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SSC CGL Quantitative Aptitude · Formula sheet

Lines, Angles and Polygons formulas

6 formulas and 14 common traps for SSC CGL Quantitative Aptitude Lines, Angles and Polygons, grouped by subtopic.

Full notes with worked examples

Angles, Parallels and Polygon Sums

Learn this subtopic in the notes

Angles at crossing lines and at parallel lines

Angles at lines

linear pair=180∘round a point=360∘co-interior pair=180∘\text{linear pair} = 180^\circ \qquad \text{round a point} = 360^\circ \qquad \text{co-interior pair} = 180^\circ

Interior and exterior angle sums of a polygon

Polygon angle sums

interior sum=(n−2)×180∘exterior sum=360∘interior+exterior=180∘\text{interior sum} = (n - 2) \times 180^\circ \qquad \text{exterior sum} = 360^\circ \qquad \text{interior} + \text{exterior} = 180^\circ
  • nnthe number of sides, which is also the number of corners

Number of diagonals of a polygon

Diagonals of an n-sided polygon

diagonals=n(n−3)2\text{diagonals} = \frac{n(n - 3)}{2}

Common traps

Calling alternate angles supplementary

With parallel lines, alternate interior angles are equal. The pair that adds to 180∘180^\circ is the co-interior pair, on the same side of the transversal. If one alternate angle is 115∘115^\circ, the other is 115∘115^\circ, and 65∘65^\circ is a wrong option.

Using the parallel-line rules on lines that are not parallel

Corresponding and alternate angles are equal only when the two lines cut by the transversal are parallel. If the question does not say so, or give equal angles that prove it, do not assume it.

Using n × 180° for the interior sum

The interior sum is (n−2)×180∘(n - 2) \times 180^\circ, because the polygon splits into n−2n - 2 triangles, not n. A decagon's angles add to 1440∘1440^\circ, not 1800∘1800^\circ.

Thinking the exterior sum grows with the sides

One exterior angle at each corner of any convex polygon adds to 360∘360^\circ, for a triangle and for a 20-sided figure alike. 'Depends on the number of sides' is a printed wrong option.

Giving the interior angle when the exterior one is asked

When a question finds a missing interior angle and then asks for the exterior angle at that corner, take one more step: exterior =180∘−= 180^\circ - interior. The interior value is usually printed as a wrong option.

Counting every pair of corners as a diagonal

n(n−1)2\frac{n(n - 1)}{2} counts every pair of corners, sides included. Take away the n sides: a hexagon has 15−6=915 - 6 = 9 diagonals, not 15.

Forgetting to halve

n(n−3)n(n - 3) counts each diagonal from both ends. Divide by 2: a hexagon has 6×32=9\frac{6 \times 3}{2} = 9 diagonals, not 18.

Regular Polygons

Learn this subtopic in the notes

Each interior and exterior angle of a regular polygon

One angle of a regular polygon

exterior=360∘ninterior=180∘−360∘n=(n−2)×180∘n\text{exterior} = \frac{360^\circ}{n} \qquad \text{interior} = 180^\circ - \frac{360^\circ}{n} = \frac{(n - 2) \times 180^\circ}{n}
  • nnthe number of equal sides

Number of sides of a regular polygon from one angle

Sides from an angle

n=360∘exteriorexterior=180∘−d2n=2(k+1)n = \frac{360^\circ}{\text{exterior}} \qquad \text{exterior} = \frac{180^\circ - d}{2} \qquad n = 2(k + 1)
  • ddinterior angle minus exterior angle
  • kkhow many times the exterior angle the interior angle is

Centre, side and area of a regular polygon

Centre angle and hexagon area

angle at centre=360∘nhexagon area=6×34a2=332a2\text{angle at centre} = \frac{360^\circ}{n} \qquad \text{hexagon area} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2
  • aathe side of the regular hexagon

Common traps

Taking 360°/n as the interior angle

360∘n\frac{360^\circ}{n} is the exterior angle. The interior angle is 180∘180^\circ minus it. For a regular octagon, 45∘45^\circ is the exterior angle and 135∘135^\circ the interior one.

Using these formulas on a polygon that is not regular

The angles of an irregular polygon are not equal, so 360∘n\frac{360^\circ}{n} is only their average. Without the word 'regular', use the angle sums instead.

Dividing 360 by the interior angle

n=360exteriorn = \frac{360}{\text{exterior}}. If the interior angle is 150∘150^\circ, first find the exterior angle, 30∘30^\circ, and only then divide: 12 sides.

Forgetting to halve when the difference is given

If the interior angle exceeds the exterior one by d, the exterior angle is 180−d2\frac{180 - d}{2}, not 180−d180 - d. With d=100d = 100 it is 40∘40^\circ, not 80∘80^\circ.

Accepting a polygon that cannot exist

The number of sides must be a whole number. An exterior angle that does not divide 360 exactly, such as 50∘50^\circ or 70∘70^\circ, belongs to no regular polygon.

Giving the interior angle for the angle at the centre

A side of a regular pentagon subtends 72∘72^\circ at the centre. Its interior angle, 108∘108^\circ, is a different angle, at a corner, and is printed as a wrong option.

Treating the hexagon's centre distance as different from its side

The six triangles in a regular hexagon are equilateral, so the distance from the centre to any corner equals the side. A circle through the corners of a hexagon of side 5 cm has radius 5 cm.

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