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Geometry

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  1. Aptitude
  3. Quantitative
  5. Geometry
  7. Theory

Geometry: Concepts, Formulas, and Clear Examples

Plane geometry MCQs test angle chasing, similar triangles, and circle theorems with minimal drawing. Mark given lengths on a sketch before choosing a formula.

Campus items favor triangles (Pythagoras, area), parallel lines, and polygons whose interior angles sum to (n−2)⋅180∘(n-2)\cdot180^\circ(n−2)⋅180∘.

Keep π\piπ factors exact until the end; options often differ only by 2\sqrt{2}2​ or 3\sqrt{3}3​.

Lines and angles

Triangle angle sum

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^\circ∠A+∠B+∠C=180∘
  • Linear pair sums to 180∘180^\circ180∘; vertically opposite angles equal.
  • Parallel lines: alternate interior / corresponding angles equal.

Triangles and Pythagoras

Pythagoras

c2=a2+b2c^2 = a^2 + b^2c2=a2+b2

Area

Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}Area=21​×base×height

Heron

A=s(s−a)(s−b)(s−c), s=a+b+c2A = \sqrt{s(s-a)(s-b)(s-c)},\ s=\frac{a+b+c}{2}A=s(s−a)(s−b)(s−c)​, s=2a+b+c​

Example — Right triangle hypotenuse

Legs 9 and 12. Hypotenuse = 81+144=225=15\sqrt{81+144}=\sqrt{225}=1581+144​=225​=15.

Similar triangles

Equal angles ⇒ side ratios match: aa′=bb′=hh′\frac{a}{a'}=\frac{b}{b'}=\frac{h}{h'}a′a​=b′b​=h′h​.

Ratio of areas

[ABC][DEF]=(side1side2)2\frac{[ABC]}{[DEF]} = \left(\frac{\text{side}_1}{\text{side}_2}\right)^2[DEF][ABC]​=(side2​side1​​)2

Circles

Circumference

C=2πrC = 2\pi rC=2πr

Area

πr2\pi r^2πr2

Inscribed angle

Inscribed=12×central angle\text{Inscribed} = \frac{1}{2}\times\text{central angle}Inscribed=21​×central angle

Quadrilaterals

Parallelogram area

A=b×hA = b \times hA=b×h

Trapezoid

A=12(a+b)hA = \frac{1}{2}(a+b)hA=21​(a+b)h

Polygons

Interior sum

(n−2)×180∘(n-2)\times 180^\circ(n−2)×180∘

Each interior (regular)

(n−2)⋅180∘n\frac{(n-2)\cdot180^\circ}{n}n(n−2)⋅180∘​

Step-by-step method for exam questions

  1. Draw and label from the stem.
  2. Spot right/similar/isosceles patterns.
  3. Write one key relation (Pythagoras or ratio).
  4. Compute; keep surds until final compare.
  5. Check units and whether diameter vs radius.
  6. Pick option.

Formula cheat sheet

SituationFormulaWhen to use
Right trianglec2=a2+b2c^2=a^2+b^2c2=a2+b2Perpendicular given
Similar sidesa/a′=b/b′a/a'=b/b'a/a′=b/b′Equal angles
Circle areaπr2\pi r^2πr2Radius known
Regular n-gon angle(n−2)180n\frac{(n-2)180}{n}n(n−2)180​Interior angle MCQ
Area scalek2k^2k2 times areaSides scale by k

Frequently asked questions

Chord vs diameter?

Diameter is longest chord; don't confuse with radius.

30-60-90 triangle?

Sides 1:3:21:\sqrt{3}:21:3​:2 opposite 30°, 60°, 90°.

Assume parallel lines?

Only if marked or stated; don't guess from figure alone in text-only items.

π in answers?

Match format in options (decimal vs multiple of π).

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.

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