Geometry is one of the main areas of Mathematics. Basically it is a study of shapes. The word 'Geo' means 'Earth' and 'Metry' means 'Measure'. So Geo - metry is literally 'Measurement of earth'. Traditional geometrical concepts are called Euclidian Geometry (mainly in two-dimension). Geometry is a sure area in all Placement Test & B-School entrance exams. In this article, you can learn Important Concepts & Formulas of Geometry and Mensuration along with the examples
- Plane geometry: Lines, angles, polygons and their properties, Mensuration etc.
- Solid geometry: This is basically a three dimensional geometry derived in three dimensional spaces. This area is dealing with different kinds of prisms, pyramids, spheres and other various three dimensional figures.
- Lines and Angles.
- Properties of angles.
- Basic properties of triangles.
- Geometric centre's of a triangle.
- Similarity and congruency in triangles.
- Area of a triangle (Various forms)
- Important theorems and useful results on triangles..
- Properties of right-angled triangles.
- Classifications of quadrilaterals
- Properties of various quadrilaterals
- Sectors & Segments
- Important properties and results related to circles
- Tangents & Secants
Lines and angles
Classification of angles
- Acute angle:
An angle greater than 0° but less than 90° is called an acute angle.
i.e. 0° < θ < 90°
Eg: 30°, 45°, 89° etc.
- Right angle:
An angle exactly equal to 90 ° is called a right angle.
i.e. θ = 90°
- Obtuse angle:
An angle greater than 90° but less than 180° is called an obtuse angle.
i.e. 0° < θ < 180°
Eg: 110°, 175° etc.
- Straight angle:
An angle exactly equal to 180° is called a straight angle.
- Reflex angle:
An angle greater than 180° but less than 360° is called a reflex angle.
i.e. 0° < θ < 360°
Eg: 190°, 270° etc.
- Complete angle:
An angle exactly equal to 360° is called a complete angle.
- Complementary angles:
If the sum of two angles equal to 90°, then the angles are complementary.
i.e. θ1 + θ2 = 90°
Eg: complementary of 30° is 60°.
Complementary of 70° is 20°.
- Supplementary angles:
If the sum of two angles equal to 180°, then the angles are supplementary.
i.e. θ1 + θ2 = 180°
Eg: supplementary of 30° is 150°.
Supplementary of 110° is 70°.
- Adjacent angles:
If two angles have a common hand and a common vertex then they are adjacent.
In diagram, ∠AOB and ∠BOC are adjacent. Therefore∠AOC = ∠AOB + ∠BOC.
Properties of angles
Case I. When two lines intersecting:
Two lines 'l' and 'm' are intersecting, and then there are four different angles formed. Each of the four angles is mutually related in the following manner.
- Linear pairs:
If two angles are adjacent and supplementary too, then they are called Linear Pairs.
(1, 2), (2, 3), (3, 4) and (4,1) are linear pairs.
i.e. ∠1 + ∠2 = 180°
∠2 + ∠3 = 180° and so on.
- Vertically opposite angles:
Out of the four angles forms by intersecting two lines, opposite angles are called vertically opposite angles, and they are equal in measure.
i.e. ∠1 = ∠3 and ∠2 = ∠4
- Angle around a point:
Around the point of intersection of two lines, there are four angles and the sum of these angles equal to 360°.
i.e. ∠1 + ∠2 + ∠3 + ∠4 = 360°.
Case II: A transversal passes through two parallel lines
In diagram lines 'l' and 'm' are parallel and a transversal 't' passes through both the lines, then there are eight different angles formed.
|By observation, it is possible to categorize these eight angles as acute and obtuse.
Acute angles: ∠2, ∠4, ∠6 and ∠8.
All acute angles in the diagram are equal in measure.
Obtuse angles: ∠1, ∠3, ∠5 and ∠7.
All obtuse angles in the diagram are equal in measure.
In this diagram, sum of any one of the acute angles and any one of the obtuse angles is equal to 180°.
- Alternate interior angles: (Z type angles)
For easiness to understand, these are the Z shape angles.
Both the corners of Z are equal in both directions.
i.e. ∠4 = ∠6 and ∠3 =∠5.
