# Formula Sheets for General Aptitude (Part A): Geometry !

By Astha Singh|Updated : February 28th, 2022

General Aptitude Formula Sheets: During the preparation, the candidates study different formulas to solve problems, but at the last moment, these formulas might not be remembered by the candidates due to exam fear or pressure. We at BYJU'S Exam Prep do not want our students to lag anywhere during the preparation, so we have come up with a concept of a Formula Sheet that will help them revise the important formulas at the last moment. This formula sheet will be a short revision tool and contain only important formulas that need to be studied at the last minute to boost the score. Our experienced subject-matter experts have meticulously designed this CSIR NET General Aptitude Formula Sheet to provide you with the best authentic material.

In this article, we will cover the CSIR NET General Aptitude Most Important Formulas of Geometry. Aspiring candidates can check all the most important formulas of Geometry for the last-minute revision. Scroll down the full article to find out!

## Formula Sheet On Geometry

In the previous paper, many questions were asked from this section.
Geometry is a branch of mathematics that deals with sizes, shapes, angles &
dimensions of figures
The figure that is in flat shapes like squares, circles, and triangles are called 2D
shapes. These figures have only two-dimension length and breadth.

Basic Information:

Line segment: A line segment has two endpoints with a definite length.

Supplementary angles: In the figure above, ∠AOC + ∠COB = ∠AOB = 180° If the sum of two angles is 180° then the angles are called supplementary angles. Two right angles always supplement each other. The pair of adjacent angles whose sum is a straight angle is called a linear pair.

Complementary angles: ∠COA + ∠AOB = 90° If the sum of two angles is 90° then the two angles are called complementary angles.

Adjacent angles: The angles that have a common aim and a common vertex are called adjacent angles. In the figure above, ∠ABC and ∠CBD are adjacent angles. Their common arm is BC and their common vertex is ‘B’.

In the figure above, x and y are two intersecting lines. ∠A and ∠C make one pair of vertically opposite angles and ∠B and ∠D make another pair of vertically opposite angles.

Perpendicular lines: When there is a right angle between two lines, the lines are said to be perpendicular to each other.

Here, the lines OA and OB are said to be perpendicular to each other.

Parallel lines: Here, A and B are two parallel lines, intersected by a line p. The line p is called a transversal, which intersects two or more lines (not necessarily parallel lines) at distinct points. As seen in the figure above, when a transversal intersects two lines, 8 angles are formed. Let us consider the details in a tabular form for easy reference.

When a transversal intersects two parallel lines,
1. The corresponding angles are equal.
2. The vertically opposite angles are equal.
3. The alternate interior angles are equal.
4. The alternate exterior angles are equal.
5. The pair of interior angles on the same side of the transversal is supplementary.
We can say that the lines are parallel if we can verify at least one of the aforementioned conditions.

Polygon: Any closed figure, having 3 or more than sides is called a polygon. Ex.: Triangle, quadrilaterals, hexagon etc.
Types of Polygons:
It should be known that polygons are categorized as different types depending on the number of sides together with the extent of the angles. Some of the prime categories of polygons include regular polygons, irregular polygons, concave polygons, convex polygons, quadrilateral polygons, pentagon polygons and so on.
Some of the most well-known polygons are triangles, squares, rectangles, parallelograms, pentagons, rhombuses, hexagons etc.

Regular polygon: Considering a regular polygon, it is noted that all sides of the polygon tend to be equal.

Furthermore, all the interior angles remain equivalent.

Irregular polygon: These are those polygons that aren’t regular. Be it the sides or the angles, nothing is equal as compared to a regular polygon.

Convex polygon: The measure of interior angle stays less than 180 degrees for a convex polygon. Such a polygon is known to be the exact opposite of a concave polygon. Moreover, the vertices associated to a convex polygon are always outwards.
Quadrilateral polygon: Four-sided polygon or quadrilateral polygon is quite common. There are different versions of a quadrilateral polygon such as square, parallelogram and rectangle.
Pentagon polygon: Pentagon polygons are six-sided polygons. It is important to note that, the five sides of the polygon stay equal in length. A regular pentagon is a prime type of pentagon polygon.

Formula Related to Polygon:
(N = Number of sides and S = Distance from centre to a corner)
The number of diagonals
Summation of the interior angles of the polygon = (N – 2) × 180o
The count of triangles (while drawing all the diagonals through a single vertex) in a polygon = (N – 2)

Angle: The study of angles is very important whenever we are trying to understand polygons and their properties. To be precise, when two rays hold a common endpoint, in this case, the two rays together form an angle. Therefore, an angle is formed by two rays initiating from a shared endpoint. These two rays creating it are termed as the sides or arms of the angle. For representing an angle, the symbol “∠” is used in geometry.
Angles of Polygon: One must keep in mind that all polygons possess internal angles and external angles. In addition, a polygon’s external angle can be termed as that which is extended on one side. Here are certain rules which are followed regarding angles of a polygon.

