Cheatsheet: Geometry & Mensuration | General Aptitude for GATE - Mechanical Engineering PDF Download

Theory

• When two angles A and B are complementary, Sum of A and B = 90 degrees
• When two angles A and B are supplementary, Sum of A and B = 180
• When two lines intersect, opposite angles are equal. Adjacent angles are supplementary.
• When any number of lines intersect at a point, the sum of all angles formed = 360 degree
  

Formula

Scalene Triangle
Perimeter (P) = a + b + c

where s = P/2  

Isosceles Triangle
Perimeter (P) = a +a + b = 2a+b

Equilateral Triangle
Perimeter (P) = a + a + a = 3a  

Right Angled Triangle
Perimeter (P) = a + b + c

Any closed two dimensional figure formed by four straight lines is called a quadrilateral.
The sum of the angles of a quadrilateral is equal to 360°.  

In the above quadrilateral Perimeter (P) = p + q + r + s

Square
Square is a quadrilateral in which all the four sides are equal and the angle between any two adjacent sides is 90°.  

Perimeter (P) = 4a

d² = a² + a²   
→ d = √2 a
Cost of fencing a square plot/field is equal to the product of perimeter and unit cost of fencing i.e. Total Cost = Perimeter × Unit Cost per meter. Similarly the cost of levelling a square plot/field is equal to the product of its area and unit cost.  
∴ Cost of levelling = Area × Unit Cost per square meter

Rectangle
A quadrilateral having two pairs of equal opposite sides and a right angle between any two adjacent sides is called a rectangle.
The longer dimension is called the length (l) and the shorter dimension is called the breadth (b).  

The line joining any two opposite sides is called the diagonal (d).  
Perimeter (P) = 2 (l + b)
Area (A) = l × b = lb
l² + b² = d²

Circle
A circle is a geometrical figure consisting of all those points in a plane which are at a given distance from a fixed point in the same plane. The fixed point is called the centre and the constant distance is known as the radius.

A circle with centre O and radius r is generally denoted by C (O, r) 

Circumference
The perimeter of a circle is called its Circumference.
The ratio of the Circumference (C) of a circle and its diameter (D) is always constant, which is π.
C/D = π  
⇒ C = πD = 2πr

Area of a Circle
The Area (A) of a circle of radius (r) is given by A = πr²

Solved Example