How to Identify Circumscribed Figures and Their Properties
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 What Does Circumscribed Mean in Maths?
1. What does it mean to circumscribe a shape in mathematics?
In geometry, to circumscribe means to draw a geometric figure around another one so that it touches the inner figure at several points without cutting through it. For example, a circle is circumscribed about a triangle when the circle passes through all three vertices of the triangle. The outer figure is called the circumscribed figure, and its centre is known as the circumcenter.
2. What is the main difference between an inscribed figure and a circumscribed figure?
The main difference lies in their positioning relative to another shape:
- An inscribed figure is drawn inside another figure, touching its boundary from within. For instance, a circle is inscribed in a triangle if all three sides of the triangle are tangent to the circle.
- A circumscribed figure is drawn outside another figure, completely enclosing it. For example, a triangle is circumscribed about a circle if all its sides are tangent to the circle.
Essentially, if shape A is inscribed in shape B, then shape B is circumscribed about shape A.
3. How is the centre of a circle circumscribing a triangle (the circumcenter) located?
The centre of a circumscribed circle, known as the circumcenter, is found by constructing the perpendicular bisectors of the sides of the triangle. The point where all three perpendicular bisectors intersect is the circumcenter. This point is unique for any given triangle and is equidistant from all three vertices, with this distance being the radius of the circumscribed circle (the circumradius).
4. Does the circumcenter of a triangle always lie inside the triangle?
No, the position of the circumcenter depends on the type of triangle:
- For an acute-angled triangle, the circumcenter lies inside the triangle.
- For a right-angled triangle, the circumcenter is located at the midpoint of the hypotenuse.
- For an obtuse-angled triangle, the circumcenter lies outside the triangle.
5. Why is the circumcenter of a triangle equidistant from all three of its vertices?
This property arises from the definition of a perpendicular bisector. Any point on the perpendicular bisector of a line segment is equidistant from the two endpoints of that segment. Since the circumcenter is the intersection point of the perpendicular bisectors of all three sides (say AB, BC, and CA), it must be equidistant from points A and B, from B and C, and from C and A. Therefore, it is equidistant from all three vertices A, B, and C.
6. Can a circle be circumscribed around any quadrilateral?
No, a circle can only be circumscribed around a special type of quadrilateral known as a cyclic quadrilateral. A key property of a cyclic quadrilateral is that the sum of its opposite angles is always 180 degrees (i.e., ∠A + ∠C = 180° and ∠B + ∠D = 180°). If this condition is not met, it is impossible to draw a single circle that passes through all four vertices.
7. What is a real-world application of finding a circumcenter?
A common application is in logistics and infrastructure planning. For instance, if you need to place a service facility, like a cell phone tower, a water sprinkler, or a public well, so that it is at an equal distance from three different locations (e.g., three houses or three towns), you would find the circumcenter of the triangle formed by these three locations. Placing the facility at the circumcenter ensures it serves all three points equally.