Therefore (4, 6) and (3, 5) are alternate interior angles.
- Corresponding angles (F type angles)
These angles formed as the corners of alphabet 'F' in different directions and corresponding angles are equal in measure.
∠1 = ∠5
∠2 = ∠6
∠3 = ∠7
∠4 = ∠8
- Co - interior angles (C type angles)
Co-interior formed as the corners of alphabet 'C'. These angles are supplementary.
∠3 + ∠6 = 180° and ∠4 +∠5 = 180°
Basic properties of triangles
- Sum of all interior angles of a triangle = 180°
- Largest angle is opposite to the longest side in a triangle or the longest side is opposite to the largest angle.
- Smallest angle is opposite to the shortest side or the shortest side is opposite to the smallest angle.
i.e. if x° > y° > z° then a > b > c.
- Angles opposite to the equal sides are equal and the sides opposite to the equal angles are equal.
- Sum of any two sides is greater than the third side.
- Difference of any two sides is lesser than the third side.
- Exterior angles of a triangle.
|Exterior angle theorem:
An exterior angle of a triangle is equal to the sum of its opposite interior angles.
∠PRS = ∠P + ∠Q
- Sum of all exterior angles = 360°
Classification of triangles as per angles
- Acute angled triangle:
In a triangle if all angles are less than 90° is called an acute angle. In an acute angled triangle the square of the side opposite to the acute angle is less than the sum of the squares of the other two sides.
- Right angled triangle:
In a triangle if exactly one angle is equal to 90° and rest two are acute, then it is a right angled triangle. In a right angled triangle the square of the side opposite to right angle is equal to the sum of the squares of the other two sides. This result is known as Pythagoras theorem and we will discuss it in detail later.
- Obtuse angled triangle:
In a triangle if exactly one angle is obtuse and rest two angles are acute, then it is an obtuse angled triangle. In an obtuse angled triangle, the square of the side opposite to the obtuse angle is greater than the sum of the squares of the other two angles.
Classification of triangles as per sides
- Equilateral triangle:
If all the three sides of a triangle are equal in length, then the triangle is equilateral.
Each angle of an equilateral triangle = 60°.
- Isosceles triangle:
Any two sides of a triangle are equal, then the triangle is isosceles.
In diagram sides FG = FH, so ∠G = ∠H.
∠G and ∠H are called the 'base angles' and ∠F is called the vertical angle.
- Scalene triangle:
If all the three sides of a triangle are different in length then the triangle is scalene.
Geometrical centers of a triangle
- Circum center(S)
The point of concurrency of the perpendicular bisectors of the three sides of a triangle is called the Circum center of the triangle.
In the diagram, S is the circum center and SA = SB = SC = R, where R is the circum radius.
In an acute angled triangle circum center lies inside the triangle. In a right angled triangle, circum centre is the midpoint of the hypotenuse, hence the circum radius = 1/2 (hypotenuse). In an obtuse angled triangle circum centre lies outside the triangle.
- In centre of a triangle (I)
In center is the point of concurrency of all the angle bisectors of a triangle.
In the diagram, 'I' is the in centre and ID is a perpendicular drawn to the side BC.
ID is the in-radius of the triangle and that is denoted by 'r'.
∠BIC = 90° + 1/2 ∠A
AIC = 90° + 1/2 ∠B
AIB = 90° + 1/2 ∠C
Point of concurrency of the internal bisector of any one angle and the external bisectors of the other two angles is called 'Ex-center' of the triangle.
- Ortho center of a triangle (O)
Point of concurrency of the three altitudes of a triangle is called the ortho-center and it is denoted by 'O'.
In an acute angled triangle ortho-center lies inside the triangle.
In a right angled triangle ortho center is the vertex where right angle formed.
In an obtuse angled triangle ortho center lies outside the triangle.
AOC = 180° - ∠B
BOC = 180° - ∠A
AOB = 180° - ∠C
- Centroid of a triangle (G)
Centroid is the point of concurrency of all the medians.
The line segment joining one vertex and the midpoint of its opposite side is called median.
