Table of contents  
3. Polygons  
4. Quadrilaterals  
4.1 Different types of Quadrilaterals  
5. Hexagon (Regular)  
6. Circles 
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.
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
= ½ x Base x Height
= ½ x Product of sides x Sine of included angle
= here s is the semi perimeter [s = (a+b+c)/2 ]
= r x s [r is radius of incircle]
[R is radius of circumcircle]
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.
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.
2. Basic Proportionality Theorem: If DE  BC, then AD/DB = AE/EC
3. Apollonius’ Theorem: AB^{2} + AC^{2} = 2 (AD^{2} + BD^{2})
4. Interior Angle Bisector Theorem: AE/ED = BA/BD
1. Right Angled Triangle:
∆ABC ≈ ∆ ADB ≈ ∆ BDC
BD^{2} = AD x DC and AB x BC = BD X DC
2. Equilateral Triangle:
All angles are equal to 60°. All sides are equal also.
Height =
Area =
Inradius = 1/3 Height
Circumradius = 2/3 Height.
3. Isosceles Triangle:
Angles equal to opposite sides are equal.
Area
30°60°90° Triangle
Area
45°45°90° Triangle
Area = x^{2}/2
30°30°120° Triangle
Area =
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 a^{2} : b^{2}
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
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 = Exterior = 360°/n
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 = where s is the semi perimeter
EduRev's Tip:
 Sum or product of opposite sides = Product of diagonals
 If a circle can be inscribed in a quadrilateral, its area is given by = √abcd
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
⇒ AC^{2} + BD^{2} = AB^{2} + BC^{2} + CD^{2} + DA^{2} A Rectangle is formed by intersection of the four angle bisectors of a parallelogram.
A parallelogram with all sides equal is a Rhombus. Its diagonals bisect at 90°.
Perimeter = 4a; Area = ½ d_{1} d_{2} ; Area = d x
A parallelogram with all angles equal (90°) is a Rectangle. Its diagonals are congruent. Perimeter = 2(l+b); Area = lb
A parallelogram with sides equal and all angles equal is a square. Its diagonals are congruent and bisect at 90°.
Perimeter = 4a; Area = a^{2}; 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.
EduRev's Tip: Sum of the squares of the length of the diagonals = Sum of squares of lateral sides + 2 Product of bases.
⇒ AC^{2} + BD^{2} = AD^{2} + BC^{2} + 2 x AB x CD
[Intext Question]
The nonparallel 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.
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 a^{2}
Area of an Octagon = 2(√2 + 1) a^{2}
Properties (contd.)
PA x PB = PC2
θ = ½ [ m(Arc AC)  m(Arc BC) ]
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
⇒ AD = BC =
Length of the Transverse Common Tangent (TCT)
⇒ RT = SU =
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 r_{1} : r_{2} internally whearea P divides OO’ in the ratio r_{1} : r_{2} externally.
 There are 4 body diagonals in a cube / cuboid of length (√3 x side) and respectively.
215 videos139 docs151 tests

1. What are the properties of a triangle? 
2. What are the different types of quadrilaterals? 
3. How do you calculate the area of a regular hexagon? 
4. What is the formula to find the perimeter of a polygon? 
5. How do you find the circumference of a circle? 

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