Perimeter and Area

Class 09 Maths

Perimeter is the total length around its border. A square with side a units has perimeter 4a units. An equilateral triangle with side a units has perimeter 3a units.

The perimeter of a rectangle with length a units and width b units is 2(a + b) units.

Circumference of Circle

In ancient days people realised that the ratio of the circumference to the diameter of the circle does not change if we change the size of the circle. This ratio is called the ‘C/D ratio’ of the circle.

This constant is now called π. π is the constant circumference to diameter ratio for all circles, and is approximately equal to 22/7 or 3.14.

$$ C = 2\pi r $$

Length of an Arc of a Circle

The circumference of a circle of diameter d is πd. Since the diameter is twice the radius, we can also write the circumference of a circle as 2πr, where r is the radius.

If the arc is AB, and it subtends an angle θ° at the centre O of the circle,

$$ \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r $$

Area of a Rectangle

The area of a rectangle with sides a units and b units is ab sq. units.

Area of a Parallelogram

The area of the parallelogram is equal to base × height = bh

Area of a Triangle

The formula for the area a triangle is

$$ A = \frac{1}{2}bh $$

Theorem: A median of a triangle divides it into two triangles with equal area.

Heron’s formula

If ΔABC has side lengths BC = a, CA = b, and AB = c, then its area can be found by first computing the semi-perimeter.

$$ s = \frac{a+b+c}{2} $$

$$ A = \sqrt{s(s-a)(s-b)(s-c)} $$

Area of a Circle

$$ A = \pi r^2 $$

Area of Sector of a Circle

A sector of a circle is the region bounded by an arc and the two radii containing the endpoints of the arc.

$$ A = \frac{\theta}{360^\circ}\,\pi r^2 $$

A segment of a circle is the region bounded by an arc of the circle and the chord joining the endpoints of the arc.