Circles

Definition Of Circle

A circle is a simple closed shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.

Simply put, a circle is a plane figure bounded by a curved line called circumference, which is always equidistant from the centre.

Types and Parts of a Circle

Below are the different parts of a circle

• circumference: the distance around the edge of the circle.
• diameter: the distance from one side of the circle to the other, passing through the centre.
• radius: the distance from the centre of the circle to the edge.

• Tangent: a line that intersects the circumference of a circle in exactly one point. A tangent is perpendicular to the radius from the point where the tangent touches the circle.
• Arc: part of the circumference of the circle.
• Chord: a line that goes from one point to another on the circle’s circumference, without passing through the centre.
• Sector: a section of the circle defined by two lines from the centre to the circumference. It looks like a piece of pizza. However, there are different kinds of “slices.” A minor sector has an angle at the centre of the circle of less than 180°. A major sector has an angle at the centre of the circle of more than 180°. There are are some special kinds of sectors:
  • Semicircle: a sector made from half the area of a circle.
  • Quadrant: a sector made from a quarter of the area of a circle.
• Segment: an area made from a chord and an arc of the circle. Each chord produces two segments: a major segment (the large shape) and the minor segment (the small shape, like the one shaded in the picture above).
• Secant: a line that touches and passes through two points on a circle’s circumference.
• Cyclic quadrilateral: a four-sided shape whose four points touch the circumference of a circle.

Bisection, construction of tangent and normal to given circle

How to construct a Centre of Circle using just a compass and a straightedge

Steps:

• Draw a line across the circle to make a “chord”
• Construct the perpendicular bisector of that chord to make a diameter of the circle
• Construct the perpendicular bisector of that diameter to get the centre of the circle

How to construct a Tangent from a Point to a Circle using just a compass and a straightedge

Steps:

• Draw a line connecting the point to the centre of the circle
• Construct the perpendicular bisector of that line
• Place the compass on the midpoint, adjust its length to reach the end point, and draw an arc across the circle
• Where the arc crosses the circle will be the tangent points.

How to Inscribe a Circle in a Triangle using just a compass and a straightedge

Steps:

• Bisect one of the angles
• Bisect another angle
• Where they cross is the centre of the inscribed circle
• Construct a perpendicular from the centre point to one side of the triangle
• Place compass on the centre point, adjust its length to where the perpendicular crosses the triangle, and draw your inscribed circle!

How to Circumscribe a Circle on a Triangle using just a compass and a straightedge

• Construct the perpendicular bisector of one side of triangle
• Construct the perpendicular bisector of another side
• Where they cross is the centre of the Circumscribed circle
• Place compass on the centre point, adjust its length to reach any corner of the triangle, and draw your Circumscribed circle!

How to construct a Circle touching 3 Points using just a compass and a straightedge

Steps:

• Join up the points to form two lines
• Construct the perpendicular bisector of one line
• Construct the perpendicular bisector of the other line
• Where they cross is the centre of the circle
• Place compass on the centre point, adjust its length to reach any point, and draw your circle!

ASSESSMENT

1. What is a circle?
2. What are the parts of a circle?
3. What are the steps taken to inscribe a circle in a triangle?
4. What is the difference between a circumference and a diameter?