Geometry & Mensuration Tricks for RRB NTPC, Group D & Section Controller

Geometry & Mensuration Tricks

Triangle Formulas & Tricks

Basic Triangle Formula

A triangle has 3 sides, 3 angles and 3 vertices.

Sum of Angles of a Triangle

The sum of all three interior angles is always: A + B + C = 180°

Quick Trick

If two angles are given: Third angle = 180° − Sum of the two given angles. Example If two angles are 65° and 45°: Third angle: 180° − (65° + 45°) = 70°

Area of a Triangle

The most basic and frequently used formula is: Area = ½ × Base × Height or Area = (Base × Height) / 2. Example: Base = 20 cm, Height = 12 cm. Area: = ½ × 20 × 12 = 120 cm²

Quick Trick

If base and height are given, simply: Base × Height ÷ 2

Heron's Formula

This is very important when all three sides of a triangle are given but height is not given. Suppose the sides are a, b and c. First find the semi-perimeter: s = (a + b + c) / 2
Then: Area = √[s(s − a)(s − b)(s − c)]

Example

Sides = 13 cm, 14 cm and 15 cm.
First: s = (13 + 14 + 15) / 2 = 21 cm
Now: Area = √[21 × 8 × 7 × 6]. Area = 84 cm²

Equilateral Triangle

In an equilateral triangle, all three sides are equal.

If each side = a:

Perimeter = 3a
Area = (√3/4)a²
Height = (√3/2)a
Inradius: r = a√3/6
Circumradius: R = a√3/3

Important Trick

Every angle of an equilateral triangle is: 60°. If the perimeter is given: Side = Perimeter ÷ 3

Example

Perimeter = 36 cm
Side: 36 ÷ 3 = 12 cm
Area: = (√3/4) × 12² = 36√3 cm²

Isosceles Triangle

In an isosceles triangle, two sides are equal. Suppose the equal sides are a and a, and the base is b. The height can be found using Pythagoras: Height = √[a² − (b/2)²]

Example: Equal sides = 10 cm, Base = 12 cm

Half of base: 12 ÷ 2 = 6 cm
Height: √(10² − 6²) = √(100 − 36) = √64 = 8 cm
Now area: = ½ × 12 × 8 = 48 cm²

Right-Angled Triangle

A triangle containing a 90° angle is called a right-angled triangle. The side opposite the 90° angle is called the hypotenuse.

Pythagoras Theorem

Hypotenuse² = Perpendicular² + Base² or c² = a² + b²

Example

Sides = 6 cm and 8 cm
Hypotenuse: c = √(6² + 8²) = √100 = 10 cm

Important Pythagorean Triplets

These are extremely useful for Railway exams.

Memorise these:

Triplet	Relation
3, 4, 5	3² + 4² = 5²
5, 12, 13	5² + 12² = 13²
7, 24, 25	7² + 24² = 25²
8, 15, 17	8² + 15² = 17²
9, 12, 15	9² + 12² = 15²
12, 16, 20	12² + 16² = 20²

Super Trick

If you see numbers like 7 and 24, don't waste time calculating. Immediately recognise: 7-24-25. So the hypotenuse is 25.

Area of a Right-Angled Triangle

If the two perpendicular sides are a and b: Area = ½ × a × b

Example: Base = 9 cm, Height = 12 cm

Area: = ½ × 9 × 12 = 54 cm²

Inradius of a Triangle

The inradius is the radius of the circle that fits inside the triangle and touches all three sides. The most useful formula is: r = Area / s where: s = semi-perimeter. Example: For a triangle with: Area = 84 cm², Semi-perimeter = 21 cm. Then: r = 84 ÷ 21 = 4 cm

Quick Trick

If Area and semi-perimeter are given: Inradius = Area ÷ Semi-perimeter

Circumradius of a Triangle

The circumradius is the radius of the circle passing through all three vertices of the triangle. Formula: R = abc / 4A

where:

a, b, c = three sides
A = area of triangle

Example

For sides 13, 14 and 15: Area = 84
Therefore: R = (13 × 14 × 15) / (4 × 84) = R = 2730 / 336= R = 65/8 cm

Triangle Inequality Rule

For any triangle: Sum of any two sides must be greater than the third side.

For sides a, b and c:

a + b > c
b + c > a
c + a > b

Example

Can 3 cm, 4 cm and 8 cm form a triangle?

Check: 3 + 4 = 7
But: 7 < 8

Therefore, these sides cannot form a triangle.

Exam Trick: Check the two smallest sides first. If their sum is not greater than the largest side, the triangle is impossible.