D, E and F are the mid points of the sides BC, AC and AB respectively.
AD, BE and CF are the medians.
G is the Centroid.
- In a right angled triangle the length of the median drawn to the hypotenuse is equal to half the length of the hypotenuse.
- Centroid divides medians in the ratio 2:1.
i.e. Ag/DG = BG/EG = CG/FG = 2/1
Median divides the triangle into two equal areas.
i.e. Area of ABE = Area of ΔCBE = Area of ΔACF = Area of ΔBCF = Area of ΔABD = Area of ΔACD = 1/2 & Area of ΔABD
Similarity and congruency of triangles
When two angles are similar in shape, then the triangles are similar triangles.
If two triangles are similar, then their sides are in proportion.
Then; AB/PQ = BC/QR = AC/PR = Perimeter of ΔABC/Perimeter of ΔPPerimeter of ΔPQR = √Area of ΔABC/√Area of ΔPQR
When two triangles are equal in size and shape, then they are congruent.
ΔABC ≅ ΔDEF
AB = DE, BC = EF and AC = DF
Perimeter of ΔABC = Perimeter of DEF
Area of ΔABC = Area of ΔDEF
Area of a triangle
There are so many different approaches for finding the area of a triangle. A suitable result is applicable as per the availability of data.
- General formula
- Heron's formula
- Trigonometric application
A, B and C → measure of three angles
- Application of circum radius (R)
Area = abc/4r
a, b and c → lengths of three sides
R → circum radius
- Application of in-radius (r)
Area = r.S
r → in-radius
S → semi perimeter
- Equilateral Triangle
Area = √3 / 4 * a2
a → length of each side.
Height of an equilateral triangle = √3/4 * a
- Isosceles triangle
Area = b/4 * √(4a2 - b2)
a→ length of equal sides
b → length of third side
Important theorems and useful results
If AB/BC= m/n and AE/ED = p/q
then; Area of ΔABE/Area of ΔACD = pm/(p+q)(m+n)
In the given diagram ΔBDF is inscribed in ΔACE.
AB/BC= 12, CD/DE = 35 and AF/FE = 1/4
Find the ratio of the area of shaded region to the area of non-shaded region.
Area of ΔABF/Area of ΔACE = 1*1/((1+2)(1+4)) = 1/15
Area of ΔBCD/Area of ΔACE= 2*3/((2+1)(3+5)) = 14
Area of ΔDEF/Area of ΔACE= 5*4/((5+3)(4+1)) = 12
Area of (ΔABF + ΔBCD + ΔDEF)/Area of ΔACE= 1/15+ 1/4 + 1/2= 49/60
i.e. Area of non-shaded region = 49/60 * Area of ΔACE
i.e. Area of shaded region = Area of ΔACE - Area of non-shaded region
Area of shaded region = [1- 49/60]Area of ΔACE = 11/60 * Area of ΔACE
Area of shaded region/Area of non-shaded region= 11/49
Hence, the required ratio = 11 : 49
Result 2: Properties of right triangles
ΔABC is a right angled triangle, AD is the altitude drawn from vertex A to hypotenuse BC.
ΔADB ΔCDA ΔCAB (similar triangles)
|AD2 = BD * CD|
|AB2 = BD * BC|
|AC2 = BC * CD|
- In the given diagram AB = 6 cm, AC = 8 cm. Find AD.
AB = 6 cm and AC = 8 cm
BC = 10 cm
Area of ΔABC = 1/2 (AC*AB) = 1/2 (BC *AD)
AC*AB = BC *AD
8 * 6 = 10 * AD
AD = 4.8 cm.
- In right triangle ABC, AB : BC = 1: . Find AD : CD.
Let AB = 1 and BC = √3
AC = √(12 + √32) = 2
Let AD = a
CD = 2 - a
AB2 = AD * AC
12 = a * 2
a = 1/2
i.e. AD = 1/2
CD = 2 - 1/2 = 3/2
AD : CD = 1/2 : 3/2
= 1 : 3
Result 3: Internal angle bisector theorem
In ΔABC, AD is the angle bisector of ∠A, then AD/AC = BD/CD and BD * AC - AB * CD = AD2
Result 4: External angle bisector theorem
BE is the angle bisector of ∠CBD. then BC/AB = CE/AE
Result 5: Apollonius theorem
In triangle ABC, AD is the median from vertex A to side BC.
