Words you need first
A tangent is a line that touches a circle at exactly one point. The point is the point of contact. A tangent is always at 90° to the radius that reaches the point of contact.
A secant is a line that cuts the circle at two points. The centre is O, the radius is r. In this lesson D means the distance from a point to the centre O.
Two shapes are tangent to each other when they touch and do not cross. We say they have a tangency there.
Tangents from a point outside the circle
How to draw them. 1) Join the point P to the centre O. 2) Find the middle M of OP. 3) With centre M and radius MO, draw a circle through O and P. 4) It cuts the first circle at two points, T1 and T2. 5) Join P to T1 and P to T2. These are the tangents.
Why it works. The angle in a half-circle is 90°. So angle OT1P = 90°. A line at 90° to the radius at T1 is a tangent.
Length. Triangle OT1P has a right angle at T1. So PT² = D² - r². The two tangents from P are equal in length and make equal angles with OP.
Tangent at a point on the circle. If the point is already on the circle, just draw the line at 90° to the radius there.
Common tangents of two circles
A line that touches two circles is a common tangent. There are two kinds. Outer (direct) tangents leave both circles on the same side. Crossing (transverse) tangents pass between the circles, with the circles on opposite sides.
With radii r1, r2 and centre distance d:
- d > r1 + r2 (apart): 4 tangents.
- d = r1 + r2 (touch outside): 3 tangents.
- |r1 - r2| < d < r1 + r2 (cut each other): 2 tangents.
- d = |r1 - r2| (touch inside): 1 tangent.
- d < |r1 - r2| (one inside the other): 0.
Length of the part between the touching points: outer = √(d² - (r1 - r2)²), crossing = √(d² - (r1 + r2)²).
Construction idea. For an outer tangent, draw a helper circle of radius r1 - r2 round the bigger centre. Draw a tangent to it from the other centre. Then move that line outwards by r2. For a crossing tangent use r1 + r2 instead.
This is the shape of a belt on two pulleys. A crossing belt makes a figure 8.
Joining lines and circles with tangent arcs
To round a corner between two lines with radius R: draw a line parallel to each, at distance R, on the inside. They meet at C, the centre of the arc. Drop perpendiculars from C to the two lines. The feet are the touching points. Draw the arc between them.
If the corner has half-angle φ, then the centre is R / sin φ from the corner, and the touching points are R / tan φ from the corner. For a 90° corner, φ = 45°, so both distances are easy: R and R√2.
The same idea joins two circles with an arc. If the new arc touches the outside of a circle, the centres and the touching point are on one line, and the distance between centres is the sum of the radii. If it touches the inside, it is the difference.
Rule to remember: the point of contact always lies on the line of the two centres (or on the perpendicular to the line).
Technical curves: ovals, ovoids and spirals
Drawers build smooth curves from circle arcs. Each joint is a tangent, so there is no corner.
- Spiral with two centres. Take two points A and B. Draw a half-circle with centre A, then one with centre B, and so on. Radii grow by the distance AB each time.
- Oval. Four arcs and four centres. Two big arcs and two small arcs, joined at points on the lines of centres. It looks like an ellipse but is made only of circle arcs.
- Ovoid (egg). Like an oval but with a different small arc, so one end is narrower, like an egg.
These are not true ellipses. They are quick to draw with a compass, and that is why technical drawing uses them.
Power of a point
Take a point P and a circle with centre O and radius r. Draw any line through P that cuts the circle at A and B. Then PA x PB is the same number for every line. This number is the power of P.
Power of P = D² - r², where D = PO.
- P outside: power is positive. Also equals PT² for a tangent PT.
- P on the circle: power is 0.
- P inside: power is negative. We use r² - D² as the product of the two parts of any chord through P.
Why. Two triangles PAT and PTB share an angle at P and have equal angles by the alternate segment idea, so they are similar. That gives PA / PT = PT / PB, so PA x PB = PT². For a point inside, triangles PAC and PDB are similar in the same way.
Two lines from P: PA x PB = PC x PD. This helps find a missing length.
Radical axis and radical centre
For two circles, the radical axis is the set of points that have equal power to both. For a point outside both circles this means equal tangent lengths.
Because power is D² - r², equal power gives a straight line, at 90° to the line of centres. If the circles cut at two points, the axis is the line through those points (the common chord). If they touch, it is the common tangent at the contact point. If the circles are apart, it lies between them, and its distance from the first centre is (d² + r1² - r2²) / (2d).