Types of Triangle Based on Angles

Acute-Angled Triangle: All angles are: < 90°
Right-Angled Triangle: One angle is: = 90°
Obtuse-Angled Triangle: One angle is: > 90°

Types of Triangle Based on Sides

Equilateral Triangle

All three sides equal: a = a = a

Isosceles Triangle

Two sides equal: a = a ≠ b

Scalene Triangle

All three sides different: a ≠ b ≠ c

Median of a Triangle

A median joins a vertex to the midpoint of the opposite side.
Important Property: A median divides the triangle into two triangles of equal area.

Exam Trick

If a median is drawn: Area of one half = Area of the other half. This is often useful in diagram-based questions.

Angle Bisector

An angle bisector divides an angle into two equal angles.

For example: If an angle is 80°:
Each part: 80° ÷ 2 = 40°

Angle Bisector Theorem

If an angle bisector divides the opposite side into two parts: BD/DC = AB/AC. This can appear in higher-level geometry questions.

Similar Triangles

If two triangles are similar, their corresponding sides are proportional.
If: Side ratio = a : b
Then: Perimeter Ratio: a : b
Area Ratio: a² : b²

Golden Trick

If similar triangles have sides in ratio: 2 : 3 then their areas are: 4 : 9. Not 2 : 3.

Important Triangle Percentage Trick

If the base increases by x% while height remains the same: Area also increases by x%. Similarly, if height increases by x% while base remains the same: Area increases by x%.

Example

Area = 120 cm²
Base increases by 20%, height unchanged.
New area: 120 × 1.20 = 144 cm²
Increase: 24 cm² = 20%

Triangle Quick Revision Box

Concept	Formula / Trick
Angle Sum	180°
Area	½ × base × height
Semi-perimeter	(a+b+c)/2
Heron's Formula	√[s(s-a)(s-b)(s-c)]
Inradius	Area/s
Circumradius	abc/4A
Right Triangle	a²+b²=c²
Equilateral Area	√3a²/4
Equilateral Height	√3a/2
Isosceles Height	√[a²-(b/2)²]
Similar Triangle Area Ratio	Square of side ratio
Median	Divides area into two equal parts
Triangle Inequality	Sum of any 2 sides > third side

Railway Exam में Triangle के ये Tricks जरूर याद रखें

Three sides given → Heron's Formula
Base + height given → ½bh
Two perpendicular sides → Pythagoras
Isosceles triangle → base को half करके Pythagoras
Equilateral triangle → 60° each
Similar triangles → Area ratio = square of side ratio
Area + semi-perimeter → Inradius = A/s
Median → equal areas
Radius/height/base percentage change → check whether area is directly or square proportional
3-4-5, 5-12-13, 7-24-25 → instant recognition

Circle Formulas & Tricks

Before solving questions, remember these basic terms:

Radius (r): Centre से circle की boundary तक की दूरी.
Diameter (d): Circle के centre से होकर जाने वाली सबसे बड़ी chord.
Chord: Circle के दो points को join करने वाली line segment.
Circumference: Circle की boundary की total length.
Sector: Two radii और an arc से बना region.
Segment: Chord और arc के बीच का region.
Tangent: Circle को केवल एक point पर touch करने वाली line.

Most Important Relation

Diameter = 2 × Radius : d=2r

Therefore:

Geometry & Mensuration Tricks for RRB NTPC, Group D & Section Controller

Circumference of a Circle

The circumference is the boundary length of a circle.

Formula

C=2πr. Since $d = 2 r$ : C=πd

Example: Radius = 7 cm. Using:

Quick Trick

If radius is 7 and $π = 22/7$ : Circumference = 44 cm
If diameter is given directly: C=πd

Area of a Circle

Formula

Example: Radius = 7 cm

Circle Based on Diameter

अगर diameter d दिया है:

Semicircle

A semicircle is half of a circle.

Common Railway Trap

अगर question area of semicircle पूछ रहा है:

Quadrant

A quadrant is one-fourth of a circle.

Trick

Circle को 4 equal parts में divide करो: Quadrant = 1/4 Circle

Arc Length

If central angle is θ:

Quick Trick

Sector of a Circle

A sector is the region formed by two radii and an arc.

Important Sector Shortcuts

Chord of a Circle

A chord joins any two points on a circle.

Most Important Fact

Diameter is the longest chord of a circle.
Therefore: Maximum chord=Diameter=2r

Perpendicular from Centre to Chord

This is a very important Railway concept. If a perpendicular is drawn from the centre of a circle to a chord, it bisects the chord.
In simple words: Centre से chord पर perpendicular → chord दो बराबर parts में divide हो जाती है.

Example

Chord = 16 cm
Half chord: 8 cm

Chord + Centre Distance Trick

Suppose:

Chord = L
Distance from centre = x

Then half chord:

Therefore diameter: 20 cm

Railway Shortcut

Half chord + centre distance = right triangle. बस Pythagoras लगाओ.