Properties of Polygons: If “n” is the total number of sides in a polygon. Then,
1. The sum of all interior angles of a polygon = (n - 2)×180°

2. Each interior angle of a regular polygon

3. The sum of all exterior angles of a polygon = 360°

4. Each exterior angle of a regular polygon

5. The total number of diagonals in a regular polygon is

6. The ratio of the sides of a polygon to the diagonals of a polygon = 2:(𝑛−3)

7. The ratio of the interior angle of a regular polygon to its exterior angle  =(𝑛 −2): 2

In a regular polygon with “n” sides and the length of each side is “a” unit, “r” is the inradius of the polygon and “R” is the circumradius of the polygon then,
1. Perimeter of the polygon = an

2. Area of a polygon (Perimeter of the polygon) × in radius na × r

3. Area of a polygon

4. Area of a polygon

5. Area of a polygon

Regular Hexagon: Let ABCDEF be a regular hexagon with each side of length “a” unit and “O” is the centre of the given hexagon.
1. The sum of the interior angles of the Hexagon = (6 – 2)×180o = 720o

2. Each exterior angle

3. Each interior angle

4. Area of the Hexagon  Area of an equilateral triangle

Regular Octagon: Let ABCDEFGH be a regular octagon with each side of length “a” unit.

1. The sum of the interior angles of the Octagon = (8 −2) × 180° = 1080°

2. Each exterior angle

3. Each interior angle

4. Area of the Octagon =

Triangle
A triangle is the simplest polygon enclosed by three sides. As in the figure below three sides AB, BC, and CA represent triangle ABC or denoted by ∆ABC. A B and C represent the vertices of the ∆ABC. Also, a, b and c represent the lengths of the sides BC, AC and AB respectively and A, B and C represent the values of angles of the ∆ABC.

Properties of a Triangle:

1. The sum of all the angles of a triangle = 180°

2. The sum of lengths of two sides > length of the third side of the triangle i.e.  a + b + > c b + c > a c + a > b

3. The difference of lengths of two sides < length of the third side of the triangle i.e. a – b < c b – c < a c – a < b4.

Area of the triangle:

1. If in a triangle ∆ABC, a perpendicular is drawn from vertex A to side BC which meets the side BC at point D. If the length of the perpendicular AD is ‘h’, then the Area of the ∆ABC –

Classification of Triangles:
1. on the basis of Sides of the Triangle
(i) Scalene Triangle: If all the sides of a triangle are of unequal lengths then the triangle is termed as Scalene Triangle. In the figure, is a scalene triangle.

ABC is an Equilateral Triangle. Where, AB = BC = CA = a
Then the area of Triangle ABC

(iii) Isosceles Triangle: If any two sides of a triangle are of equal length “a” unit and the third side is of “b” unit length then the triangle is said to be an Isosceles Triangle.

2. On the basis of Angle of a Triangle:
(i). Obtuse-angled Triangle: In a triangle, if one angle is more than 90° or the other two angles are less than 90° then the triangle is said to be an obtuse-angled triangle

iii). Right Angled Triangle: If in a triangle, two sides are perpendicular to each other or make 90° to each other then the triangle is said to be a Right-Angled Triangle. Here in the figure Δ ABC, AB is perpendicular, BC is base and AC is the hypotenuse of the Right-Angled triangle.

Pythagoras Theorem: In a Right-Angled triangle, the sum of the squares of base and perpendicular is equal to the square of the hypotenuse. i.e. (Hypotenuse)2 = (Base)2 + (Perpendicular)2 AC2 = BC2 + AB2

Ex.: A triangle with sides 3 cm, 4cm and 5cm form a right – angle triangle as 32 + 42 = 52.Pythagoras Triplets: It is a set of three positive whole numbers that represent the length of the three sides of the right-angle triangle.List of some Known and commonly used Pythagorean Triplets:(3,4,5), (5,12,13), (6,8,10), (8,15,17), (7,24,25), (9,40,41), (11,60,61), (20,21,29) etc.

Application of Pythagoras Theorem:

• If a2 + b2 = c2 then the triangle is a right-angled triangle.

• If a2 + b2 < c2 then the triangle is an obtuse-angled triangle.

• If a2 + b2 > c2 then the triangle is an acute-angled triangle.

