Circles — SSC CGL Quantitative Aptitude Formulas and Traps

Chords and Perpendiculars

Parts of a circle and the equal-chord facts

Longest chord

\text{Diameter} = 2r

$r$ the radius

Half-chord, distance from the centre and radius

Chord, distance and radius

r^2 = p^2 + \left(\frac{c}{2}\right)^2 \qquad c = 2\sqrt{r^2 - p^2}

$r$ the radius
$p$ the perpendicular distance of the chord from the centre
$c$ the length of the chord

A chord of the larger of two concentric circles

Concentric circles

\text{Chord} = 2\sqrt{R^2 - r^2} \qquad R^2 - r^2 = \left(\frac{C}{2}\right)^2 - \left(\frac{c}{2}\right)^2

$R$ the radius of the outer circle
$r$ the radius of the inner circle
$C$ the chord the line makes on the outer circle
$c$ the chord the same line makes on the inner circle

Two parallel chords and the radius

Two parallel chords

\text{gap} = p_1 - p_2 \text{ (same side)} \qquad \text{gap} = p_1 + p_2 \text{ (opposite sides)}

$p_1, p_2$ the distances of the two chords from the centre

Common traps

Calling the radius the longest chord

A radius ends at the centre, so it is not a chord at all. The longest chord is the diameter, twice the radius.

Mixing up a secant and a tangent

A secant cuts the circle at two points; a tangent touches it at one. A chord extended both ways is a secant.

Putting the whole chord in the right triangle

The triangle uses half the chord. For a chord of 70 cm at 12 cm from the centre, the radius is

\sqrt{144 + 1225} = 37

cm, not

\sqrt{144 + 4900}

.

Stopping at the half-chord

The triangle gives half the chord. When the question asks for the chord, double that answer before choosing an option.

Treating the inner radius as half the chord

The inner radius is the chord's distance from the centre. The half-chord is

\sqrt{R^2 - r^2}

, found from the right triangle.

Taking the point of contact as an end of the chord

A chord of the outer circle that touches the inner circle is touched at its midpoint, so each piece is half the chord.

Subtracting the distances for chords on opposite sides

When the centre lies between two parallel chords, the gap is the sum of their distances from the centre, not the difference.

Placing the longer chord farther from the centre

The longer chord is always the nearer one. If your working puts it farther away, a half-chord has been paired with the wrong distance.

Angles in a Circle

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Angle at the centre is twice the angle at the circumference

Centre and circumference

\angle AOB = 2\,\angle ACB \qquad \text{reflex } \angle AOB = 360^\circ - \angle AOB

$O$ the centre
$C$ a point on the arc that does not contain the arc AB

Angle in a semicircle and the circle through three points

Diameter and right angle

\angle ACB = 90^\circ \text{ when } AB \text{ is a diameter} \qquad R = \frac{\text{hypotenuse}}{2}

$R$ the radius of the circle through the three corners

Opposite angles of a cyclic quadrilateral

Cyclic quadrilateral

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

A chord's length from the angle it subtends

Chord from the centre angle

c = 2r\sin\frac{\theta}{2}

$c$ the chord
$\theta$ the angle the chord subtends at the centre

Common traps

Doubling instead of halving

The angle at the centre is the larger one. From the centre angle you halve to reach the circumference; from a circumference angle you double to reach the centre.

Halving the centre angle for a point on the minor arc

A point on the minor arc sees half the reflex angle, which is 180° minus half the centre angle. For a centre angle of 100°, that point sees 130°, not 50°.

Taking a shorter side of a right triangle as the diameter

Only the hypotenuse, the side facing the right angle, is a diameter of the circle through the three corners.

Assuming any three points lie on a circle

Three points have a circle through them only when they are not in a line. If two of the distances add up to the third, the answer is zero circles.

Adding adjacent angles to 180°

Only opposite angles of a cyclic quadrilateral add to 180°. Two neighbouring angles can add to anything.

Matching an exterior angle with the wrong interior angle

An exterior angle of a cyclic quadrilateral equals the interior angle at the opposite corner, not the interior angle beside it.