Tangent to a Circle

A tangent touches a circle at only one point.
The radius drawn to the point of contact is perpendicular to the tangent.
Radius⊥Tangent
Therefore: ∠OTP=90∘

Tangents from an External Point

If two tangents are drawn from the same external point: PA=PB

Example

If: PA=12 cm
Then: PB=12 cm

Super Trick

Same external point → Tangents are equal.

Angle Between Radius and Tangent

At the point where tangent touches the circle: 90∘. This is frequently used in diagram-based questions.

Percentage Change in Circumference

Circumference: C=2πr
Therefore: C∝r
If radius increases by 20%, circumference also increases by: 20%

Golden Rule

Radius changes by x% → Circumference also changes by x%.

Percentage Change in Circle Area

Radius Decreases Important Trap

Suppose radius decreases by 20%.

Ratio of Circumferences

Ratio of Areas

Circle Inside a Square

If a circle is perfectly inscribed inside a square: Diameter of circle = Side of square

Square Inside a Circle

If a square is inscribed inside a circle: Diagonal of square = Diameter of circle

Ring / Annulus

If a large circle has radius R and a smaller circle has radius r, the area between them is:

Circular Path Trick

If a circular path exists around a circular garden:

Circle Questions में ये 10 Tricks जरूर याद रखें

$d = 2 r$
Circumference $= 2 π r$
Area $= π r^{2}$
Diameter is the longest chord.
Centre से chord पर perpendicular → chord bisect.
Chord question → half chord + distance → Pythagoras.
Same external point से tangents → equal.
Radius : circumference → direct ratio.
Radius : area → square ratio.
Circular path → outer area − inner area.

Quadrilateral

A quadrilateral is a closed figure with 4 sides, 4 angles and 4 vertices.

Sum of Interior Angles

The sum of all four interior angles is: 360∘

Quick Trick

If three angles are given: Fourth angle=360∘−sum of three angles

Example

Three angles are: $8 0^{\circ}, 9 0^{\circ}, 11 0^{\circ}$ .
Fourth angle: $360 - (80 + 90 + 110)$ $= 360 - 280 =$ $8 0^{\circ}$

Square

A square has:

4 equal sides
4 right angles
Equal diagonals
Diagonals bisect each other at $9 0^{\circ}$

If side = $a$ :

Square Diagonal Trick

If the diagonal is given:

Rectangle

A rectangle has:

Opposite sides equal
All angles $9 0^{\circ}$
Diagonals equal

Length = $l$
Breadth = $b$

Rectangle Diagonal Trick

Example

Length = 15 cm
Breadth = 8 cm

Rectangle vs Square Important Trap

Suppose a rectangle has: l=20,b=10
Area: 20×10=200
But if the question asks perimeter: 2(20+10)=60

Remember

Area and perimeter are completely different quantities.

Area → square units
Perimeter → linear units

Parallelogram

A parallelogram has:

Opposite sides parallel
Opposite sides equal
Opposite angles equal

Area

$A = ba se \times h e i g h t$

Perimeter

If adjacent sides are $a$ and $b$ : P=2(a+b)

Important Trick

A parallelogram's slant side is NOT necessarily its height.
For area, always use: Base × perpendicular height

Rhombus

A rhombus has:

All four sides equal
Opposite sides parallel
Diagonals perpendicular
Diagonals bisect each other

Rhombus Diagonal Trick

The diagonals of a rhombus bisect each other at 90∘.
Therefore, if half diagonals are known, they form a right triangle.
If diagonals are d1,d2, side:

Kite

A kite has two pairs of adjacent equal sides.

Trick

Kite and Rhombus → same diagonal-area formula

Trapezium

A trapezium has one pair of parallel sides.
Parallel sides = a,b
Height = h

Example

Parallel sides = 10 cm and 16 cm
Height = 8 cm

Midpoint Theorem in Trapezium

The line joining the midpoints of the non-parallel sides is called the median/midline

Polygon Basic Concept

A polygon is a closed figure made of straight line segments.

PolygonNumber of Sides

Triangle	3
Quadrilateral	4
Pentagon	5
Hexagon	6
Heptagon	7
Octagon	8
Nonagon	9
Decagon	10

Sum of Interior Angles of a Polygon

Quick Trick

Number of triangles formed from one vertex: n−2. Then multiply by $18 0^{\circ}$ .

