Point, Line, Ray
A point has position; a line extends both ways; a ray has one endpoint.
S4 · Topic 11 · Reasoning & Quant
Geometry & Mensuration
Notes for UPSC Principal & Vice-Principal examinations only.
01Basics
The Foundation — Shapes, Measures and Relationships
Learn the geometric language first; formulas become reliable only after the correct figure is identified.
AEssential geometric vocabulary
Angle
Two rays with a common endpoint; measured in degrees or radians.
Polygon
A closed plane figure made of straight line segments; sides meet only at vertices.
Circle
The set of plane points at a fixed distance from one centre.
Geometry versus mensuration
Geometry studies properties and relationships of figures. Mensuration calculates lengths, perimeters, areas, surface areas and volumes. A diagram may not be drawn to scale unless explicitly stated.
BAngles and line relationships
Complementary and supplementary
\[\alpha+\beta=90^\circ\quad\text{(complementary)}\]\[\alpha+\beta=180^\circ\quad\text{(supplementary)}\]
Linear pair and vertically opposite
- Angles in a linear pair sum to \(180^\circ\).
- Vertically opposite angles are equal.
- Angles around a point sum to \(360^\circ\).
Parallel lines and transversal
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Co-interior angles are supplementary.
CTriangles: the central framework
Universal triangle facts\[A+B+C=180^\circ\]\[\text{Exterior angle}=\text{sum of two remote interior angles}\]
- Triangle inequality: the sum of any two sides is greater than the third.
- The larger side lies opposite the larger angle.
- Area uses the perpendicular height, not a sloping side.
By sides
- Scalene: all sides unequal.
- Isosceles: two equal sides; base angles equal.
- Equilateral: all sides equal; every angle \(60^\circ\).
By angles
- Acute-angled
- Right-angled
- Obtuse-angled
\[a^2+b^2=c^2\]
Pythagoras applies only to a right triangle, with \(c\) as hypotenuse.
Area formulas
\[\Delta=\frac12bh\]\[s=\frac{a+b+c}{2},\quad \Delta=\sqrt{s(s-a)(s-b)(s-c)}\]
The second expression is Heron’s formula.
DCongruence and similarity
| Concept | Valid tests | Result | Not sufficient |
|---|---|---|---|
| Congruence | SSS, SAS, ASA, AAS, RHS | Same shape and same size; corresponding sides and angles equal. | AAA gives similarity only; SSA is generally ambiguous. |
| Similarity | AA, SAS proportionality, SSS proportionality | Same shape; corresponding sides proportional and angles equal. | Equal area alone does not prove similarity. |
| Similar perimeter ratio | Corresponding side ratio \(k\) | \(P_1:P_2=k\) | Do not square the perimeter ratio. |
| Similar area ratio | Corresponding side ratio \(k\) | \(A_1:A_2=k^2\) | Do not use \(k\) directly. |
| Similar volume ratio | Corresponding linear ratio \(k\) | \(V_1:V_2=k^3\) | Applies to similar solids. |
EQuadrilaterals and polygons
Rectangle
- Opposite sides equal and parallel.
- All angles \(90^\circ\).
- Diagonals equal and bisect each other.
Rhombus
- All sides equal.
- Opposite angles equal.
- Diagonals bisect at right angles.
Trapezium
- One pair of opposite sides parallel.
- Area uses average of parallel sides times height.
Parallelogram
- Opposite sides and angles equal.
- Diagonals bisect each other.
- Adjacent angles supplementary.
Kite
- Two pairs of adjacent equal sides.
- Diagonals perpendicular.
- One diagonal bisects the other.
All quadrilaterals
\[\text{Interior angle sum}=360^\circ\]
A square is simultaneously a rectangle, rhombus and parallelogram; the reverse statements are not always true.
Interior-angle sum
\[S=(n-2)180^\circ\]
Regular polygon
\[\text{Each interior angle}=\frac{(n-2)180^\circ}{n}\]\[\text{Each exterior angle}=\frac{360^\circ}{n}\]
Number of diagonals
\[D=\frac{n(n-3)}2\]
FCircle vocabulary and core theorems
- Diameter \(d=2r\); it is the longest chord.
- Circumference \(C=2\pi r=\pi d\).
- A tangent touches the circle at one point.
- A secant cuts the circle at two points.
- An arc is part of the circumference; a sector is bounded by two radii and an arc.
Tangent facts
- Radius is perpendicular to tangent at contact.
