Area Questions

FACTS  AND  FORMULAE  FOR  AREA  QUESTIONS

FUNDAMENTAL CONCEPTS :

I. Results on Triangles:

1. Sum of the angles of  a triangle is ${180}^o$

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

3. Pythagoras Theorem : In a right - angled triangle,

${\left(Hypotenuse\right)}^2={\left(Base\right)}^2+{\left(Height\right)}^2$

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

5. The point where the three medians of a triangle meet, is called 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.

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.

II.Results on Quadrilaterals :

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.

IMPORTANT FORMULAE

I. 

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

$\therefore Length =\frac{Area}{Breadth} and Breadth=\frac{Area}{Length}$

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

II. Area of square = ${\left(side\right)}^2=\frac{1}{2}{\left(diagonal\right)}^2$ 

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

IV.

1. Area of a triangle =$\frac{1}{2}\times base\times height$

2. Area of a triangle = $\sqrt{s(s-a)(s-b)(s-c)}$, where a, b, c are the sides of the triangle and $s=\frac{1}{2}\left(a+b+c\right)$

3. Area of an equilateral triangle =$\frac{\sqrt{3}}{4}\times {\left(side\right)}^2$

4. Radius of incircle of an equilateral triangle of side $a=\frac{a}{2\sqrt{3}}$

5. Radius of circumcircle of an equilateral triangle of side $a=\frac{a}{\sqrt{3}}$

6. Radius of incircle of a triangle of area $∆$ and semi-perimeter $s=\frac{∆}{s}$

V.

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

2. Area of a rhombus = $\frac{1}{2}\times \left(Product of diagonals\right)$

3. Area of a trapezium = $\frac{1}{2}\times (sum of parallel sides)\times dis\tan ce between them$

VI.

1. Area of a cicle = ${\mathrm{\pi R}}^2$, where R is the radius.

2. Circumference of a circle = $2\mathrm{\pi R}$.

3. Length of an arc = $\frac{2\mathrm{\pi R\theta }}{360}$, where $\theta$ is the central angle.

4. Area of a sector = $\frac{1}{2}\left(arc\times R\right)=\frac{{\mathrm{\pi R}}^2\mathrm{\theta }}{360}$ 

VII.

1. Area of a semi-circle = $\frac{{\mathrm{\pi R}}^2}{2}$

2. Circumference of a semi - circle = $\mathrm{\pi R}$