = 2πr(r+h)
Example: r=7, h=10 → volume = 22/7·49·10 = 1540; CSA = 440.
Common Mistake: Curved surface area (2πrh) and total surface area (which adds the two circular ends) are not the same thing.
14Cone
Volume and surface area of a circular pyramid.
V = ⅓πr2h | slant l = √(r2+h2) | CSA = πrl
Example: r=3, h=4 → l=5, volume = ⅓·22/7·9·4 = ≈37.7.
CAT Insight: The slant height is not the same as the vertical height — find it first with Pythagoras before using it in CSA or TSA.
15Sphere & Hemisphere
Volume and surface area of a ball and half a ball.
Sphere: V=⃀πr3, SA=4πr2 | Hemisphere: V=⅔πr3, TSA=3πr2
Example: sphere of radius 3 → volume = ⃀·π·27 = 36π ≈ 113.1.
CAT Favourite: A hemisphere's total surface area is 3πr² — the curved half (2πr²) plus its flat circular base (πr²), a frequently tested combination.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples)
- Similar figures: area ratio = (side ratio)², volume ratio = (side ratio)³
- A tangent is always perpendicular to the radius at the point of contact
- Parallel slopes are equal; perpendicular slopes multiply to −1
- Find the cone's slant height (Pythagoras) before computing CSA or TSA
- Hemisphere total surface area = 3πr², not 2πr²
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Most Common CAT Traps
- Confusing curved surface area with total surface area for a cylinder or cone.
- Using the side ratio directly as the area ratio for similar figures, instead of squaring it.
- Mixing up radius and diameter in area or circumference formulas.
- Forgetting that exterior angles of any convex polygon always sum to 360°.
- Treating the slant height of a cone as if it were the vertical height.
- Skipping the segment = sector − triangle relationship and computing it from scratch.
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30-Second Revision Box
- Triangle angle sum = 180°; polygon interior sum = (n−2)×180°
- Circle: Area=πr2; Circumference=2πr; Sector=(θ/360)πr2
- Distance = √[(x2−x1)2+(y2−y1)2]; slope=(y2−y1)/(x2−x1)
- Cylinder: V=πr2h; Cone: V=⅓πr2h; Sphere: V=⃀πr3
- Cone slant height l=√(r2+h2); Hemisphere TSA=3πr2
- Similar figures: side ratio k ⇒ area ratio k², volume ratio k³
This topic rewards spotting the right relationship over grinding through algebra — once you see a right angle at a tangent, a squared area ratio in similar figures, or a slant height waiting to be found with Pythagoras, the answer is usually one substitution away. Drill this sheet until the standard triples and the 3D formulas are reflex, then test them on full sets and track progress with the CAT score predictor. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