Median & Centroid:

Median: It is the line segment that is drawn from a vertex of the triangle and joins the midpoint of the opposite side. There can be three medians that can be drawn from three vertices of the triangle

In the above diagram, AD is the median of ∆ABC which divides BC into two equal parts i.e. BD = DC

Centroid or Gravity Centre: The intersection points of all the three medians of a Triangle is called Centroid or Gravity Centre. It is denoted by ‘G’. It is also called as “Gravity Centre”. It divides each of the medians in the ratio 2:1.

Property 4: The sum of sides of a triangle is always greater than the sum of medians of the same triangle. AB+BC+CA > AD+BE+CF

Property 5: If m1, m2, and m3 are the lengths of medians of a Triangle. Then, The area of Triangle Where,

Property 6: If medians of Triangle intersect at 90° then,

Angle Bisector and Incentre:

Angle Bisector: It is a line segment that originates from a vertex and also bisects the same angle into equal parts.

Incentre (I): It is the point of intersection of all the three angle bisectors of the triangle.

Property 1: If in a triangle ABC with sides AB(=c), BC(=a) and AC(=b) also AD, BE and CF are the angle bisectors and I is Incentre. Then,

It can also be written as: AB × CD = AC × BD = AD2

Exterior Bisector Theorem: In a triangle, an external angle bisector divides the opposite side of the external angle in the ratio of two other sides of the triangle.

Circumcentre: It is the point of intersection of all three perpendicular bisectors of the triangle.
(i) For the Acute-angled triangle, the Circumcentre lies inside the triangle.
(ii) For the Obtuse-angled triangle, the Circumcentre lies outside the triangle.
(iii) For the Right-angled triangle, the Circumcentre lies on the midpoint of the hypotenuse of the given triangle.

Circum-Radius: The distance between the circumcentre and the three vertices of the triangle is always equal in length and this length is said as the Circumradius (R) of the triangle.
Circum-Circle: The circle drawn with circumradius as the radius of the circle, is called the circumcircle of the triangle. And it passes through all the three vertices of the triangle

For any triangle: 𝑅 = 𝑎𝑏𝑐/4𝐴; where a, b, c are the lengths of the sides and “A” is the area of the Triangle.

In the case of the Right-Angle Triangle, Circumradius will be half the length of the hypotenuse of the right-angled triangle.

BSC = 2A
ASC = 2B
ASB = 2C
Sine Rule: In a Δ ABC, the lengths of the sides AB, BC and CA are a, b, and c respectively and the angles on the vertices A, B, and C are ∠A, ∠B, and ∠C. Then according to the Sine Rule,

Altitude and Orthocentre:

Altitude or Height: It is the line segment that is drawn from a vertex perpendicularly on the opposite side of the triangle. Thus, there are three altitudes in the triangle

BOC = 180°-A BOC = 180°-A AOC = 180°-B AOB = 180°-C

Note:1. For the right-angle triangle, the orthocentre lies at the vertex containing the right angle.

2. In obtuse angle triangle it lies opposite to largest side and outside the triangle.

3. In an acute angle triangle, it lies inside the Triangle.

If “O” and “I” are Circumcentre and Incentre of the circle respectively and “R” and “r” are Circumradius and Inradius of the circle respectively. Then, Distance between O and I:  d2 = R (R – 2r)

Concept of Congruency of Triangles
Any two triangles ABC and XYZ are said to be congruent when every corresponding side has the same length, and every corresponding angle has the same measure. It is denoted by.

Two triangles are said to be similar if –
(i) Their corresponding angles are equal
(ii) Their corresponding sides are in proportion.
In two triangles, ΔABC and ΔXYZ are similar triangles when

and
It is denoted by.

Similarity Rules: 1. AAA Rule or AA Rule: If in two triangles, all three (two) corresponding angles are equal then the two triangles are termed as similar triangles by AAA (or AA) Rule.

2. SAS Rule: If in two triangles, two corresponding sides are in proportion and the corresponding angles contained between the two sides are equal then the two triangles are termed as similar triangles by SAS Rule.

3. SSS Rule: If in two triangles, all three sides and the corresponding sides are in proportion then the two triangles are termed as similar triangles by SSS Rule.

Properties of Similar Triangles – If Triangle ABC and XYZ are similar, then

1. The Ratio of their perimeters: Also, If m1, m2 are the length of medians of ∆ABC and ∆XYZ respectively and similarly h1 and h2 are Altitudes, I1 and I2 are angle bisectors, r1 and r2 are inradii and R1 and R2 are Cricumradii of respective triangles then –

2. The Ration of their Areas:
𝐀𝐥𝐬𝐨, If m1, m2 are the length of medians of ∆ABC and ∆XYZ respectively and similarly h1 and h2 are Altitudes, I1 and I2 are angle bisectors, r1 and r2 are inradii and R1 and R2 are Circumradii of respective triangles then –

The similarity in Right-Angle Triangle: In a Right Angle Triangle ABC, B = 90° and a perpendicular are drawn on Hypotenuse AC from vertex BC.