AB2 + AC2 = 2(AD2 + BD2)
Lengths of the diagonals of a parallelogram are 20cm and 24 cm. if one of its sides is 16 cm, find the length other non-parallel side.
Diagonals of a parallelogram are bisecting each other. Therefore E is the midpoint of BD (and AE).
In ΔABD, AE is a median.
As per Apollonius theorem;
AB2 + AD2 = 2(AE2 + BE2)
162 + x2 = 2(122 + 102)
x2 = 232
x = 2√58
Hence, the length of the other non-parallel side = 2√58 cm
Pythagoras theorem and its application
"In a right angled triangle, hypotenuse2 = base2 + altitude2 "
8 : 15 : 17
Special Pythagorean triplets
In an isosceles right angled triangle, sides are in the ratio 1 : 1: √2
Exact half of an equilateral triangle is a right triangle with angles 30°, 60° and 90°. In such a triangle sides are in the ratio 1 : √3 : 2
We can find out most of the Pythagorean Triplet by using the following general expressions;
2m, m2 - 1 and m2 + 1, where m > 1.
Polygons - a general approach
Convex polygon: In a polygon if all the interior angles are less than 180°, then it is a convex polygon. In a convex polygon, all the diagonals lie inside the polygon itself.
Concave polygon: If at least one of the angles of a polygon is a reflex angle, then the polygon is concave. At least one of its diagonals passes outside the polygon.
Almost all the B-school entrance exams limited the testing points only in convex polygons. Very rarely only exams like CAT and XAT testing the properties of concave polygons. As a general requirement, here we are considering convex polygons only.
Name of polygons as per number of sides
|Number of sides (n)||Name|
In any 'n' sided polygon;
- Sum of interior angles = (n - 2) 180°
- Sum of exterior angles = 360°
- Number of diagonals from any particular vertex to rest all vertices's = n - 3
- Total number of diagonals = n/( n-3)2
- Number of triangles formed by joining vertices's = n ( n-1)(n-2)/6
In a polygon if all sides are equal in length and all the angles are equal in measure, then it is a regular polygon.
Eg: square, equilateral triangle etc.
GEOMETRY & MENSURATION
In this section, we are mainly discussing about the basic properties of various quadrilaterals and circles.
One pair of opposite sides are parallel.
i.e. AB || CD.
Height of a trapezium is equal to the distance between the parallel sides.
Area = 1/2 * h(a+b), where a and b are the lengths of the parallel sides and h is distance between the parallel sides.
In a trapezium if the lengths of the non-parallel sides are equal, then it is an isosceles trapezium.
Pair of opposite sides are parallel. i.e. AB || CD and BC || AD .
Height of a parallelogram is the distance between two parallel sides.
Diagonals bisect each other. i.e. M is the midpoint of AC and BD.
Δ AMB ≅ Δ CMD and ΔAMD ≅ ΔCMB.
Δ AMB, Δ CMD, ΔAMD and ΔCMB are all equal in area. i.e. Area of each triangle =
1/4 * Area of the parallelogram
Area = b * h, where 'b' is base and 'h' is height.
All sides are equal
Diagonals bisect each other at 90°.
ΔAMB ≅ ΔCMD ≅ ΔAMD ≅ ΔCMB. area of each triangle = 1/4 * Area of the rhombus
Area = 1/2 * (d1 * d2), where d1 and d2 are the diagonals.
Opposite sides are equal in length.
Opposite sides are parallel.
Diagonals are equal and bisect each other.
Four triangles formed by the intersection of two diagonals are all equal in area. i.e. area of each triangle =
1/4 * Area of the rectangle.
Out of the above mentioned triangles, opposite triangles are congruent.
Area = length * breadth
Perimeter = 2 ( length + breadth)
All sides are equal.
All angles are right angles.
Diagonals are equal and bisect each other at 90°.