Each Interior Angle of a Regular Polygon

Exterior Angle of a Regular Polygon

Interior Angle + Exterior Angle

At a vertex: Interior+Exterior=180∘

Example

Interior angle = $15 0^{\circ}$

Exterior angle: $180 - 150 =$ $3 0^{\circ}$

Number of sides: 360/30=12

Number of Diagonals of a Polygon

Important Diagonal Values

Polygon	Sides	Diagonals
Triangle	3	0
Quadrilateral	4	2
Pentagon	5	5
Hexagon	6	9
Heptagon	7	14
Octagon	8	20
Nonagon	9	27
Decagon	10	35

Regular Polygon Important Formula Set

Regular Polygon Super Trick

Remember: Interior angle → 180 − Interior = Exterior → 360 ÷ Exterior = Sides

Important Quadrilateral Properties

Shape	Key Property
Square	All sides equal, all angles 90°
Rectangle	Opposite sides equal, all angles 90°
Rhombus	All sides equal, diagonals perpendicular
Parallelogram	Opposite sides parallel and equal
Trapezium	One pair of parallel sides
Kite	Two pairs of adjacent equal sides

Area Formula Quick Table

Common Railway Traps

Quick Revision Box

One-Minute Revision

Cylinder Formulas, Tricks

A cylinder has:

Two circular bases
One curved surface
Radius = r
Height = h

Think of objects like a water tank, pipe, drum or cylindrical container.

Curved Surface Area (CSA)

The curved surface is the side portion of the cylinder, excluding the two circular bases.

Formula: CSA=2πrh

Total Surface Area (TSA)

A closed cylinder has:

One curved surface
Two circular bases

Volume of Cylinder

This is one of the most important formulas.

CSA vs TSA vs Volume Don't Get Confused

Golden Rule

CSA → side only
TSA → side + both circular ends
Volume → space inside

Open Cylinder

This is a very common Railway trap.

Suppose a cylindrical tank is open at the top.

Then it has:

Curved surface
Bottom

But no top.

Don't use TSA

Cylinder Open at Both Ends

If both circular ends are open: Only curved surface remains.
Area=2πrh
This is simply CSA.
Example: A cylindrical pipe without circular ends: 👉 Use CSA.

Hollow Cylinder / Pipe

A hollow cylinder has:

Outer radius = R
Inner radius = r
Height = h

Shortcut

Outer cylinder volume − Inner cylinder volume

Hollow Cylinder Curved Surface Area

If both inner and outer curved surfaces are included: $CS A = 2 πh (R + r)$ because: $2 π R h + 2 π r h$

Hollow Cylinder - Total Surface Area

For a hollow cylinder open at both ends, including both inner and outer curved surfaces plus the two ring-shaped ends:

This is a more advanced formula and can appear in higher-level questions.

Cylinder Ratio Trick

Cylinder Radius Ratio → Volume Ratio

Same Volume Cylinder Trick

Important

Radius increases → height must decrease to maintain same volume.

Water Tank Questions

Suppose water is transferred from one cylindrical tank to another.

Shortcut

Cylinder Recasting / Melting

If a solid cylinder is melted and converted into another solid:

Volume remains constant.

Therefore: V1=V2

Example

A cylinder is melted and converted into smaller cylinders.
If one original cylinder produces n identical smaller cylinders: Voriginal=n×Vsmall

Cylinder → Sphere Recasting

Suppose a cylinder is converted into a sphere.

Cylinder → Cone Recasting

Same Radius and Height - Cylinder vs Cone

If cylinder and cone have the same radius and height:

Shortcut

Same $r, h$ → Cone volume is exactly one-third of cylinder.

Percentage Change in Cylinder Volume

Radius Decrease in Cylinder

If radius decreases by 20%

Radius + Height Both Change

Suppose radius increases by 20% and height increases by 10%

Tough Question Shortcut

Never add percentages directly when radius changes because radius is squared.

Cylinder Surface Area Ratio

CSA: CSA=2πrh
Therefore: CSA∝rh
If radius doubles and height remains same: CSA=2 times
But volume becomes: 4 times

Important Difference

CSA depends on $r$
Volume depends on $r^{2}$

Cylinder with Diameter Given

Sometimes the question gives diameter instead of radius.

Cylinder Height from Volume

If volume V and radius r are given:

Cylinder Radius from Volume

Units -Very Important

Cylinder Water Capacity

If a cylindrical tank has:

Radius = r
Height = h

Capacity:

Railway Exam's Most Important Cylinder Trick

Cylinder Formula Revision Table

Cone

A cone has:

Radius = r
Height = h
Slant height = l

Quick Trick

Whenever a cone question gives radius + height and asks for slant height, immediately think: Pythagoras Theorem.

Slant Height of Cone

Curved Surface Area of Cone

Total Surface Area of Cone

A closed cone consists of:

Curved surface
Circular base

Common Trap

Don't use h in CSA.

Wrong:πrh
Correct: πrl

because surface follows the slant height.

Volume of Cone

The most important cone formula:

Cone vs Cylinder

For the same radius and same height

Shortcut

Same $r, h$ → Cone = 1/3 Cylinder
This is a very useful Railway shortcut.

