Radius and diameter are the first vocabulary learned by you related to circles. The radius or the diameter determines the size of a circle. A circle is defined to be the set of all points in a plane that are equidistant from a fixed point called the center of the circle.Radius and Diameter

Radius, chord and diameter are different types of line segments that are defined in relation to a circle.

  • A radius of a circle is the line segment joining the center of the circle and any point on the circle. In the above diagram OA is a radius of the circle.
  • A chord of a circle is the line segment joining two points on the circle. Segments PQ and BC are both chords of the circle shown.
  • A diameter is a chord of the circle which passes through the center of the circle. Chord PQ is a diameter of the circle as it contains the center O of the point.

The term radius refers both to the line segment joining the center of a circle and a point on the circle as well its length.
Thus radius is the distance between the center and a point on the circle.



In the diagram the line segments OA, OB and OC are the radii (plural of radius) of the circle. As per the definition of the circle, all the points A, B and C are equidistant from the center O.
Hence, OA = OB = OC. Thus OAOBOC as equal line segments are congruent.
Theorem:
All radii of the circle are congruent.
A circle is thus defined by a unique radius. The size of a circle is determined by the measure of the radius, as both the circumference and the area are calculated using formulas containing radius 'r' as the variable.

Circumference of the circle= C = 2πr and Area of the circle = A = πr2.

Solved Example

Question: Find the circumference and area of the circle whose radius = 5 cm.
Solution:
 
Circumference of the circle C = 2πr = 2 x π x 5 = 10π cm ≈ 31.42 cm
Area if the circle A = πr2 = π x 5 x 5 = 25π cm2 = 78.54 cm2.
Radius measure is also useful in finding the location of a point with reference to a circle.  If C be a circle with center O and radius = r, and P any point then.
P lies inside the circle if OP < r.
P lies on the circle if OP = r
P lies outside the circle if OP > r.
 

Diameter is a chord of the circle made up of collinear radii.



In the above diagram the diameter AB is made up of two collinear radii OA and OB. As A and B are points on the circle. the diameter AB also satisfy the definition of a chord. Indeed diameters are the longest chords in a circle.
As the diameter of a circle is made up of two radii, its length is twice the length of a radius.
Diameter d = 2r.

The formula for the circumference of a circle is indeed derived from the definition of the irrational number π.
For any circle the circumference of the circle bears a constant ratio to its diameter, which is represented by π.
Thus, $\frac{C}{d}$ = π.
and so C = πd.
Substituting d = 2r, we get another version of the formula for the circumference as C = 2πr units.

Radius
Diameter
Radius is the line segment joining
the center of the circle to a point on
the circle.
Diameter is a special chord joining two
points of the circle and contain the center.
Radius is half the measure of diameter.
r = `1/2` d
The diameter of a circle is twice its radius.
d = 2r.
Two collinear radii join to form a diameter.
The diameter is bisected at the center to get
two radii.

The known diameter is divided by 2 to get the radius.
r = `1/2``1/2`d.

Solved Example

Question: A square is circumscribed by a circle as shown, Find the radius of the circle.

Diameter to Radius
Solution:
 
The diameter of the circle = Diagonal of the square.
Diagonal of the square AC = √2 side of the square AB = √2 x 6 = 6√2 cm.    
(Using special right angle (45º , 45º, 90º) relationship)
Diameter of the circle = 6√2 cm.
Radius of the circle = `1/2`d = `1/2` x 6√2 = 3√2 cm ≈ 4.24 cm.
 

The known radius is multiplied by 2 to get the diameter of the circle.
d = 2r.

Solved Example

Question: Find the diameter of the circle whose area is given to be = 16π sq.inches.
Solution:
 
The formula used to find the area of a circle A = πr2.
Solving this equation for r, we get r = $\sqrt{\frac{A}{\pi }}$.
                                                       = $\sqrt{\frac{16\pi }{\pi }}$ = √16 = 4 inches.
Hence the diameter of the circle d = 2r = 2 x 4 = 8 inches.