Then,

CIRCLE:
Definition: A plane 2- dimensional shape bounded by a single curved line, every point of which is equally distant from the point at the centre of the shape.

⦁ The distance from the centre is called the radius of the Circle. Normally, it is denoted by ‘r’.

⦁ Every point on the circumference of the circle is equidistant from its centre

⦁ The Circumference of the Circle is given as

⦁ The Ares bounded by the circumference of the Circle is

⦁ Area of Circle in terms of its Diameter

⦁ Area of Circle in terms of its Perimeter  Chord and TangentChord: Any line that joins two arbitrary points on the circumference of the Circle is called Chord. Ex. PQ is a chord in the below diagram.

• Maximum four tangents can be drawn between two Circles.

Length of Transverse Tangent
Where, d = distance between centres of two circles, r1, r2 = radius of the two circles
(ii) If both the circles are touching each other

Sector and Segment

Definition: The area between two radii and the connecting arc of a circle is called a sector.

For any circle, the angle between a tangent and a chord through the point of contact of the tangent is equal to the alternate segment.
Special cases:
(i)

Definition: A Quadrilateral is a closed shape formed by joining four non-linear points to each other

And

Definition: A four-sided shape that is made up of two pairs of parallel lines and that has four right angles; especially a shape in which one pair of lines is longer than the other pair

• Both the Diagonals divide the area into four equal parts.

• Perimeter of a Rectangle = 2 × (Length + Breadth) = 2(𝑙 + 𝑏)
• Perimeter of the rectangle if one side (l) and diagonal (d) are given.
Length of the Diagonal of a rectangle
• Area of Rectangle = Length × Breadth = 𝑙 × b
• Area of the rectangle if one side (l) and diagonal (d) are given.
• Area of the rectangle if perimeter (P) and diagonal (d) are given.
• If each diagonal of a rectangle is of length “d” and the area is “A” then the Perimeter of the rectangle
• The diagonals of the rectangle are equal to the length of the diameter of the circumcentre

Parallelogram

Definition: A quadrilateral in which opposite sides are equal and parallel is called a parallelogram. The diagonals of a parallelogram bisect each other

• A Diagonal of a Parallelogram does not bisect the Angles.
• The opposite angles are equal in a parallelogram.
• Area of a Parallelogram = Length × Height = AB x DE
• Area of a parallelogram where a and b are adjacent sides, d is the length of the diagonal connecting the ends of the two sides and
• Perimeter of a Parallelogram = 2 × (Length + Breadth)
• In a parallelogram, the sum of the squares of the diagonals = 2x(the sum of the squares of the two adjacent sides) i.e.,

Square

Definition: A four-sided shape that is made up of four straight sides that are the same length and that has four right angles. The diagonals of a square are equal and bisect each other at 900.

• The Diagonals of a Square are also angle bisectors.

Diagonals of a Square divide the area of Square in four equal parts.

• Area of a Square = Side × Side = (Side)2 = 𝑎2

• Perimeter of a square = 4 × Side = 4 × a

• Length of the Diagonal of a square

• The diagonals of the square are equal to the length of the diameter of the circumcentre.

Property: If ‘O’ is any point inside or outside a Square ABCD then,  OA2+OC2 = OB2+OD2

Rhombus

Definition: It is a flat shape with 4 equal straight sides.

• Diagonals of a Rhombus divide the area of the Rhombus into four equal parts.
• In a Rhombus the t opposite angles are equal and sum of the adjacent angles is 180°.
• Area of a rhombus
• Perimeter of a rhombus = 𝟒 × 𝐚
• Side of a rhombus Where d1 and d2 are two-diagonals.

If P, Q, R, and S are mid-points of sides AB, BC, CD, and DA respectively, then the shape formed by joining the points P, Q, R, and S will be a Rectangle.

Trapezium

Trapezium: A trapezoid is a 2-dimensional geometric figure with four sides, at least one set of which are parallel. The parallel sides are called the bases, while the other sides are called the legs.

• Area of a trapezium sum of parallel sides × distance between parallel sides

• In a Trapezium the diagonals cut each other in equal ratios.  𝑂𝐴/𝑂𝐶 = 𝑂𝐵/𝑂𝐷

• Perimeter of a Trapezium = Sum of All Sides

• If M and N are midpoints of Sides PS and QR respectively then the length of

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