What Is the Circumscribed Circle Formula and How to Find It
Circumscribed Meaning
When a shape is restricted within another shape, it is termed as circumscribed. It cannot pass the outer figure. Whenever we draw such a figure, the purpose is to draw it in a manner that it will not cross the outer figure though all the vertices will touch the figure. It is a must for the outer figure to touch the inside figure at the vertices or maximum points. If this condition is not fulfilled, then the Shape will not fall under the category of circumscribed figures. It is also necessary for the inner figure to not cross the outer figure in any case. Geometrical shapes that are included in circumscribing include square, circle, rectangle, triangle, for a quadrilateral as well.
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Circumscribed Geometry Definition
Circumscribed in geometry means where two figures are drawn, one inside the other. The figure is drawn such that it will not intersect at any point and will only touch all the maximum points of the inner figure. If the outer figure is not touching the maximum points, then it is not circumscribed in shape.
E.g. A circle drawn inside the hexagon falls under the category of circumscribed Shape. Make sure the circle is touching all the six vertices without cutting it. If the circle is cutting it, then it will not come under circumscribed shapes. The list of circumscribed shapes includes:-
Polygon
Angle
Rectangle
Circle
Triangle
Quadrilateral
Circumscribed Circle:
When a circle is drawn outside a figure in a manner that it is passing through all the vertices of the figure but not intersecting it, then it is termed as circumcircle. In the figure, all the vertices must be touched by the circle. Whether it is a triangle inside the circle or whether a hexagon or an octagon, the circle must be touching all the vertices appropriately and not intersecting at any point.
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In the above figure, a triangle is inside a circle. It is termed as a circumscribed circle.
Circumscribed Triangle:
Whenever there are any other geometrical shapes inside the triangle and the sides of the triangle touch the maximum points while not intersecting it, then the triangle is termed as a circumscribed triangle. One can consider a circle inside a triangle (like in the figure) or any other shape. But as per the condition, it should not intersect any side of the triangle.
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Circumscribed Angle:
When a circle is enclosed inside the arms of the angle, the angle is termed as the circumscribed angle. For the same as well, the condition applied that it is not passing through the circle. It is only touching the edges in the form of tangents. Tangents are straight lines touching the curved surface at a specific point.
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Circumscribed Polygon:
A polygon is a plane figure with at least three straight lines and angles. Whenever there is a geometrical shape available inside a polygon in a manner that all the vertices or maximum points are touching the inside figure then the polygon is termed as a circumscribed polygon. Both regular and irregular polygons can be circumscribed polygons. It is necessary that all the vertices or maximum points are touched and it is not intersecting any of the sides.
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Circumscribed Quadrilateral:
When a quadrilateral surrounds a circle in such a way that the sides of the quadrilateral are tangents to the circle, then it is termed as a circumscribed quadrilateral. A quadrilateral is a polygon with four vertices and four edges. As per the condition, the sides must not intersect.
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Circumscribed Hexagon:
A Hexagon is a polygon of 6 sides. A circumscribed hexagon is a hexagon with a geometrical shape enclosed inside it. The maximum points of the geometrical Shape must be touching all the sides of the hexagon. In case, any of the sides is missed, and then it will not be considered as a circumscribed hexagon. If the geometrical figure is a circle, all the sides of the hexagon will appear to be tangents to the figure. For example, if there is a circle closed inside the hexagon, all the sides will appear to be tangent to it.
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Circumscribed Rectangle:
When there is a geometrical figure available inside the rectangle and the vertices of the figure are touching the sides, then it is termed as a circumscribed rectangle. The maximum points of the inner figure must be touched by the outer figure. In case any of the points are missed, then it will not be termed as a circumscribed rectangle.
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Example:
When there is a circle enclosed inside a square, it is termed as a circumscribed square.
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When there is a circle enclose inside a triangle, it is termed as a circumscribed triangle
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When there is a circle enclosed inside a quadrilateral, it is termed as a circumscribed quadrilateral.
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FAQs on Circumscribed Circle Explained for Triangles and Polygons
1. What does circumscribed mean in geometry?
In geometry, circumscribed means a shape is drawn around another shape so that it touches it at specific points without cutting through it. For example:
- A circle is circumscribed about a polygon when it passes through all the polygon’s vertices.
- A polygon is circumscribed about a circle when all its sides touch the circle exactly once.
2. What is a circumscribed circle?
A circumscribed circle (or circumcircle) is a circle that passes through all the vertices of a polygon. In a triangle:
- The circle passes through all three vertices.
- The center of this circle is called the circumcenter.
3. How do you find the center of a circumscribed circle?
The center of a circumscribed circle is found by locating the intersection of the perpendicular bisectors of the sides of the triangle. Steps:
- Draw the perpendicular bisector of one side.
- Draw the perpendicular bisector of another side.
- Their intersection point is the circumcenter.
4. What is the formula for the radius of a circumscribed circle of a triangle?
The radius of the circumscribed circle of a triangle is given by R = a / (2 sin A). More generally:
- R = a / (2 sin A) = b / (2 sin B) = c / (2 sin C)
5. What is the difference between inscribed and circumscribed?
The difference is that an inscribed figure is drawn inside another, while a circumscribed figure is drawn around another. For example:
- An inscribed circle lies inside a triangle and touches all sides.
- A circumscribed circle lies outside the triangle and passes through all vertices.
6. Does every triangle have a circumscribed circle?
Yes, every triangle has exactly one circumscribed circle. The perpendicular bisectors of the three sides always intersect at a single point called the circumcenter. Depending on the triangle type:
- Acute triangle: circumcenter lies inside.
- Right triangle: circumcenter lies at the midpoint of the hypotenuse.
- Obtuse triangle: circumcenter lies outside.
7. What is a circumscribed polygon?
A circumscribed polygon is a polygon drawn around a circle so that each side touches the circle at exactly one point. In this case:
- The circle is called an inscribed circle or incircle.
- Each side of the polygon is tangent to the circle.
8. How do you find the radius of a circumscribed circle of a right triangle?
For a right triangle, the radius of the circumscribed circle is half the length of the hypotenuse. Formula:
- R = c / 2
9. Can you give an example of finding a circumscribed circle radius?
Yes, the radius can be calculated using the formula R = a / (2 sin A). Example:
- Let side a = 8 cm and angle A = 30°.
- sin 30° = 1/2.
- R = 8 / (2 × 1/2) = 8 / 1 = 8 cm.
10. What are the properties of a circumscribed circle?
The main properties of a circumscribed circle (circumcircle) are:
- It passes through all vertices of a polygon.
- Its center is the circumcenter, formed by perpendicular bisectors.
- The circumcenter is equidistant from all vertices.
- Every triangle has one unique circumcircle.