Cone vs Cylinder Reverse Question

If cone volume is 500 cm³ and both have the same radius and height: $V_{cy l in d er} = 3 \times 500$ $1500 cm^{3}$

Cone Radius from Volume

Cone Height from Volume

Cone Height and Slant Height

Cone Percentage Tricks

Radius Increases by 20%

Height Increases by 20%

Radius Decreases by 20%

Sphere

A sphere has only one main measurement:Radius = $r$

$Sphere Surface Area$

Sphere Volume

Sphere Diameter Given

Useful Shortcut

Sphere surface area = $π d^{2}$

when diameter is directly given.

Sphere Radius Ratio

If radii are: $r_{1} : r_{2} = a : b$

Then:

Sphere Percentage Change

SphereRadius Doubles

Hemisphere

A hemisphere is half of a sphere.

Hemisphere Volume

Sphere volume:

Curved Surface Area of Hemisphere

Only curved part:

Total Surface Area of Hemisphere

A solid hemisphere has:

Curved surface
Circular base

Therefore:

Common Trap

Don't confuse:

Hemisphere Quick Table

Recasting/Melting

This is one of the most important tough-level Mensuration concepts.
When one solid is melted and converted into another: Volume remains constant

Sphere → Small Spheres

Suppose one large sphere is melted to make n identical small spheres.

Cone → Sphere Recasting

Cylinder → Cone Recasting

Sphere → Cylinder Recasting

Cone + Hemisphere

Sometimes a solid is made by joining a cone and hemisphere.

Cylinder + Hemisphere

For a shape consisting of a cylinder with a hemisphere attached:

Cone + Hemisphere Surface Area

If the cone and hemisphere are joined at their circular bases, the common circular base is internal and should not be counted.

External surface area:

Cone + Cylinder Combination

If a cone is placed on top of a cylinder:
Total external surface area depends on which surfaces are exposed.
The common circular surface between them is not counted.

Rule

Joined surfaces are internal → don't count them.

Water Transfer Between Different Solids

If water is transferred without loss: Volume1=Volume2

For example:

Most Common Railway Traps

Cone + Sphere + Hemisphere Formula Sheet

10 Most Important Trick

Cube

Cube की सभी edges equal होती हैं। अगर side = a

Cube Diagonal

Cube की space diagonal:

Cube Diagonal से Side

Cube Diagonal से Volume

अगर diagonal d है:

Shortcut

अगर cube की space diagonal दी हो, तो पहले side निकालो

Cube - Surface Area और Volume का Relation

Cube में Side Double होने पर क्या होगा?

यह बहुत common ratio question है।

अगर:

Cube Ratio Trick

अगर दो cubes की sides:

Example - Cube Ratio

दो cubes की sides का ratio $2 : 3$ है।

Surface area ratio:

Cuboid

Cuboid के तीन dimensions होते हैं:

Length = l
Breadth = b
Height = h

Cuboid Volume

V=lbh
Example: $l = 10$ , $b = 6$ , $h = 4$
$V = 10 \times 6 \times 4$ = 240 cm3

Cuboid Total Surface Area

Cuboid की 6 faces होती हैं।
$TS A = 2 (l b + bh + h l)$

Remember

LB + BH + HL → multiply by 2

Cuboid Lateral Surface Area

अगर सिर्फ चार vertical faces चाहिए: LSA=2h(l+b)

Cuboid Diagonal

Space diagonal:

Cuboid Diagonal Trick

पहले base diagonal:

Cube vs Cuboid

Open Cuboid / Open Box

अगर ऊपर की face नहीं है: Area=lb+2lh+2bh
यानी: Bottom + 4 side faces

Trap

Closed cuboid: 2(lb+bh+hl)
Open-top cuboid: lb+2h(l+b)

Cuboid with No Bottom and No Top

अगर सिर्फ चार side walls हैं: LSA=2h(l+b). यह room/wall type questions में useful है।

Room Questions

Room को एक cuboid मान सकते हैं।

Four Walls + Ceiling

अगर room की चार walls और ceiling paint करनी है: 2h(l+b)+lb

Shortcut

Four walls = LSA
Four walls + ceiling = LSA + floor/ceiling area

Painting a Cube

अगर cube की सभी six faces paint करनी हैं:

Cube Cut Into Smaller Cubes

यह बहुत important Railway concept है।
Suppose a cube of side a is cut into smaller cubes of side x.

Number of small cubes:

Cube Cutting - Number of Cuts

अगर एक cube को $n \times n \times n$ small cubes में divide करना है:
Number of cuts: 3(n−1)

Example

A cube is divided into: 4×4×4 small cubes.
Cuts: $3 (4 - 1)$ $= 9$

यह Section Controller/NTPC के tougher questions में useful concept है।

अगर एक large cube की all six faces painted हैं और उसे equal smaller cubes में काटा जाता है, तो small cubes में painted faces की संख्या के हिसाब से classification होता है।

3 Painted Faces

केवल 8 corner cubes: 8

Painted Cube - Total Check

Total small cubes:

Example Painted Cube

A cube is divided into: 4×4×4
small cubes after all outer faces are painted.