- Tangents from the same external point are equal.
Angle facts
- Angle at centre is twice the angle at circumference on the same arc.
- Angle in a semicircle is \(90^\circ\).
- Angles in the same segment are equal.
Cyclic quadrilateral
\[A+C=180^\circ,\qquad B+D=180^\circ\]
Its opposite angles are supplementary.
GPlane mensuration formula bank
| Figure | Perimeter / circumference | Area | Useful result |
|---|---|---|---|
| Square, side \(a\) | \(4a\) | \(a^2\) | Diagonal \(=a\sqrt2\) |
| Rectangle, \(l,b\) | \(2(l+b)\) | \(lb\) | Diagonal \(=\sqrt{l^2+b^2}\) |
| Triangle | \(a+b+c\) | \(\frac12bh\) | Heron: \(\sqrt{s(s-a)(s-b)(s-c)}\) |
| Equilateral triangle | \(3a\) | \(\frac{\sqrt3}{4}a^2\) | Height \(=\frac{\sqrt3}{2}a\) |
| Parallelogram | \(2(a+b)\) | \(bh\) | Also \(ab\sin\theta\) |
| Rhombus | \(4a\) | \(\frac12d_1d_2\) | Diagonals are perpendicular bisectors. |
| Trapezium | Sum of all sides | \(\frac12(a+b)h\) | \(a,b\) are parallel sides. |
| Kite | \(2(a+b)\) | \(\frac12d_1d_2\) | Diagonals perpendicular. |
| Circle | \(2\pi r\) | \(\pi r^2\) | Use the value of \(\pi\) stated. |
| Semicircle | \(\pi r+2r\) | \(\frac12\pi r^2\) | Perimeter includes the diameter. |
| Sector, angle \(\theta\) | Arc \(=\frac{\theta}{360^\circ}2\pi r\) | \(\frac{\theta}{360^\circ}\pi r^2\) | Sector perimeter = arc + \(2r\). |
| Annulus | Outer + inner circumferences if required | \(\pi(R^2-r^2)\) | Difference of two concentric circles. |
02Expert
The Deep Dive — Composite Figures and Solids
Advance from direct formulas to scaling, paths, hollow solids, recasting and coordinate methods.
AUnits and dimensional discipline
Length
\[1\text{ m}=100\text{ cm},\quad1\text{ km}=1000\text{ m}\]
Area
\[1\text{ m}^2=10{,}000\text{ cm}^2\]\[1\text{ hectare}=10{,}000\text{ m}^2\]
Volume
\[1\text{ m}^3=10^6\text{ cm}^3\]\[1\text{ L}=1000\text{ cm}^3\]
Capacity
\[1\text{ m}^3=1000\text{ L}\]\[1\text{ mL}=1\text{ cm}^3\]
Conversion rule
If a linear conversion factor is \(k\), the area factor is \(k^2\) and the volume factor is \(k^3\). Thus \(1\text{ m}=100\text{ cm}\), but \(1\text{ m}^2=100^2\text{ cm}^2\).
BComposite and shaded figures
Additive method
Split a complex figure into non-overlapping standard shapes.
\[A_{\text{total}}=A_1+A_2+\cdots+A_n\]
Subtractive method
Start with the outer figure and remove holes or cut-outs.
\[A_{\text{shaded}}=A_{\text{outer}}-A_{\text{unshaded}}\]
Boundary warning
For perimeter, count only exposed boundary segments. Internal dividing lines do not contribute unless requested.
Square with circular cut-out
A circle of radius \(7\) is cut from a square of side \(20\).
\[A=20^2-\pi(7)^2\]
Using \(\pi=\frac{22}{7}\), shaded area is \(400-154=246\) square units.
Four quarter-circles
Four corner quadrants of radius \(r\) together form one full circle.
\[4\left(\frac14\pi r^2\right)=\pi r^2\]
Combine repeated fractions before calculating.