With three circles you get three radical axes. They meet at one point, the radical centre. From the radical centre, the tangent lengths to all three circles are equal. So a circle drawn from this point with that radius cuts all three circles at right angles.
How to find the axis of two circles that do not meet. Draw any circle that cuts both. Draw the two common chords. They meet at a point of the axis. Do it twice to get two points and join them.
Where tangencies are used in design
- Roads and railways: a straight, then a tangent arc, then a straight. No sudden turn.
- Logos and fonts: letters like S, B and P are straight lines joined by arcs.
- Gears and belts: common tangents give the belt path, the belt length is 2 x tangent length + the arcs on the wheels.
- Furniture and phone corners: rounded corners are tangent arcs (fillets).
- Architecture: arches and spiral stairs are made from arcs with shared tangents.
In a drawing, always mark the points of contact. They are where one curve ends and the next begins.
Try it: a belt on two cans
Take two cans or glasses of different sizes. Put them on a table a little apart. Wrap a string tightly round both like a belt. The straight parts of the string are the outer tangents. Predict first: if you move the cans closer, will the string get shorter? Check it. Now cross the string into a figure 8 and see the crossing tangents. Measure the length and compare it with the formula in the 3D.
Also try the tangent from a point: put a pencil dot P away from a coin, hold a ruler through P so that it only touches the coin, and mark the touching point. Do the same on the other side. Measure both lengths from P. They are equal.
Key formulas and definitions
- Tangent ⟂ radius at the point of contact
- Tangent length from P: PT = √(D² - r²), D = PO
- Outer common tangent: √(d² - (r1 - r2)²); crossing: √(d² - (r1 + r2)²)
- Arc joining two lines (half-angle φ): centre at R / sin φ, contact at R / tan φ from the corner
- Power of P = PA x PB = D² - r² (outside) = r² - D² (inside, for chord parts)
- Radical axis of two circles: points with equal power, a line ⟂ line of centres, at (d² + r1² - r2²) / (2d) from centre 1
Worked examples
1. A point P is 13 cm from the centre of a circle of radius 5 cm. Find the tangent length from P.
PT = √(D² - r²) = √(169 - 25) = √144 = 12 cm.
2. A point P is 10 cm from the centre of a circle of radius 6 cm. Find the power of P. A secant from P cuts the circle at A and B, with PA = 4 cm. Find PB.
Power = D² - r² = 100 - 36 = 64. PA x PB = 64, so PB = 64 / 4 = 16 cm.
3. Two secants from P: the first has PA = 3 and PB = 8. The second has PC = 4. Find PD.
PA x PB = 3 x 8 = 24. PC x PD = 24, so PD = 24 / 4 = 6.
4. Two chords cross at a point inside a circle. One chord is cut into parts 4 and 9. One part of the other chord is 6. Find the other part.
The product of the parts is the same: 4 x 9 = 36. So 6 x x = 36 and x = 6.
5. Two circles have radii 5 and 2, and their centres are 25 apart. Find the length of the outer and the crossing common tangent (between touching points).
Outer: √(25² - (5 - 2)²) = √(625 - 9) = √616 ≈ 24.8. Crossing: √(25² - (5 + 2)²) = √(625 - 49) = √576 = 24.
6. Circle 1 has radius 5, circle 2 has radius 3, and their centres are 8 apart. Find where the radical axis cuts the line of centres, measured from centre 1.
x = (d² + r1² - r2²) / (2d) = (64 + 25 - 9) / 16 = 80 / 16 = 5. The axis passes 5 from centre 1. This is exactly the point where the circles meet, because 5 + 3 = 8: they touch there.
7. Two straight roads meet at a right angle. Join them with an arc of radius 4 m. How far from the corner are the touching points and the arc centre?
Half-angle φ = 45°. Touching points: R / tan 45° = 4 m from the corner. Centre: R / sin 45° = 4 / 0.707 ≈ 5.66 m from the corner.
Common mistakes
- Saying a tangent from a point is only one line. From an outside point there are exactly two, and they are equal.
- Forgetting the 90° angle between tangent and radius, and drawing the tangent so that it cuts the circle.
- Using PA x PB with A and B wrong. Always measure from P to each point on the same line, not from A to B.
- Putting the point of contact of two touching circles anywhere. It must lie on the line of centres.