Aptitude - Area

FUNDAMENTAL CONCEPTS

1. Results on Triangles:

   1. Sum of the angles of a triangle is 180°.

   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.

   4. The line joining the mid-point of a side of a triangle to the positive vertex is called the median.

   5. The point where the three medians of a triangle meet, is called centroid. The centroid divided each of the medians in the ratio 2 : 1.

   6. In an isosceles triangle, the altitude from the vertex bisects the base.

   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.

2. Results on Quadrilaterals:

   1. The diagonals of a parallelogram bisect each other.

   2. Each diagonal of a parallelogram divides it into 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.

IMPORTANT FORMULAE

1. 1.   Area of a rectangle = (Length x Breadth).

   Length =		Area		and Breadth =		Area		.
   		Breadth				Length

   2.   Perimeter of a rectangle = 2(Length + Breadth).

2. Area of a square = (side)² = (diagonal)².

3. Area of 4 walls of a room = 2 (Length + Breadth) x Height.

4. 1.   Area of a triangle = x Base x Height.

   2.   Area of a triangle = s(s-a)(s-b)(s-c)
         where a, b, c are the sides of the triangle and s = (a + b + c).

   3.   Area of an equilateral triangle =	3	x (side)2.
   	4

   4.   Radius of incircle of an equilateral triangle of side a =	a	.
   	23

   5.   Radius of circumcircle of an equilateral triangle of side a =	a	.
   	3

   6.   Radius of incircle of a triangle of area and semi-perimeter s =
   	s

5. 1.   Area of parallelogram = (Base x Height).

   2.   Area of a rhombus = x (Product of diagonals).

   3.   Area of a trapezium = x (sum of parallel sides) x distance between them.

6. 1.   Area of a circle = R², where R is the radius.

   2.   Circumference of a circle = 2R.

   3.   Length of an arc =	2R	, where is the central angle.
   	360

   4.   Area of a sector =	1	(arc x R)	=	R2	.
   	2			360

7. 1.   Circumference of a semi-circle = R.

   2.   Area of semi-circle =	R2	.
   	2