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SPI Mathematics · Formula sheet

Circles formulas

8 formulas, 1 reference table and 18 common traps for SPI Mathematics Circles, grouped by subtopic.

Full notes with worked examples

Tangent and Radius

Learn this subtopic in the notes

Length of a tangent from a right triangle

Tangent length

t=d2−r2d=r2+t2t = \sqrt{d^2 - r^2} \qquad d = \sqrt{r^2 + t^2}
  • ttlength of the tangent from the outside point
  • dddistance from the outside point to the centre
  • rrradius of the circle

Angles in the radius and tangent triangle

Angles at the point of contact

∠OQP=90∘∠QOP+∠OPQ=90∘sin⁡30∘=cos⁡60∘=12\angle OQP = 90^\circ \qquad \angle QOP + \angle OPQ = 90^\circ \qquad \sin 30^\circ = \cos 60^\circ = \tfrac{1}{2}

What a tangent is and how many there are

SituationFact
A line touching the circle at one pointA tangent
A line cutting the circle at two pointsA secant
Tangents from a point inside the circleNone
Tangents from a point on the circleExactly one
Tangents from a point outside the circleExactly two
Tangents at the two ends of a diameterParallel to each other
Distance between two parallel tangentsThe diameter, 2r
It is the diameter, not the radius: a circle of radius 4 cm has parallel tangents 8 cm apart.
The tangent at a point is the one line through it that is perpendicular to the radius there.

Common traps

Giving the radius as the gap between parallel tangents

Parallel tangents touch the circle at opposite ends of a diameter, so the gap is 2r2r. For a radius of 3 cm the answer is 6 cm, not 3 cm.

Two tangents from a point on the circle

Only a point outside the circle has two tangents. A point on the circle has exactly one, and a point inside has none.

Adding the squares when one is the hypotenuse

The distance to the centre is the longest side. With distance 13 cm and radius 5 cm the tangent is 169−25=12\sqrt{169 - 25} = 12 cm; adding gives 194\sqrt{194}, which is wrong.

Subtracting the lengths instead of their squares

With distance 13 cm and radius 5 cm, 13−5=813 - 5 = 8 cm is not the tangent. Square, subtract, then take the square root.

Putting the right angle at the wrong corner

The 90∘90^\circ angle is at the point of contact, between the radius and the tangent. It is never at the centre or at the outside point.

Using sin where cos is needed

If the angle at the centre is 60∘60^\circ, the radius is the side next to it, so use cos⁡60∘=radiushypotenuse\cos 60^\circ = \frac{\text{radius}}{\text{hypotenuse}}. The radius is opposite the angle at the outside point.

Tangents from an External Point

Learn this subtopic in the notes

Two tangents from one point are equal

Equal tangents

PA=PBAB+CD=AD+BC (circle touching all four sides)PA = PB \qquad AB + CD = AD + BC \text{ (circle touching all four sides)}

The angle between two tangents and the angle at the centre

Tangents and radii

∠APB+∠AOB=180∘\angle APB + \angle AOB = 180^\circ

The circle inside a triangle

Radius of the incircle

r=a+b−c2 (right triangle)r=Areas,s=perimeter2r = \frac{a + b - c}{2} \text{ (right triangle)} \qquad r = \frac{\text{Area}}{s}, \quad s = \frac{\text{perimeter}}{2}

Common traps

Matching the wrong pair of pieces

When B lies on the tangent CQ and BR is a tangent from B, the equal pair is BR and BQ. Find BQ as CQ minus CB first; BR is not equal to CB.

Thinking only the first point gives equal tangents

Every outside point has two equal tangents, including the points where a third tangent cuts the first two. Use each of them.

Taking 90 minus the angle instead of 180 minus it

Tangents at 40∘40^\circ need radii at 140∘140^\circ, not 50∘50^\circ. The two angles add to 180∘180^\circ because the other two corners are right angles.

Forgetting the two right angles

The angles of OAPB add to 360∘360^\circ, and the corners at the points of contact are 90∘90^\circ each. Subtract both before finding the last angle.

Dividing the area by the whole perimeter

In r=Areasr = \frac{\text{Area}}{s}, ss is half the perimeter. Dividing by the whole perimeter gives half the true radius.

Halving the hypotenuse

Half the hypotenuse of a right triangle is the radius of the circle through its three corners, a different circle. The circle inside has radius a+b−c2\frac{a + b - c}{2}.

Chords, Angles and Cyclic Quadrilaterals

Learn this subtopic in the notes

The perpendicular from the centre bisects a chord

Chord length

chord=2r2−d2r2=d2+(chord2)2\text{chord} = 2\sqrt{r^2 - d^2} \qquad r^2 = d^2 + \left(\tfrac{\text{chord}}{2}\right)^2
  • rrradius of the circle
  • dddistance of the chord from the centre

Angles made by an arc

Angles in a circle

∠AOB=2 ∠ACBangle in a semicircle=90∘\angle AOB = 2\,\angle ACB \qquad \text{angle in a semicircle} = 90^\circ

Opposite angles of a cyclic quadrilateral

Cyclic quadrilateral

∠A+∠C=180∘∠B+∠D=180∘\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ

Common traps

Stopping at the half chord

Pythagoras gives half the chord. With radius 10 cm and distance 6 cm the half chord is 8 cm, and the chord is 16 cm.

Putting the whole chord in the right triangle

The right triangle has the radius, the distance from the centre and HALF the chord. Halve the chord before you square it.

Halving when you should double

The angle at the centre is the larger one. If the angle at the circle is 64∘64^\circ, the angle at the centre is 128∘128^\circ, not 32∘32^\circ.

Using an angle from the other side of the chord

Only points on the same side of a chord see it at equal angles. Two points on opposite sides make angles that add to 180∘180^\circ.

Adding neighbouring angles to 180

Only opposite angles of a cyclic quadrilateral add to 180∘180^\circ. Angles A and B are neighbours, and their sum can be anything.

Matching the exterior angle with the angle next to it

The exterior angle at C equals the interior angle at A, the opposite corner. The interior angle at C, right next to it, is the one it adds to 180∘180^\circ with.

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