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Important Formulas for CAT Geometry & Mensuration

Geometry & Mensuration hold significant weight in competitive exams, making it a crucial topic for aspirants. This document serves as a valuable resource by providing essential formulas for Geometry & Mensuration. It is designed to facilitate quick and effective revision, ensuring that candidates can reinforce their understanding of key concepts in preparation for competitive exams.

1. Lines and Angles

  • Sum of the angles in a straight line is 180°  
  • Vertically opposite angles are congruent (equal).
  • If any point is equidistant from the endpoints of a segment, then it must lie on the perpendicular bisector
  • When two parallel lines are intersected by a transversal, corresponding angles are equal, alternate angles are equal and co-interior angles are supplementary. (All acute angles formed are equal to each other and all obtuse angles are equal to each other)
    Important Formulas for CAT Geometry & Mensuration

EduRev's Tip: The ratio of intercepts formed by a transversal intersecting three parallel lines is equal to the ratio of corresponding intercepts formed by any other transversal.
⇒ a/b = c/d = e/f

2. Triangles

  • Sum of interior angles of a triangle is 180° and the sum of exterior angles is 360°. 
  • Exterior Angle = Sum of remote interior angles.
  • Sum of two sides is always greater than the third side and the difference of two sides is always lesser than the third side. 
  • Side opposite to the biggest angle is longest and the side opposite to the smallest angle is the shortest.

Area of a triangle:

Important Formulas for CAT Geometry & Mensuration

= ½ x Base x Height
= ½ x Product of sides x Sine of included angle
= Important Formulas for CAT Geometry & Mensuration here s is the semi perimeter [s = (a+b+c)/2 ]
= r x s [r is radius of incircle]
Important Formulas for CAT Geometry & Mensuration[R is radius of circumcircle]

  • A Median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. The three medians intersect in a single point, called the Centroid of the triangle. Centroid divides the median in the ratio of 2:1.
  • An Altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an extension of the opposite side. The three altitudes intersect in a single point, called the Orthocenter of the triangle. 
  • A Perpendicular Bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint. The three perpendicular bisectors intersect in a single point, called the Circumcenter of the triangle. It is the center of the circumcircle which passes through all the vertices of the triangle.
  • An Angle Bisector is a line that divides the angle at one of the vertices in two equal parts. The three angle bisectors intersect in a single point, called the Incenter of the triangle. It is the center of the incircle which touches all sides of a triangle.

EduRev's Tip: 

  • Centroid and Incenter will always lie inside the triangle.
    (i) For an acute angled triangle, the Circumcenter and the Orthocenter will lie inside the triangle.
    (ii) For an obtuse angled triangle, the Circumcenter and the Orthocenter will lie outside the triangle.
    (iii) For a right angled triangle the Circumcenter will lie at the midpoint of the hypotenuse and the Orthocenter will lie at the vertex at which the angle is 90°.
  • The orthocenter, centroid, and circumcenter always lie on the same line known as the Euler Line.
    (i) The orthocenter is twice as far from the centroid as the circumcenter is.
    (ii) If the triangle is Isosceles then the incenter lies on the same line.
    (iii) If the triangle is equilateral, all four are the same point.

Theorems

Important Formulas for CAT Geometry & Mensuration

1. Mid Point Theorem: The line joining the midpoint of any two sides is parallel to the third side and is half the length of the third side.

Important Formulas for CAT Geometry & Mensuration
2. Basic Proportionality Theorem: If DE || BC, then AD/DB = AE/EC

Important Formulas for CAT Geometry & Mensuration
3. Apollonius’ Theorem: AB2 + AC2 = 2 (AD2 + BD2)

Important Formulas for CAT Geometry & Mensuration
4. Interior Angle Bisector Theorem: AE/ED = BA/BD

Special Triangles

1. Right Angled Triangle:
∆ABC ≈ ∆ ADB ≈ ∆ BDC
BD2 = AD x DC and AB x BC = BD X DC

2. Equilateral Triangle:
Important Formulas for CAT Geometry & Mensuration
All angles are equal to 60°. All sides are equal also.
Height = Important Formulas for CAT Geometry & Mensuration
Area = Important Formulas for CAT Geometry & Mensuration
Inradius = 1/3 Height
Circumradius = 2/3 Height.

3. Isosceles Triangle:
Important Formulas for CAT Geometry & Mensuration
Angles equal to opposite sides are equal.
Area Important Formulas for CAT Geometry & Mensuration
30°-60°-90° Triangle
Important Formulas for CAT Geometry & Mensuration
Area Important Formulas for CAT Geometry & Mensuration
45°-45°-90° Triangle
Important Formulas for CAT Geometry & Mensuration
Area = x2/2
30°-30°-120° Triangle
Important Formulas for CAT Geometry & Mensuration
Area =Important Formulas for CAT Geometry & Mensuration

Similarity of Triangles

Two triangles are similar if their corresponding angles are congruent and corresponding sides are in proportion.