Using the angle at the circumference as the centre angle

A chord that subtends 30° at a point on the circle subtends 60° at the centre. Double first, then use the triangle of two radii.

Tangents from an External Point

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Tangent at right angles to the radius, and its length

Length of a tangent

t = \sqrt{d^2 - r^2}

$t$ the length of the tangent from the outside point
$d$ the distance from the outside point to the centre
$r$ the radius

Two tangents from one outside point

Two tangents from P

PA = PB \qquad \angle APB + \angle AOB = 180^\circ \qquad \text{Area}(OAPB) = r \times t

$r$ the radius
$t$ the length of each tangent

Tangent lengths from the corners of a triangle to its inscribed circle

Tangent from a corner

AF = s - a \qquad s = \frac{a + b + c}{2}

$a$ the side BC, opposite the corner A
$s$ half the perimeter of the triangle

The alternate segment theorem

Tangent and chord

\angle(\text{tangent}, CA) = \angle CBA = \frac{1}{2}\angle COA

$B$ a point on the arc on the other side of the chord CA
$O$ the centre

Common traps

Adding the squares when the distance is given

The line from the outside point to the centre is the hypotenuse. With that distance and the radius known, subtract the squares to get the tangent.

Answering with the radius when the diameter is asked

The right triangle gives the radius. A question about the diameter or the area of the circle needs one more step.

Setting the angle at the centre equal to the angle between the tangents

The two angles add to 180°, because the other two corners of OAPB are right angles. Tangents at 40° give 140° at the centre.

Using the whole angle between the tangents in triangle OAP

OP halves the angle between the tangents, so triangle OAP has half of it at P.

Subtracting a neighbouring side from the semi-perimeter

The tangent from a corner is the semi-perimeter minus the side opposite that corner, the one side that does not touch it.

Adding the tangent lengths once for the perimeter

Every tangent length appears in two sides, so the perimeter is twice the sum of the three tangent lengths.

Using the angle in the same segment instead of the alternate one

The tangent-chord angle equals the angle the chord subtends on the far side of the chord. A point on the near side sees 180° minus it.

Doubling the tangent-chord angle to get the angle at the circumference

The tangent-chord angle already equals the angle at the circumference. Only the centre angle is twice it.

Intersecting Chords and Power of a Point

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Two chords crossing inside a circle

Intersecting chords

PA \times PB = PC \times PD

$P$ the point where the two chords cross

Secants and a tangent from an outside point

Secants and a tangent from P

PA \times PB = PC \times PD \qquad PT^2 = PA \times PB

$PA$ P to the near point of the secant
$PB$ P to the far point, that is PA plus the chord AB
$PT$ the tangent from P

Power of a point from its distance to the centre

Power of a point

PA \times PB = d^2 - r^2 \text{ (P outside)} \qquad PA \times PB = r^2 - d^2 \text{ (P inside)}

$d$ the distance from P to the centre
$r$ the radius

Common traps

Multiplying a piece by the whole chord when the chords cross inside

Inside the circle both factors are pieces measured from the crossing point. The whole chord enters only when you add the pieces at the end.

Using the chord instead of the whole secant

Outside the circle the second factor is the full distance from P to the far point, the near length plus the chord, never the chord alone.

Squaring the near length instead of the tangent

The tangent squared equals near length times far length. The near length is not a tangent and is never squared on its own.

Using $r^2 - d^2$ for an outside point

For a point outside the circle d is larger than r, so the product is

d^2 - r^2

. The order flips only for a point inside.

Common Tangents and Common Chords

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Length of a direct common tangent

Direct common tangent

L = \sqrt{d^2 - (r_1 - r_2)^2}

$L$ the length of the direct common tangent
$d$ the distance between the centres
$r_1, r_2$ the two radii

Length of a transverse common tangent, and the two tangents together

Transverse common tangent

M = \sqrt{d^2 - (r_1 + r_2)^2} \qquad L^2 - M^2 = 4r_1 r_2

$M$ the length of the transverse common tangent
$L$ the length of the direct common tangent

The common chord of two intersecting circles

Common chord

h^2 = r_1^2 - x^2 = r_2^2 - (d - x)^2 \qquad \text{Chord} = 2h

$h$ half the common chord
$x$ the distance from the first centre to the chord's midpoint
$d$ the distance between the centres

