Quantitative Aptitude Area Important formulae and concepts

Area important concepts

I. Triangles

1. Sum of the angles of a triangle is 180⁰.(<A+<B+<C=180⁰)(Fig.1.)
2. The sum of any two sides of a triangle is greater than the third side.
3. Pythagoras Theorem : In a right-angled triangle,(Hypotenuse)² = (Base)² + (Height)².(Fig.2)
4. The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.(In fig 1. AF,BD and CE are medians)
5. The point where the three medians of a triangle meet, is called centroid.(In fig1 G is the centroid) The centroid divides each of the medians in the ratio 2:1
6. In an isosceles triangle, the altitude from the vertex bisects the base.(In fig.3. CD=DB)
7. The median of a triangle divides it into two triangles of the same area.
8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.      
9. 
   

Types of Triangles Based on sides

• Scalene Triangle:A triangle  having all three un equal sides
• Isosceles Triangle:A triangle having two equal sides.
• Equilateral Triangle:A triangle having three equal sides.

Types of Triangles Based on angles

• If each angle is less than 90^{0 ,}then the triangle is called an acute angled triangle
  If any one angle is right angled,then the triangle is called right angled trianle.
  If any one angle is greater than 90⁰ then the triangle is called obtuse angled triangle.

II.  Quadrilaterals (Rectangle ,square, parallelogram, rhombus, trapezium )

1.  The diagonals of a parallelogram bisect each other.
2.    Each diagonal of a parallelogram divides it into two-triangles of the same area.
3.   The diagonals of a rectangle are equal and bisect each other.
4.    The diagonals of a square are equal and bisect each other at right angles.
5. The diagonals of a rhombus are unequal  and bisect each other at right angles.
6. A parallelogram and a rectangle on the same base and between-the same parallels are equal in area.
7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

  Area -Important Formulae

•  Area of a rectangle = (Length x Breadth).
• Perimeter of a rectangle = 2 (Length + Breadth).
• Area of a square = (side)² = ½  (diagonal).
• Area of a triangle = ½ ( Base x Height)
• Area of a triangle = (This formula is known as herons Formula)
• where a, b, c are the sides of the triangle and s = ½  (a + b+ c).
• Area of an equilateral triangle =                                                            
• Radius of incircle of an equilateral triangle of side a =
• Radius of circumcircle  of an equilateral triangle of side a = 
• Radius of incircle of triangle of area Δ and semi-perimeter s =
• Area of a parallelogram = (Base x Height).
• Area of a rhombus = 1 /2  (Product of diagonals).
• Area of a trapezium = 1 /2  (sum of parallel sides) x distance between them.
•  Area of a circle =, where r. is the radius.
• Circumference of a circle = = 2πr
• Length of an arc =                                             
• Area of a sector = ½ (Arc × r )=