3 faces painted: 8

Recasting Cube

जब एक cube को melt करके दूसरे cubes/cuboids में बनाया जाता है: Volume remains constant

Example

एक cube की side 6 cm है। उसे 8 equal cubes में convert किया गया।

Cube → Cuboid Recasting

अगर cube side a को cuboid l,b,h में convert किया जाए

Water Tank - Cuboid

Cuboid tank capacity:

Water Level Change

Water Transfer - Cuboid to Cuboid

No water loss: l1b1h1=l2b2h2
अगर सिर्फ water level बदल रहा है: A1h1=A2h2

Percentage Change in Cuboid Volume

$V = l bh$

अगर length 20% बढ़े और बाकी same: 20% increase
अगर length और breadth दोनों 20% बढ़ें: 1.2×1.2=1.44
Therefore: 44% increase
अगर तीनों dimensions 20% बढ़ें: 1.23=1.728
Therefore: 72.8% increase

Dimension Doubles

अगर l,b,h तीनों double: $2 \times 2 \times 2$ 8 times volume
Surface area: 4 times

Cuboid Volume Ratio

Two cuboids: $l_{1} : l_{2} = a : b$ $b_{1} : b_{2} = c : d$ $h_{1} : h_{2} = e : f$
Then: V1:V2=ace:bdf

Example

Lengths ratio = $2 : 3$
Breadths = $3 : 4$
Heights = $4 : 5$
Volume ratio: $2 \times 3 \times 4 : 3 \times 4 \times 5 =$ $24 : 60 =$ 2:5

Cube Surface Area vs Volume

Most Important Traps

Cube & Cuboid Formula Sheet

Mixed Mensuration का Golden Rule

Question देखते ही पहले identify करो: अगर पूछा है - कितना space? 👉 Volume
अगर पूछा है - कितना paint/material लगेगा? 👉 Surface Area
अगर पूछा है - कितनी boundary है? 👉 Perimeter
अगर पूछा है - पानी transfer हुआ?👉 Volume remains same
अगर पूछा है - melted/recasted?👉 Volume remains same
अगर दो solids join हुए? 👉 Common surface को count मत करो

Master Formula Chart

Same Volume Questions

अगर दो different solids का volume equal है: V1=V2

Example

A cylinder and cone have equal volume.

Same Radius & Same Height

यह सबसे useful shortcut है।

Cylinder → Cone Water Transfer

एक cylindrical tank का water cone में transfer किया गया।

Cylinder → Hemisphere

Sphere → Cylinder

Sphere → Small Spheres

Large sphere radius = R
Small sphere radius = r

Number of small spheres:

Cube → Small Cubes

Large cube side = A
Small cube side = a

Cube → Cuboid

Recasting means:

Cylinder → Smaller Cylinders

Percentage Change - Master Rule

Always remember the power of dimension.

Quantity | Depends on

Length ∝ r
Area ∝ r2
Volume ∝ r3

Radius Increased by 10%

Radius Decreased by 10%

Radius Increased by 20%

Dimension Change in Cuboid

Cuboid: V=lbh

Suppose:

Length increases 10%
Breadth increases 20%
Height unchanged

New volume factor: $1.1 \times 1.2$ $= 1.32$

Therefore: 32% increase

All Three Dimensions Change

Length +10%
Breadth +20%
Height +30%
New volume: $1.1 \times 1.2 \times 1.3$ $= 1.716$
Increase: $1.716 - 1$ $71.6%$

Surface Area Percentage

If all dimensions of a similar solid are multiplied by k:

Length: k

Similar Solids Ratio

If corresponding lengths are: 2:3.
Then: Surface areas:4:9
Volumes:8:27

Combined Solid - Cylinder + Hemisphere

Suppose a solid consists of a cylinder with a hemisphere attached on top.

Same radius r, cylinder height h.

Cylinder + Hemisphere Surface Area

If hemisphere is attached to the top of cylinder: The common circular surface is internal.

External surface:

Cylinder curved surface
Cylinder bottom
Hemisphere curved surface

Cone + Hemisphere

If a cone and hemisphere have same radius and are joined at their circular bases:

Volume:

Open vs Closed Solid

This is a very common exam trap.

Tank Capacity Conversion

Water Level Formula

Water Transfer Ratio

If same volume of water is transferred:

Painting Cost Questions

If painting cost per square metre = ₹x:

Painting Four Wall

Flooring Cost

Floor area: lb
Cost: lb×Rate
Don't use TSA.

Cubical Room - Air Capacity

A room is a cuboid.
Therefore: Volume=lbh

If dimensions are in metres:

This gives the volume of air inside the room.