Positions of two circles and the number of common tangents

Position	Condition	Common tangents
One outside the other	$d > r_1 + r_2$	4: two direct, two transverse
Touching externally	$d = r_1 + r_2$	3: two direct, one at the point of contact
Cutting at two points	$r_1 - r_2 < d < r_1 + r_2$	2: both direct
Touching internally	$d = r_1 - r_2$	1: at the point of contact
One inside the other	$d < r_1 - r_2$	0 Concentric circles (d = 0) belong here.

Work out

r_1 + r_2

and

r_1 - r_2

first, then see where

d

falls.

Common traps

Giving intersecting circles a transverse tangent

Two circles that cut at two points have exactly two common tangents, and both are direct.

Counting three tangents for internal touching

Circles touching externally have 3 common tangents. Circles touching internally have only 1, the tangent at the point of contact.

Using the sum of the radii for a direct tangent

The direct tangent uses the difference of the radii, because both radii point the same way from the tangent. The sum belongs to the transverse tangent.

Treating the tangent as the hypotenuse

The distance between the centres is the longest side. A direct common tangent is always shorter than that distance unless the radii are equal.

Swapping the sum and the difference

Direct tangent:

d^2 - (r_1 - r_2)^2

. Transverse tangent:

d^2 - (r_1 + r_2)^2

. The transverse one is always the shorter.

Finding a transverse tangent for circles that cut

If

d

is less than the sum of the radii,

d^2 - (r_1 + r_2)^2

is negative: there is no transverse tangent.

Putting the chord's midpoint halfway between unequal circles

Only equal circles have the common chord halfway between the centres. For unequal radii, find the split of d from the two right triangles.

Forgetting to double half the chord

The right triangles give half the common chord. Double it before matching the options.

Touching Circles

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Direct common tangent of two circles touching externally

Touching circles, direct tangent

L = 2\sqrt{r_1 r_2}

$L$ the direct common tangent between its two points of contact
$r_1, r_2$ the radii of the two touching circles

Centres, the point of contact, and a circle touching two others

Distance between touching centres

d = r_1 + r_2 \text{ (external)} \qquad d = r_1 - r_2 \text{ (internal)}

$d$ the distance between the centres

Common traps

Using $2\sqrt{r_1 r_2}$ for circles that do not touch

The shortcut assumes the centres are exactly

r_1 + r_2

apart. When the distance is given and is different, use

\sqrt{d^2 - (r_1 - r_2)^2}

.

Taking the sum of the radii as the tangent

r_1 + r_2

is the distance between the centres of touching circles. The tangent is

2\sqrt{r_1 r_2}

, which is never longer than that.

Adding the radii for circles touching internally

Circles touching internally have one inside the other, so the distance between their centres is the difference of the radii.

Using a radius alone for the distance to a third touching circle

A circle of radius

r

touching a circle of radius

r_1

externally has its centre

r_1 + r

away, the sum of both radii.

Circles formulas

Chords and Perpendiculars

Parts of a circle and the equal-chord facts

Half-chord, distance from the centre and radius

A chord of the larger of two concentric circles

Two parallel chords and the radius

Common traps

Angles in a Circle

Angle at the centre is twice the angle at the circumference

Angle in a semicircle and the circle through three points

Opposite angles of a cyclic quadrilateral

A chord's length from the angle it subtends

Common traps

Tangents from an External Point

Tangent at right angles to the radius, and its length

Two tangents from one outside point

Tangent lengths from the corners of a triangle to its inscribed circle

The alternate segment theorem

Common traps

Intersecting Chords and Power of a Point

Two chords crossing inside a circle

Secants and a tangent from an outside point

Power of a point from its distance to the centre

Common traps

Common Tangents and Common Chords

Length of a direct common tangent

Length of a transverse common tangent, and the two tangents together

The common chord of two intersecting circles

Positions of two circles and the number of common tangents

Common traps

Touching Circles

Direct common tangent of two circles touching externally

Centres, the point of contact, and a circle touching two others

Common traps

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