CPaths, borders and frames
Path inside rectangle
For outer dimensions \(l,b\) and uniform inside width \(x\):
\[A_{\text{path}}=lb-(l-2x)(b-2x)\]
Path outside rectangle
\[A_{\text{path}}=(l+2x)(b+2x)-lb\]
Circular ring
If path width is \(x\) outside radius \(r\):
\[A=\pi[(r+x)^2-r^2]=\pi x(2r+x)\]
DThree-dimensional mensuration
| Solid | Curved / lateral surface | Total surface area | Volume |
|---|---|---|---|
| Cube, side \(a\) | \(4a^2\) | \(6a^2\) | \(a^3\) |
| Cuboid, \(l,b,h\) | \(2h(l+b)\) | \(2(lb+bh+hl)\) | \(lbh\) |
| Right prism | Perimeter of base \(\times h\) | LSA + \(2\times\) base area | Base area \(\times h\) |
| Cylinder | \(2\pi rh\) | \(2\pi r(h+r)\) | \(\pi r^2h\) |
| Cone | \(\pi rl\) | \(\pi r(l+r)\) | \(\frac13\pi r^2h\) |
| Sphere | Not separated | \(4\pi r^2\) | \(\frac43\pi r^3\) |
| Hemisphere | \(2\pi r^2\) | \(3\pi r^2\) | \(\frac23\pi r^3\) |
| Pyramid | Sum of triangular face areas | LSA + base area | \(\frac13\times\text{base area}\times h\) |
| Frustum of cone | \(\pi(R+r)l\) | \(\pi(R+r)l+\pi(R^2+r^2)\) | \(\frac13\pi h(R^2+Rr+r^2)\) |
Cone slant height
\[l=\sqrt{r^2+h^2}\]
Curved area uses \(l\); volume uses vertical height \(h\).
Cuboid space diagonal
\[d=\sqrt{l^2+b^2+h^2}\]
Frustum slant height
\[l=\sqrt{h^2+(R-r)^2}\]
EOpen, closed and hollow solids
Open cylinder
- Open at one end: \(2\pi rh+\pi r^2\).
- Open at both ends: \(2\pi rh\).
- Closed: \(2\pi rh+2\pi r^2\).
Hollow cylinder material
Outer radius \(R\), inner radius \(r\), height \(h\):
\[V_{\text{material}}=\pi h(R^2-r^2)\]
Painting rule
Paint cost depends on exposed surface area, not volume.
\[\text{Cost}=\text{area painted}\times\text{rate per unit area}\]
FRecasting, displacement and water level
Recasting principle
\[V_{\text{before}}=V_{\text{after}}\]
Number of smaller solids \(=\frac{\text{volume of original}}{\text{volume of one small solid}}\).
Immersion displacement
\[\text{Volume displaced}=\text{immersed volume}\]
For complete immersion, displaced water equals the object’s volume.
Rise in a uniform tank
\[\Delta h=\frac{V_{\text{displaced}}}{A_{\text{base}}}\]
Use consistent cubic and square units.
Sphere recast into cones
A sphere of radius \(R\) is recast into cones of radius \(r\) and height \(h\).
\[n\left(\frac13\pi r^2h\right)=\frac43\pi R^3\]
\(n=\frac{4R^3}{r^2h}\).
Water-rise model
A solid of volume \(2400\text{ cm}^3\) is immersed in a tank with base area \(600\text{ cm}^2\).
\[\Delta h=\frac{2400}{600}=4\text{ cm}\]
Water rises by \(4\) cm.
GScaling of similar figures and solids
Linear scale\(k\)
Perimeter scale\(k\)
Area scale\(k^2\)
Volume scale\(k^3\)
Dimension rises by \(x\%\)
\[A\text{ multiplier}=\left(1+\frac{x}{100}\right)^2\]
All dimensions rise by \(x\%\)
\[V\text{ multiplier}=\left(1+\frac{x}{100}\right)^3\]
Area determines side
\[\frac{L_1}{L_2}=\sqrt{\frac{A_1}{A_2}}\]
HCoordinate geometry essentials
Distance
\[AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]
Midpoint
\[M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\]
Slope
\[m=\frac{y_2-y_1}{x_2-x_1}\]
Parallel lines have equal slopes; perpendicular slopes satisfy \(m_1m_2=-1\), when defined.
Section formula
Point dividing internally in ratio \(m:n\):
\[P=\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)\]
Triangle area
\[\Delta=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|\]
Collinearity
Three points are collinear if their triangle area is zero or their pairwise slopes agree.
\[\Delta=0\]
IHigh-value triangle results
Median and centroid
The three medians meet at the centroid. It divides each median in ratio:
\[2:1\]
Measured from vertex to midpoint side.
Angle-bisector theorem
\[\frac{BD}{DC}=\frac{AB}{AC}\]
An internal angle bisector divides the opposite side in the adjacent-side ratio.
