If R is the circumradius of the triangle,
Inscribed Circle
A circle which touches the three sides of a triangle a, b, and c internally is called an inscribed circle or incircle. The centerof this circleis called the incenter and the radius of the circle is called the inradius.
If r be the radius, then
Radii of Inscribed and Circumscribed Circles
The radius of the inscribed circle r of a regular polygon is given by
The Radius of a circumscribed circle of a regular polygon is given by
Area of a Regular Polygon
]]>Then
(i) If you fix
and vary , the sum of the squares of the diagonals is constant. Indeed .(ii) If you fix
and vary , the ratio of the diagonals is constant. Indeed .]]>Length of the side, s is given by
This is wrong! It is
]]>Area, A is given by
The perimeter, P, is given by the formula
Where n is the number of sides of the n-gon, and s is the side length.
Area of a regular n-gon if the diagonal length is known (formulated myself ):
Where n is the number of sides of the n-gon and R is the length of a diagonal.
(Oh I see ganesh has the first formula in a different form)
]]>The radius of a circle inscribed in a triangle of sides a, b, and c is given by
where the semiperimeter s of the triangle is given by
Circle Circumscribing a Triangle
The radius of a circle circumscribing a triangle of sides a, b, and c is given by
where once again s is the semiperimeter of the triangle.
Regular n-gon Inscribed in a Circle
The area of a regular n-gon inscribed in a circle of radius r is given by
The perimeter of the n-gon is given by
Regular n-gon Circumscribing a Circle
The area of a regular n-gon circumscribing a circle of radius r is given by
The perimeter of the n-gon is given by
Ellipse
The area of an ellipse of semi-major axis a and semi-minor axis b is given by
The perimeter of the ellipse is given by
or approximately
]]>Length of an Arc
Sector of a circle is the area of a circle between two radii.
Area of a sector of a circle.
Segment:- A sector minus the triangle formed by the two radii is called the segment. Area of the segment,
Perimeter of the segment :- Length of the arc + length of the chord
Area = 1/2 x (Product of diagonals) x (sine of the angle between them)
Area = 1/2 x diagonal x sum of the perpendiculars to it from opposite sides.
Parallelogram
Area = Base x Height
Area = Product of any two adjacent sides x (sine of the included angle)
Perimter = 2 (a+b) where a and b are the adjacent sides.
Rhombus
Area = 1/2 x Product of the diagonals
Area = Product of the adjacent sides x sine of the angle between them.
Trapezium
Area = 1/2 x (Sum of the parallel sides) x (height)
Regular Hexagon
A polygon is formed by a closed series of straight lines (segments).
A regular polygon is one with all sides and angles equal.
An equilateral triangle, a square etc. are examples of regular polygons.
The Sum of all angles of a polygon with 'n' sides is
Sum of all the exterior angles = 360°
No. of sides = 360°/Exterior angle.
Perimeter = n x s
No. of diagonals =
]]>There is also a formula for the area of a trapezoid, where the length of all sides are known:
Another formula for the area can be used when all that is known are the lengths of the four sides. If the sides are a, b, c and d, and a and c are parallel (where a is the longer parallel side), then:
Square:-
A square consists of four equal sides and each side is perpendicular to the other, and opposite sides are parallel.
If a is the length of the side of a square, the are A is
Perimeter = 4a
Length of the diagonal is equal to
Rectangle.
A rectangle is a quadrilateral with opposite sides parallel to each other and each side perpendicular to the adjacent side. The opposite sides are equal.
Area = l x b (where l and b are the sides)
Perimeter = 2(l+b)
Length of the diagonal =