All the four triangles formed by the intersection of diagonals are congruent.
Area = side2 = a2
Perimeter = 4a
Length of diagonal = √2 * a
Circle is the collection of all points which are equidistant from a particular point on a plane. This particular point is called the center of the circle and constant distance is the radius of the circle.
O - center
r - radius
Diameter(d) = 2r
Area = π r2
Circumference = 2 πr
Angle around the centre = 360°
Area and arc length of a sector:
If the central angle of a sector = θ°, then;
θ/360 = arc length of the sector/ 2 πr = area of the sector/ π r2
Area of sector = θ3/60 * π r2
Arc length of the sector = θ/360 * 2πr
- Perpendicular from the center of a circle to a chord will bisect the chord.
- If two chords are intersecting in a circle, then;
PT * QT = RT * ST
- If any two secants of a circle are intersecting then;
CA * CB = CE * CD
- Angle inscribed in a semi circle is right angle.
- Central angle property:
Angle subtended at the center of a circle by an arc is double the angle subtended by it at any point on the remaining part of the circumference.
Angles in the same segment of a circle are equal.
Angles lie in the same arc are equal.
Tangents and secants to a circle
If a line touches the circle at exactly one point, then the line is a tangent to the circle.
- AB is a tangent and T is the point of tangency.
- There are infinitely many tangents are possible to a circle.
- Through one point on the circumference of a circle, there is exactly one tangent is possible to a circle.
- From an outside point, there are two different tangents are possible to a circle.
In the diagram PQ and PR are the tangents from an outside point P.
PQ = PR
Radius to the point of tangency is perpendicular to the tangent.
- In the below diagram AB is a tangent and ACD is a secant to the same circle.
AB2 = AC * AD
- In the below diagram PBA and PDC are two secants to the same circle.
PB * PA = PC * PD
Alternate segment theorem:
The angle between a tangent and a chord through the point of contact of the tangent and is equal to the angle made by the chord in the alternate segment.
∠PQR = ∠QSR
A quadrilateral inscribed in a circle is called Cyclic Quadrilateral.
In diagram ABCD is a cyclic quadrilateral.
Area of a cyclic quadrilateral = √((S-a)(S-b)(S-c)(S-d))
Where a, b, c and d are the four sides of the quadrilateral and S = a+b+c+d / 2
Opposite angles of a cyclic quadrilateral are supplementary. i.e. ∠A+∠C= ∠B+ ∠D=180°
If ABCD is a cyclic quadrilateral, then the product of the two diagonals is equal to the sum of the product of the opposite sides.
AC * BD = (AB * CD) + (AD * BC)
In this section on Geometry and Mensuration, we are dealing with 'Three Dimensional Geometry'. The term Three Dimensional Geometry represents a vast area of mathematical application, even in higher mathematical concepts too. However our requirement is as simple as to deal with some of the important concepts about Solid Geometry. The section 'Solid Geometry' mainly contains the concepts of Surface Areas and Volumes of various solid figures. Again there is a limitation in this regard; here we are dealing only with some particular type of solid figures, such as Prisms, Pyramids and Spheres. Most of the questions from this section of Geometry in MBA entrance exams are of application type, neither theoretical nor property related.
- Right Prisms
- Classification of Prisms
- Properties and results related to prisms
- General properties and results of Pyramids
- Cone and Frustum of a cone
- Sphere and Hemi-sphere
Areas and Volumes of solids
Basically solid figures are three dimensional objects which have surface area and volume too. As per the required testing area of the competitive exams, here we are dealing with three well defined categories of three dimensional objects such as Right Prisms, Pyramids and Spheres.
A right prism is solid object which has two identical and parallel faces (named base and top) connected by a lateral surface. The number of lateral faces is equal to the number of sides of the base polygon and each lateral face is in the shape of a rectangle.
The distance between the base and top faces of a prism is known as the length or height of the prism.
The name of a prism is based on the shape of the base polygon. i.e. if the base of a prism is in the shape of circle then it is a circular prism or cylinder or the base is a triangle then it is a triangular prism and so on.
Above diagrams are examples for right prisms.