Midpoint theorem
The segment joining midpoints of two sides is parallel to the third side and half its length.
\[DE=\frac12BC\]
03Accuracy
The UPSC Trap / Watch Out
Eliminate formula confusion, unit errors and misleading diagrams.
AClassic distractors and common errors
Diagram assumed to scale
Use stated lengths and angle facts, not visual appearance.
Sloping side used as height
Area requires the perpendicular distance to the chosen base.
Radius and diameter mixed
Replace \(d\) by \(2r\) before using a circle formula.
Semicircle perimeter incomplete
Perimeter is arc plus diameter: \(\pi r+2r\).
Sector perimeter incomplete
Add both radii to the arc length.
Area and perimeter ratios mixed
For similar figures, side ratio \(k\) gives area ratio \(k^2\).
Surface area used for capacity
Capacity depends on volume, not surface area.
CSA and TSA confused
Check whether bases are exposed, covered or open.
Cone height confusion
Use slant height for curved area and vertical height for volume.
Squared unit converted linearly
Area and volume conversions require squared and cubed factors.
Path width counted once
A uniform border changes both ends of each dimension, hence \(2x\).
Interior lines counted in perimeter
Count only the exposed boundary unless otherwise stated.
Recasting loses material
Unless wastage is stated, equate total volumes exactly.
Tank not uniform
Height and volume fractions are equal only with constant cross-sectional area.
Wrong congruence test
AAA proves similarity, while SSA is not a general congruence criterion.
Polygon exterior sum multiplied
One exterior angle of a regular polygon is \(360^\circ/n\); the total is always \(360^\circ\).
Square hierarchy reversed
Every square is a rectangle and rhombus; not every rectangle or rhombus is a square.
Value of \(\pi\) changed mid-solution
Use the supplied value consistently; otherwise retain \(\pi\) until the last step.
BHigh-speed shortcuts
Pythagorean triples
Recall \((3,4,5)\), \((5,12,13)\), \((7,24,25)\), \((8,15,17)\), \((9,40,41)\) and their multiples.
Regular polygon
Exterior angle \(=360^\circ/n\); invert to get \(n=360^\circ/e\).
Four quadrants
Four quarter-circles of the same radius equal one full circle.
Area scaling
If every length doubles, area becomes \(4\) times and volume \(8\) times.
Ring factorisation
\(\pi(R^2-r^2)=\pi(R-r)(R+r)\).
Diagonal recognition
Rectangle diagonal uses two-dimensional Pythagoras; cuboid diagonal uses three squares.
Path expansion
Outside rectangle: add \(2x\) to both dimensions. Inside: subtract \(2x\).
Recasting cancellation
Cancel common \(\pi\), fractional constants and powers before multiplying.
Dimensional check
Perimeter ends in units, area in square units, volume in cubic units. Wrong dimension exposes a wrong formula.
Fast accuracy rule
Sketch and label the figure, mark the requested dimension, convert all units, choose only exposed surfaces or boundaries, and retain exact values such as \(\pi\) and \(\sqrt3\) until the final step.
04Revise
Last-Minute Master Sheet
Compact relationships for rapid revision.
Triangle\[\Delta=\frac12bh\qquad a^2+b^2=c^2\]
Heron\[s=\frac{a+b+c}{2},\quad\Delta=\sqrt{s(s-a)(s-b)(s-c)}\]
Polygon\[S=(n-2)180^\circ,\quad D=\frac{n(n-3)}2\]
Circle\[C=2\pi r,\quad A=\pi r^2\]
Sector\[L=\frac{\theta}{360^\circ}2\pi r,\quad A=\frac{\theta}{360^\circ}\pi r^2\]
Rectangle / square\[A=lb,\quad d=\sqrt{l^2+b^2},\quad A_{\square}=a^2\]
Cylinder\[V=\pi r^2h,\quad CSA=2\pi rh,\quad TSA=2\pi r(h+r)\]
Cone\[V=\frac13\pi r^2h,\quad CSA=\pi rl,\quad l=\sqrt{r^2+h^2}\]
Sphere / hemisphere\[V_s=\frac43\pi r^3,\quad S_s=4\pi r^2\]\[V_h=\frac23\pi r^3,\quad TSA_h=3\pi r^2\]
Cuboid\[V=lbh,\quad TSA=2(lb+bh+hl)\]
Similarity scale\[L:k,\qquad A:k^2,\qquad V:k^3\]
Coordinates\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]
One-line memory rule
Lengths scale once, areas scale twice, volumes scale three times—and every calculation must end in the dimension requested.