The Biggest Railway Trap: Area vs Volume

Question: How much paint? → Area
How much water? → Volume
How much material required to make? Usually → Volume
How much sheet needed to cover? → Surface area
How many smaller solids can be made? → Volume/recasting

Formula Selection Trick

Question में ये words दिखें:

Keyword	Formula
Paint	Surface Area
Polish	Surface Area
Sheet required	Surface Area
Capacity	Volume
Water	Volume
Melted	Volume
Recast	Volume
Number of small cubes	Volume
Floor	(lb)
Four walls	(2h(l+b))
Diagonal	Pythagoras
Open tank	Exposed surfaces
Joined solids	Don't count common surface

Trick

Same radius: Cone height= 3×Cylinder height

Ultimate Formula Revision

Tough Practice Set

A cylinder of radius 7 cm and height 15 cm is converted into a cone of the same radius. Find the height of the cone.

A) 30 cm
B) 35 cm
C) 45 cm
D) 50 cm

The radius of a sphere is increased by 20%. The percentage increase in its volume is:

A) 44%
B) 60%
C) 72.8%
D) 80%

A cube of side 18 cm is cut into smaller cubes of side 3 cm. Number of smaller cubes is:

A) 36
B) 108
C) 216
D) 324

A cuboid has dimensions 9 cm, 12 cm and 20 cm. Its space diagonal is:

A) 23 cm
B) 24 cm
C) 25 cm
D) 26 cm

The radius of a cylinder is increased by 10% and its height is decreased by 10%. The percentage change in volume is:

A) 8.9% increase
B) 10% increase
C) 1% decrease
D) No change

Two spheres have radii in the ratio $3 : 5$ . Their volumes are in the ratio:

A) 9:25
B) 27:125
C) 3:5
D) 125:27

A cube of side 12 cm is melted and converted into a cuboid of length 18 cm and breadth 6 cm. The height of the cuboid is:

A) 12 cm
B) 14 cm
C) 16 cm
D) 18 cm

A cube is divided into $5 \times 5 \times 5$ smaller cubes after all its faces are painted. How many smaller cubes have exactly two faces painted?

A) 24
B) 36
C) 48
D) 54

A cylindrical tank has radius 5 m and contains water to a height of 8 m. This water is transferred to a cylindrical tank of radius 4 m. The new height of water is:

A) 10.5 m
B) 12.5 m
C) 14.25 m
D) 16 m

A sphere of radius 6 cm is melted to form identical cones, each having radius 2 cm and height 6 cm. How many cones can be formed?

A) 6
B) 8
C) 10
D) 12

Mixed Geometry & Mensuration

Circle से Solid बनने वाले Questions

Trick

Question में अगर लिखा हो: "A circular sheet/base of radius..." तो तुरंत: r=base radius

Circle + Cylinder

Suppose cylinder का base radius 7 cm है।

Circle + Cone

Circle Rolled Into Cylinder

Example - Sheet Rolled Into Cylinder

Circle + Hemisphere

Cylinder + Hemisphere

Cylinder + Hemisphere - Surface Area

Triangle + Circle

Triangle का Circumradius

Right Triangle + Circle

Equilateral Triangle + Circle

Circle Inside Square

Circle Around Square

Square + Circle Area Difference

अगर circle square के अंदर perfectly fit है:

Cube + Cylinder

Cylinder Inside Cube

Cube side a.
Cylinder diameter = a
Cylinder height = a

Therefore:

Cube + Sphere

Sphere Inside Cube -Empty Space

Water Displacement

Water Level Rise

Two Solids Displaced

Hollow Solid

Hollow Cylinder

Outer radius = R
Inner radius = r
Height = h

Material volume:

Hollow Cylinder CSA

Curved surface consists of:

Outer curved surface
Inner curved surface

Pipe Questions

अगर pipe की:

Outer radius = R
Inner radius = r
Length = h

Material volume:

Recasting Hollow/Multiple Solids

अगर एक solid को melt करके दूसरे solid में बनाया:

Mixed Ratio Shortcut

Master Ratio Rule

Tough Railway-Level Mixed Questions

Ultra-Quick Revision

Railway Exam Traps

Advanced Geometry & Mensuration

Percentage Change - सबसे Important Trick

Radius में Percentage Change

Quick Percentage Table

Radius Change	Area Change	Volume Change
+10%	+21%	+33.1%
+20%	+44%	+72.8%
+25%	+56.25%	+95.3125%
+50%	+125%	+237.5%
+100%	+300%	+700%

Remember

Radius double: $A re a = 4 \times$ or $V o l u m e = 8 \times$

Radius Decrease

Radius और Height दोनों Change

Opposite Percentage Changes

Keeping Volume Constant

Sphere का Volume Constant

Recasting - Master Concept

जब कोई solid melt करके दूसरे shape में बनाया जाता है: Volume before=Volume after Surface area की जरूरत नहीं होती।

