Circle, radius, chord and diameter
A circle is the set of all points that are at the same distance from a fixed point, the centre. That distance is the radius (r). All radii of a circle are equal.
A chord is a straight line joining two points of the circle. The diameter is a chord that passes through the centre. It is the longest chord and is 2 × r.
A chord is shorter when it is farther from the centre. If a chord is d away from the centre, its length is 2 × √(r² − d²). The line from the centre perpendicular to a chord cuts the chord in half.
A line and a circle
Let d be the distance from the centre to a line. Compare d with the radius r.
- d < r: the line cuts the circle at 2 points. It is a secant.
- d = r: the line touches at exactly 1 point. It is a tangent.
- d > r: the line and circle have no common point.
Tangent and secant
The point where a tangent touches is the point of contact. The tangent is perpendicular to the radius drawn to that point (90°).
From a point outside the circle you can draw two tangents. They are equal in length. If the outside point is D from the centre, each tangent length is √(D² − r²).
A secant is a line that crosses the circle at two points. Its piece inside the circle is a chord.
Circle inscribed in an angle
A circle can sit inside an angle and touch both arms. Its centre is the same distance from both arms, so the centre lies on the angle bisector. The two touching points are the same distance from the corner (equal tangents). Move the centre along the bisector and the circle grows or shrinks: it still touches both arms.
If the angle is 2φ and the centre is t from the corner, the radius is t × sin φ.
Incircle and circumcircle of a triangle
The incircle sits inside the triangle and touches all three sides. Its centre is where the three angle bisectors meet. For a right triangle with legs a, b and hypotenuse c, its radius is (a + b − c) / 2.
The circumcircle goes through all three corners. Its centre is where the three perpendicular bisectors of the sides meet. For a right triangle the centre is the middle of the hypotenuse and the radius is c / 2.
Try it: tangent with a coin
Roll a coin along a ruler. Where the coin touches the ruler is the point of contact. Hold a pencil from the middle of the coin to that point and check with a set-square: the pencil (radius) and the ruler (tangent) make a right angle. Predict first: if you lift the ruler a little, will it touch at one point or two? Check.
Key formulas and definitions
- Diameter = 2 × radius
- Chord length = 2 × √(r² − d²), d = distance from centre
- Line and circle: d < r secant, d = r tangent, d > r none
- Tangent ⟂ radius; tangent length = √(D² − r²)
- Right triangle: circumradius = c/2, inradius = (a + b − c)/2
Worked examples
1. The radius of a circle is 7 cm. Find the diameter.
Diameter = 2 x 7 = 14 cm.
2. A circle has radius 5 cm. A chord is 3 cm from the centre. Find the chord length.
Half chord = √(5² − 3²) = √16 = 4. Chord = 8 cm.
3. A circle has radius 6 cm. A line is 6 cm from the centre. What is the line?
d = r, so the line is a tangent.
4. A point is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from the point.
The tangent is perpendicular to the radius, so tangent² = 13² − 5² = 169 − 25 = 144. Tangent = 12 cm.
5. From an outside point P, tangent PA = 9 cm. Find the other tangent PB.
Tangents from one outside point are equal, so PB = 9 cm.
6. An angle of 60° has a circle inside it with centre 10 cm from the corner. Find the radius.
The centre is on the bisector, so the half angle is 30°. Radius = 10 x sin 30° = 10 x 0.5 = 5 cm.
7. A right triangle has sides 6, 8, 10. Find the radii of its circumcircle and incircle.
Circumradius = c/2 = 10/2 = 5. Inradius = (6 + 8 − 10)/2 = 2.
Common mistakes
- Thinking any chord is a diameter. Only a chord through the centre is.
- Saying a tangent touches at two points. It touches at exactly one.
- Forgetting the tangent is at 90° to the radius, not to the diameter line in general.
- Mixing the circles: incircle is inside and touches sides; circumcircle goes through the corners.