Aptitude:: Area

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FUNDAMENTAL CONCEPTS

1. Results on Triangles:

i. Sum of the angles of a triangle is 180.

ii. The sum of any two sides of a triangle is greater than the third side.

iii. Pythagoras Theorem:

In a right-angled triangle,

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

v. 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.

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

vii. The median of a triangle divides it into two triangles of the same area.

viii. 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:

i. The diagonals of a parallelogram bisect each other.

ii. Each diagonal of a parallelogram divides it into triangles of the same area.

iii. The diagonals of a rectangle are equal and bisect each other.

iv. The diagonals of a square are equal and bisect each other at right angles.

v. The diagonals of a rhombus are unequal and bisect each other at right angles.

vi. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

vii. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

IMPORTANT FORMULAE

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

Length = and Breadth =.

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

II. Area of a square =

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

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

2. Area of a triangle =

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

3. Area of an equilateral triangle =

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

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

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

V. 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.

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

2. Circumference of a circle = 2R.

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

4. Area of a sector = .

VII. 1. Circumference of a semi-circle = .

2. Area of semi-circle = .