Example

Cube side = 6 cm
Cone बनाना है:
Radius = 3 cm
Height = h

Cube volume:

Multiple Recasting

एक बड़ा cube melt करके 8 identical cubes बनाए।

अगर बड़े cube की side: A
छोटे cube की side: a

Number of Smaller Solids

Water Tank - Advanced Formula

Cylindrical Tank

Rectangular Tank

Water Transfer Between Cylinders

Water + Solid Displacement

Two Different Solids

अगर दो solids डाल दिए:

Hollow Solid in Water

अगर hollow solid पानी में पूरी तरह submerged है, displacement generally उसके external displaced volume पर depend करता है, न कि केवल material volume पर—जब तक पानी अंदर भी न भर जाए।

Hollow Cylinder Advanced

Outer radius = R
Inner radius = r
Height = h

Material volume:

Thickness of Pipe

Pipe Material Ratio

Painting Cost

Four Walls

Ceiling

Four Walls + Ceiling + Floor

Painting a Cylinder

Cost of Polishing a Sphere

Area Ratio → Cost Ratio

Volume Ratio → Material Cost

Diagonal - Advanced Revision

Surface Area vs Volume - Final Decision Rule

Question पूछता है:

Capacity? → Volume
Water? → Volume
Material required to make? → Volume
Melted/recast? → Volume
Paint? → Surface Area
Polish? → Surface Area
Sheet required? → Surface Area
Flooring? → Area
Boundary? → Perimeter

Advanced Railway Practice

The radius of a cylinder is doubled while its volume remains constant. Its height becomes:

A) Half
B) One-fourth
C) Double
D) Four times

The radius of a sphere is increased by 25%. The percentage increase in its volume is:

A) 56.25%
B) 72.5%
C) 95.3125%
D) 100%

A cylindrical tank of radius 4 m contains water to a height of 12 m. This water is transferred to another cylindrical tank of radius 8 m. Find the new height.

A) 2 m
B) 3 m
C) 4 m
D) 6 m

A pipe has outer diameter 14 cm and inner diameter 10 cm. Its thickness is:

A) 1 cm
B) 2 cm
C) 3 cm
D) 4 cm

A room is 12 m long, 8 m wide and 5 m high. Find the area of its four walls.

A) 160 m²
B) 180 m²
C) 200 m²
D) 240 m²

A cube of side 10 cm is melted and converted into 8 identical cubes. The side of each small cube is:

A) 2.5 cm
B) 5 cm
C) 6 cm
D) 8 cm

A cylinder has radius 7 cm and height 10 cm. If the cost of painting its curved surface is ₹2 per cm2, find the total cost. Use $π = \frac{22}{7}$ .

A) ₹880
B) ₹1540
C) ₹1760
D) ₹2200

Two spheres have radii in ratio $2 : 5$ . Their surface areas are in ratio:

A) 2:5
B) 4:25
C) 8:125
D) 25:4

The radius of a cylinder is increased by 20% and its height is decreased by 30%. The percentage change in its volume is:

A) 8.4% decrease
B) 0.8% increase
C) 8.4% increase
D) 10% decrease

A solid sphere of radius 6 cm is melted and recast into cones of radius 3 cm and height 4 cm. How many cones are formed?

A) 6
B) 8
C) 10
D) 12

Question में पूछा गया	तुरंत सोचो
Area	(Area) formula
Boundary	Perimeter
Paint/Polish	Surface Area
Capacity	Volume
Water	Volume
Melted/Recast	Volume same
Water level rise	(V/A)
Small solids	Volume ratio
Radius change	(r^2) / (r^3)
Similar figures	(k,k^2,k^3)
Four walls	(2h(l+b))
Diagonal	Pythagoras

Which Formula

"Capacity" ➡️ Volume
"Water" ➡️ Volume
"Melted" ➡️ Volume
"Recast ➡️ Volume
"Painted"➡️ Surface Area
"Polished"➡️ Surface Area
"Sheet required" ➡️ Surface Area
"Flooring" ➡️ Area
"Boundary" ➡️ Perimeter

Geometry & Mensuration Complete Formula Sheet

Triangle

Heron's Formula

Equilateral Triangle

Right-Angled Triangle

Triangle का Inradius & Circumradius

Quadrilateral

Parallelogram

Rhombus

Trapezium

Circle Complete Formula

Semicircle

Sector

Circle में Important Ratios

Cylinder

Cylinder Special Cases

Cone

Cone Golden Trick

Sphere

Hemisphere

Cube

Side = a

Cuboid

Hollow Cylinder/Pipe

Water Tank

Water Level Rise

Water Transfer

Recasting/Melting

Percentage Change Master Table

Painitng Question

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Geometry & Mensuration Tricks for RRB NTPC, Group D & Section Controller