General results for any right prisms:
Lateral surface area = Base perimeter * Height of the prism
Total surface area = Lateral surface area + 2 (Base area)
Volume = Base area * Height of the prism.
Classification of prisms
A right prism whose base is a rectangle is called cuboid (rectangular prism)
Base area = Length * Breadth = lb
Base perimeter = 2(l + b)
Lateral surface area = 2(l + b) * h
Total surface area = 2(lh + bh + lb)
Volume = lbh
Longest diagonal of a cuboid = √(l2 + b2 + h2)
A right prism whose all faces are identical squares is called a cube. i.e. the height of cube is equal to the side of base.
Base area = a2
Base perimeter = 4a
Lateral surface area = 4a2
Total surface area = 6a2
Volume = a3
Longest diagonal = a√3
A right prism whose base is a circle is known as cylinder.
Base area = πr2
Base perimeter = πr
Lateral surface area = πrh
Total surface area = 2πr(h+r)
Volume = πr2h
Pyramids are solid objects (three dimensional figures) whose base is a polygon and lateral faces are triangles. The lateral faces have a common vertex and this vertex is not co-planar with the base.
Line segments from the common vertex to the vertices of the base are called the lateral edges (e).
Pyramids are named according to the shape of its base. i.e. if the base is a triangle, then it is a triangular pyramid. If the base is a square, then it is a square pyramid, and so on.
If the base has a center, then the line joining the common vertex to the center of base is called axis.
A pyramid that has the axis perpendicular to the base is a right pyramid; otherwise it is an oblique pyramid.
If the base is a regular polygon and the pyramid is a right pyramid, then it is called a regular pyramid.
The altitude (h) of a pyramid is the perpendicular distance from the vertex to the base.
The slant height (l) of a pyramid is the altitude of the lateral face.
General results for pyramids:
Lateral surface area =1/2 * Permieter of the base *slant height
Total surface area = Lateral surface area + Area of the base.
Volume = 13 * Area of base * Height
Cone is a right pyramid which has its base as a circle.
Radius of base → r
Height of the cone → h
Slant height of the cone → l
l2 = r2 + h2
Volume = 13πr2h
Curved surface area = πrl
Total surface area = πr(l+r)
Cone cut in to two parts:
In the diagram, a cone is cut into two parts by a plane parallel to the base of the cone. There are two similar cones forms, one is the larger cone and the other is the smaller cone formed by cutting the lager cone.
Measures of the larger cone:
Radius = R
Height = H
Slant Height = L
Measures of the smaller cone:
Radius = r
Height = h
Slant Height = l
Following relations can furnish on the measures of both the cones;
r/R = h/H = l/L = 2πr/2πR = √(πr2)/√(πR2) = ∛(πr2 h)/∛(πR2 H) = √(πrl)/√(πRL)
Frustum of a cone:
When cutting a cone into two parts by a plane parallel to the base of the cone, the portion that contains the base is called the frustum of the cone.
Bottom radius = R
Top radius = r
Height = h
Slant height = l
l2 = (R-r)2 + h2
Volume = 1/3 * πh (R2 + Rr+r2)
Lateral surface area = πl ( R+r)
Total surface area = π(R2+ r2+ Rl+rl)
Frustum of any pyramid:
If a pyramid cut into two portions by a plane parallel to the base, then the part that containing the base is called the frustum of a pyramid.
Area of the base = A
Area of the top = a
Height of the pyramid = h
Slant height = l
Volume = 1/3 h (a + A + √(a * A))
Lateral surface area = 1/2 * (sum of the perimeters of base and top)*slant height
Total surface area = Lateral surface area + a + A
Sphere is a collection of all the points which are equidistant a particular point in a space. This particular point is called the center of the sphere and the distance between from the center to all the points in a sphere is the radius of the sphere.
Volume = 4/3 * π r3
Surface area = 4πr2
This is exactly half portion of a sphere, or the portion of a sphere formed by cutting a sphere through its center by a plane in any direction.
A hemisphere has two faces, one curved face and another circular plane face.
Volume = 2/3 * πr3
Curved surface area = 2πr2
Total surface area = 3πr2