Tests of similarity: (AA / SSS / SAS)

For similar triangles, if the sides are in the ratio of a:b

⇒ Corresponding heights are in the ratio of a:b

⇒ Corresponding medians are in the ratio of a:b

⇒ Circumradii are in the ratio of a:b

⇒ Inradii are in the ratio of a:b

⇒ Perimeters are in the ratio of a:b

⇒ Areas are in the ratio a2 : b2

Congruency of Triangles

Two triangles are congruent if their corresponding sides and angles are congruent.

Tests of congruence: (SSS / SAS / AAS / ASA)
All ratios mentioned in similar triangle are now 1:1

3. Polygons

  • Sum of interior angles = (n-2) x 180° = (2n-4) x  90° Sum of exterior angles = 360°
  • Number of diagonals = Important Formulas for CAT Geometry & Mensuration
  • Number of triangles which can be formed by the vertices = nC3

Regular Polygon:

If all sides and all angles are equal, it is a regular polygon. All regular polygons can be inscribed in or circumscribed about a circle.
Area = ½ x Perimeter x Inradius {Inradius is the perpendicular from centre to any side}
Each Interior Angle =Important Formulas for CAT Geometry & Mensuration Exterior = 360°/n

4. Quadrilaterals


Important Formulas for CAT Geometry & Mensuration

  • Sum of the interior angles = Sum of the exterior angles = 360°
  • Area for a quadrilateral is given by ½ d1 d2 Sinθ.

Cyclic Quadrilateral


Important Formulas for CAT Geometry & Mensuration
If all vertices of a quadrilateral lie on the circumference of a circle, it is known as a cyclic quadrilateral.
Opposite angles are supplementary
Area = Important Formulas for CAT Geometry & Mensuration  where s is the semi perimeter Important Formulas for CAT Geometry & Mensuration

EduRev's Tip:

  • Sum or product of opposite sides = Product of diagonals
    Important Formulas for CAT Geometry & Mensuration
  • If a circle can be inscribed in a quadrilateral, its area is given by = √abcd

4.1 Different types of Quadrilaterals

(a) Parallelogram


Important Formulas for CAT Geometry & Mensuration

  • Opposite sides are parallel and congruent.
  • Opposite angles are congruent and consecutive angles are supplementary.
  • Diagonals of a parallelogram bisect each other.
  • Perimeter = 2(Sum of adjacent sides); 
  • Area = Base x Height = AD x BE

EduRev's Tip: 

  • A parallelogram inscribed in a circle is always a Rectangle. A parallelogram circumscribed about a circle is always a Rhombus.
  • Each diagonal divides a parallelogram in two triangles of equal area.
  • Sum of squares of diagonals = Sum of squares of four sides
    ⇒ AC2 + BD2 = AB2 + BC2 + CD2 + DA2
  • A Rectangle is formed by intersection of the four angle bisectors of a parallelogram.

(b) Rhombus


Important Formulas for CAT Geometry & Mensuration
A parallelogram with all sides equal is a Rhombus. Its diagonals bisect at 90°.
Perimeter = 4a;   Area = ½ d1 d2 ;   Area = d xImportant Formulas for CAT Geometry & Mensuration

(c) Rectangle

A parallelogram with all angles equal (90°) is a Rectangle. Its diagonals are congruent. Perimeter = 2(l+b); Area = lb

(d) Square


A parallelogram with sides equal and all angles equal is a square. Its diagonals are congruent and bisect at 90°.
Perimeter = 4a; Area = a2; Diagonals = a√2

EduRev's Tip: From all quadrilaterals with a given area, the square has the least perimeter. For all quadrilaterals with a given perimeter, the square has the greatest area.

(e) Kite


Important Formulas for CAT Geometry & Mensuration

  • Two pairs of adjacent sides are congruent. 
  • The longer diagonal bisects the shorter diagonal at 90°.
  • Area = Product of Diagonals / 2

(f) Trapezium / Trapezoid

  • A quadrilateral with exactly one pair of sides parallel is known as a Trapezoid. The parallel sides are known as bases and the non-parallel sides are known as lateral sides.
  • Area = ½ x (Sum of parallel sides) x Height
  • Median, the line joining the midpoints of lateral sides, is half the sum of parallel sides.

EduRev's Tip: Sum of the squares of the length of the diagonals = Sum of squares of lateral sides + 2 Product of bases.
⇒ AC2 + BD2 = AD2 + BC2 + 2 x AB x CD

Isosceles Trapezium


Important Formulas for CAT Geometry & Mensuration
The non-parallel sides (lateral sides) are equal in length. Angles made by each parallel side with the lateral sides are equal.


