# Formulas and Theories: Figures

1) For any regular polygon , the sum of the exterior angles is equal to 360 degrees
hence measure of any external angle is equal to 360/n. ( where n is the number of sides)

(2) If any parallelogram can be inscribed in a circle , it must be a rectangle.

(3) If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sies equal).

(4) For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .

(5) Area of a regular hexagon : root(3)*3/2*(side)*(side)

(6) For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is
0.5*d1*d2, where d1,d2 are the lenghts of the diagonals.

(7) For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2

(8) For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the internal opposite angle.

(9) the quadrilateral formed by joining the angular bisectors of another quadrilateral is
always a rectangle.

(10) Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.

(11) Area of a trapezium = 1/2 * (sum of parallel sids) * height = median * height
where median is the line joining the midpoints of the oblique sides.

(12) For a cyclic quadrilateral , area = root( s* (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2

Here are some neat shortcuts on Simple/Compound Interest.

(13) Area of a hexagon = root(3) * 3 * (side)^2

(14) for similar cones , ratio of radii = ratio of their bases.

(15)  Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for
[(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]

If a1/b1 = a2/b2 = a3/b3 = ………….. , then each ratio is equal to
(k1*a1+ k2*a2+k3*a3+…………..) / (k1*b1+ k2*b2+k3*b3+…………..) , which is also equal to
(a1+a2+a3+…………./b1+b2+b3+……….)

(16) Let W be any point inside a rectangle ABCD .
Then
WD^2 + WB^2 = WC^2 + WA^2