Regular Polygons


A many sided closed figure is called a polygon. If all the sides of a polygon are equal, it is called a regular polygon.




1. Interior Angle + Exterior angle = 180°
2. \(P = n \times a\) (n = number of sides, a = side)
3. Sum of exterior angles \(= 360°\)
4. Sum of Interior angles \(= (2n – 4) 90°\)
5. Each Interior angle \(= [(\frac{2n – 4)}{n}] \times 90°\)
6. Each exterior angle \(= \frac{360}{n}\)
7.\( A = \frac{1}{2} P\)
\( r = \frac{1}{2 }\) n a r
P = perimeter (r = ⊥ from centre to any one side = radius of encircle)


Regular Hexagon

1. Each interior angle = 120°
2. Sum of Interior angles = 720°
3. Each exterior angle = 60°
4. Area \(= (\frac{3\sqrt{3}}{2}) \times a^2 (a = side)\)
5. P = 6a


Circle

A circle is the path traversed by a point which moves in such a way that its distance from a fixed point always remains constant.





C is called centre and R the radius of the circle. Circumference \(= 2 \pi R = \pi D\)

1. Area \(= A = \pi R^2 = \frac{\pi D^2}{4} (D = Diameter = 2R) \pi = \frac{22}{7} or 3.14\)
2. Length of arc of a circle is given by \(\ell (AB) = (\frac{θ°}{360°}) \times 2 \pi R\)
3. Area of Sector \(ABC = (\frac{θ}{360}) \times \pi R^2 = \frac{\frac{1}{2}}{(AB) \times R}\)
4. Distance travelled by a wheel in n revolutions \(= n\)\( \times\) circumference
5. Angle at the centre made by an arc = twice the angle made by the arc at any point on the remaining part of the circumference.



We have \(\angle APB =\frac{1}{2} \angle AOB = 30° = \angle AQB\)
6. Angle in a semicircle is a right angle.