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A circumcircle or circumscribed circle refers to that circle of a polygon that moves through all the vertices of the polygon. The middle of the circle is known as circumcentre, its radius is known as circumradius. A polygon that has a circumcircle is known as a cyclic polygon and concyclic polygon as all the vertices of the polygon are concyclic. E.g., trapezoids, simple polygons are cyclic polygons. A related concept is the least bounding circle, which contains the polygon inside it as it is the smallest circle.
Though even a polygon has a circumscribed circle, it cannot match with the smallest circle with the least bounding circle. E.g. for an obtuse triangle, the least bounding circle has a long diameter and does not make its way through the opposite vertex. Circumcircle not only comes under a circle but it is also found in a triangle, quadrilaterals, etc. all the cyclic polygons.
CIRCUMCIRCLE OF A TRIANGLE:
As we know that every triangle is cyclic so all triangle has circumscribed circle. When there is a meeting of 3 lines which that makes a 90-degree angle on one side of the triangle and it crosses that side from the middle.
The position of the circumscribed circle lies on a variety of triangles:
- If it is an acute triangle, then the circumcentre of that triangle will lie inside it only.
- If it is an obtuse triangle, the circumcentre of that triangle will lie outside the triangle.
- If it is a right triangle, then the circumcentre will be seen on either 1 of the sides of the triangle
DIAMETER OF THE TRIANGLE:
The diameter of a triangle can be calculated by the height of one of the side of the triangle. So the diameter of a circle let’s take ΔABC is:
D = a b c/2 area (since area=abc/(4r))
= a b c/ 2 (√s(s-a)(s-b)(s-c) )
= 2a b c/ √(a+b+c)(-a+b+c)(a-b+c)(a+b-c)
= 2a b c /√ (a²+b²+c²)² -2(a4+b4+c4)
Here, s refers to the sides of the triangle so sides = (a +b +c)/2 which is the semi perimeter.
RADIUS:
The point at which the 3 lines stay is the circumscribed circle of that 3 points and is often called the circle of infinite radius. The radius can be calculated by:
R = a bc/ 4 A = a bc /√ (a²+b²+c²)² -2(a4+b4+c4 )
Radius for right handed triangle can be calculated by :
R = a²+b² / 2 = c/2
AREA:
Area can be calculated by the formula:
A=π /16(a b c/ A)2 = (abc)²/(a² + b²+ c²)2-2(a4 + b4 + c4).
PERIMETER:
The perimeter of the triangle can be evaluated by the formula:
P =π /2 (abc/A)
CIRCUMCIRCLES OF QUADILATERALS:
RADIUS:
R = √abcd(a²+b²+c²+d²)+ (ab) ² (c²+d²)+(cd)² (a²+b²)/ 4A.
R = √l²+w² / 2 .
For quadrilaterals the formula is:
R = abcd(a²+b²+c²+d²)+ (ab )² (c²+d²)+(cd)² (a²+b²) ⁄ (a² + b²+ c²+ d²)²+8 a b c d -2 (a4 + b4 + c4 +d4)
CIRCUMCIRLCLES OF POLYGONS:
RADIUS:
The radius of a normal shaped polygon can be calculated by:
R = s / 2 sin (180/n).
DIAMETER :
The diameter can be calculated by:
D = 1/ sin (180/n) s.
AREA AND PERIMETER:
The area can be calculated by the formula:
A= / 4 sin²(180/n)s²
The perimeter can be calculated by :
P = / sin (180/n ) s.
APPLICATION-BASED QUESTION:
Question.
A regular pentagon inside a circle of radius 6 cm. suppose the pentagon is named as ABCDE. Then 1(AC) = ?.
Solution:
Circle of radius = 6 cm.
Length of polygon = 2 * r * sin(180°/ n)
Here n = 5, r = 6
so, l = 2 * 6 * sin(36°)
l = 7.05 cm.
draw a polygon and then,
In ∆ AOC,<AOC = 72°*2 = 144°
<OCA = <OAC = 180°-144° / 2 = 18°.
AC = 2*r*cos(<OCA)= 2*6*0.95
AC= 11.4 cm.
CONCLUSION:
The circumcenter is known as the center of the circumscribed circle and all the vertices of a triangle are nearest from the circumcenter. Though the circumcentre stays inside the triangle in an acute-angled triangle. And circumcentre lies outside the triangle when it is an obtuse angle triangle. When it is a hypotenuse side of a right angle triangle then the circumcentre lies inside the midpoint. the circumcentre of any triangle can be made by drawing the perpendicular bisector of any of the two sides of the triangle.