EduRev's Tip: If a trapezium is inscribed in a circle, it has to be an isosceles trapezium. If a circle can be inscribed in a trapezium, Sum of parallel sides = Sum of lateral sides.


5. Hexagon (Regular)


Important Formulas for CAT Geometry & Mensuration 

  • Perimeter = 6a; Area =Important Formulas for CAT Geometry & Mensuration
  • Sum of Interior angles = 720°.
  • Each Interior Angle = 120°. Exterior = 60°
  • Number of diagonals = 9 {3 big and 6 small}
  • Length of big diagonals (3) = 2a
  • Length of small diagonals (6) = √3a

EduRev's Tip: A regular hexagon can be considered as a combination of six equilateral triangles. All regular polygons can be considered as a combination of ‘n’ isosceles triangles.
Area of a Pentagon = 1.72 a2
Area of an Octagon = 2(√2 + 1) a2

6. Circles

  • Diameter = 2r; Circumference = 2πr; Area = πr2
  • Chords equidistant from the centre of a circle are equal.
  • A line from the centre, perpendicular to a chord, bisects the chord.
  • Equal chords subtend equal angles at the centre.
  • The diameter is the longest chord of a circle.
    Important Formulas for CAT Geometry & Mensuration
  • A chord /arc subtends equal angle at any point on the circumference and double of that at the centre.
  • Chords / Arcs of equal lengths subtend equal angles.
    Important Formulas for CAT Geometry & Mensuration
  • Chord AB divides the circle into two parts: Minor Arc AXB and Major Arc AYB.
  • Measure of arc AXB = ∠AOB = θ
  • Length (arc AXB) = Important Formulas for CAT Geometry & Mensuration
  • Area (sector OAXB) = Important Formulas for CAT Geometry & Mensuration
  • Area of Minor Segment = Shaded Area in above figure
    ⇒ Area of Sector OAXB  - Area of ∆ OAB
    Important Formulas for CAT Geometry & Mensuration

Properties of Tangents, Secants and Chords


Important Formulas for CAT Geometry & Mensuration

  • The radius and tangent are perpendicular to each other.
  • There can only be two tangents from an external point, which are equal in length  PA = PB
    Important Formulas for CAT Geometry & Mensuration
    PA x PB = PC x PD
    θ = ½ [ m(Arc AC) – m(Arc BD) ]
    Important Formulas for CAT Geometry & Mensuration
    PA x PB = PC x PD
    θ = ½ [ m(Arc AC) + m(Arc BD) ]

Properties (contd.)
Important Formulas for CAT Geometry & Mensuration
PA x PB = PC2
θ = ½ [ m(Arc AC) - m(Arc BC) ]

Alternate Segment Theorem


Important Formulas for CAT Geometry & Mensuration
The angle made by the chord AB with the tangent at A (PQ) is equal to the angle that it subtends on the opposite side of the circumference.
⇒ ∠BAQ = ∠ACB

Common Tangents


Important Formulas for CAT Geometry & Mensuration
Important Formulas for CAT Geometry & Mensuration
⇒ AD = BC  = Important Formulas for CAT Geometry & Mensuration
Length of the Transverse Common Tangent (TCT)
⇒ RT = SU = Important Formulas for CAT Geometry & Mensuration

EduRev's Tip: 

  • The two centers(O and O’), point of intersection of DCTs (P)and point of intersection of TCTs (Q) are collinear.  Q divides OO’ in the ratio r1 : r2 internally whearea P divides OO’ in the ratio r1 : r2 externally.
    Important Formulas for CAT Geometry & Mensuration
  • There are 4 body diagonals in a cube / cuboid of length (√3 x side) and Important Formulas for CAT Geometry & Mensurationrespectively.
The document Important Formulas for CAT Geometry & Mensuration is a part of the UPSC Course CSAT Preparation.
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FAQs on Important Formulas for CAT Geometry & Mensuration

1. What are the properties of a regular hexagon?
Ans. A regular hexagon has six equal sides and six equal angles, each measuring 120 degrees. It can be divided into six equilateral triangles.
2. How can you find the area of a circle given its radius?
Ans. The area of a circle can be calculated using the formula A = πr^2, where r is the radius of the circle.
3. What is the sum of the interior angles of a triangle?
Ans. The sum of the interior angles of a triangle is always 180 degrees. This property holds true for all triangles, regardless of their shape or size.
4. How many diagonals can be drawn in a polygon with 6 sides?
Ans. In a polygon with 6 sides (a hexagon), the number of diagonals that can be drawn is 9. This can be calculated using the formula n(n-3)/2, where n is the number of sides of the polygon.
5. What are the different types of quadrilaterals and their properties?
Ans. Some common types of quadrilaterals include squares, rectangles, parallelograms, rhombuses, and trapezoids. Each type has its own unique properties in terms of angles and side lengths.
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