SSC CGL Maths · Circles

Chords and Perpendiculars

The perpendicular from the centre cuts a chord into two equal halves, so the radius, the half-chord and the chord's distance from the centre always make a right triangle.

All Circles notes

Why this matters

Forty questions in the bank, 34 EASY and 6 MODERATE, the largest page of the chapter. Most are one right triangle: two of radius, half-chord and distance are given and you find the third. The MODERATE ones put two chords, or two circles with one centre, into the same figure.

Concept 1 of 4: Parts of a circle and the equal-chord facts

A circle is every point at one fixed distance, the radius, from the centre. Straight lines meet it in a small number of ways, and each way has a name SSC expects you to know. Equal chords behave the same in every respect, because a chord's position is fixed by its length.

Definition

A chord joins two points of the circle. A diameter is a chord through the centre: it is $2r$ long and is the longest chord. Its distance from the centre is 0.
A radius is not a chord: one end is at the centre, not on the circle.
A secant is a line that cuts the circle at two points (a chord extended both ways). A tangent touches it at exactly one point, and the radius to that point is perpendicular to the tangent.
The perpendicular from the centre to a chord bisects the chord, and the line from the centre to the midpoint of a chord is perpendicular to it.
Equal chords subtend equal angles at the centre, cut off equal arcs and are equally far from the centre. Each of these facts works backwards too.
Of two chords, the longer one is nearer the centre.
Circles with equal radii are congruent, and since the circumference is $2\pi r$ , circumferences are in the same ratio as radii.

Longest chord

\text{Diameter} = 2r

rthe radius

Worked example

Which statement is false? (a) A chord nearer the centre is longer. (b) Equal chords subtend equal angles at the centre. (c) A secant meets a circle at exactly one point. (d) The perpendicular from the centre to a chord bisects the chord.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 3 · Q74Easy

Example 1 · Circles · Chords and Perpendiculars

In a circle with center O, chords AB and CD are equal in length. If the distance from O to AB is 8 cm, what is the distance from O to CD?

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.

Drill 7 more on parts of a circle and the equal-chord facts

Concept 2 of 4: Half-chord, distance from the centre and radius

Drop the perpendicular from the centre O to a chord AB. It meets the chord at its midpoint M. Now OM, MA and the radius OA form a right triangle with the right angle at M, and the radius is the longest side.

Definition

Let the chord be $c$ long and $p$ from the centre. The right triangle has sides $p$ , $\frac{c}{2}$ and hypotenuse $r$ .
Halve the chord first. Putting the whole chord into the triangle is the commonest slip.
"The perpendicular from the centre", "the distance of the chord from the centre" and "the segment from the centre to the midpoint" all mean $p$ .
If the question asks for the chord, double the half-chord at the end.
Fast route: spot a triple. 3, 4, 5 and 5, 12, 13 and 8, 15, 17 and 7, 24, 25 and 20, 21, 29, and their multiples, fill most of these questions. If two sides fit a triple, read off the third. Use it only when the numbers really fit: 6 and 8 give 10, but 6 and 9 are not part of any of these.
For an approximate answer, place the square between two perfect squares: $\sqrt{193}$ lies between 13 and 14, close to 14 because 193 is close to 196.

Chord, distance and radius

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

rthe radius
pthe perpendicular distance of the chord from the centre
cthe length of the chord

Worked example

A chord of a circle is 30 cm long and lies 8 cm from the centre. Find the radius.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 1 · Q74Easy

Example 2 · Circles · Chords and Perpendiculars

A circle has a radius of 13 cm. A chord is drawn at a distance of 5 cm from the center. What is the length of the chord?

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.

Drill 21 more on half-chord, distance from the centre and radius

Concept 3 of 4: A chord of the larger of two concentric circles

Concentric circles share one centre. A chord of the bigger circle that just touches the smaller circle is a tangent to it, so the small radius to the touching point is perpendicular to the chord. That small radius is the chord's distance from the centre, and the touching point is the chord's midpoint.

Definition

Concentric circles have the same centre and different radii.
A chord of the outer circle (radius $R$ ) that touches the inner circle (radius $r$ ) is $r$ from the centre, so its length is $2\sqrt{R^2 - r^2}$ .
The point of contact is the midpoint of that chord.
A line that cuts both circles makes two chords at the same distance from the centre. Write the right triangle for each and the distance cancels: $R^2 - r^2 = \left(\frac{C}{2}\right)^2 - \left(\frac{c}{2}\right)^2$ .

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

Rthe radius of the outer circle
rthe radius of the inner circle
Cthe chord the line makes on the outer circle
cthe chord the same line makes on the inner circle

Worked example

Two concentric circles have radii 13 cm and 12 cm. Find the length of the chord of the larger circle that touches the smaller circle.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 24 Sep 2024, Shift 2 · Q57Easy

Example 3 · Circles · Chords and Perpendiculars

The radii of two concentric circles are 26 cm and 10 cm. If the chord of the greater circle is a tangent to the smaller circle, then find the length of that chord.

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.

Drill 4 more on a chord of the larger of two concentric circles

Concept 4 of 4: Two parallel chords and the radius

Two parallel chords are both perpendicular to one line through the centre. Each chord makes its own right triangle with the same radius, and the gap between the chords is either the difference or the sum of their two distances from the centre.

Definition

Each chord's distance is $p = \sqrt{r^2 - \left(\frac{c}{2}\right)^2}$ . The longer chord is nearer the centre.
Same side of the centre: gap $= p_1 - p_2$ . Opposite sides: gap $= p_1 + p_2$ .
Radius unknown: call the nearer distance $p$ . Write $r^2$ twice, once with each half-chord, and set them equal. The $p^2$ terms cancel and a simple equation for $p$ is left.
If the stem does not say which side, try one. A negative distance means the chords lie the other way.
Fast route: test the options. For each radius, find both distances and see whether they differ (or add) to the gap. It is quick because the options are few, and safe because a wrong radius cannot pass both checks.

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

Worked example

Two parallel chords of a circle of radius 25 cm are 40 cm and 30 cm long. Find the distance between them when they lie (a) on the same side of the centre and (b) on opposite sides.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 1 · Q74Moderate

Example 4 · Circles · Chords and Perpendiculars

There are two parallel chords measuring 16 cm and 12 cm, both situated on the same side of the center of a circle. The space between the two chords is 2 cm. What is the radius of the circle?

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.

Drill 4 more on two parallel chords and the radius

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Parts of a circle and the equal-chord facts
Longest chord
$\text{Diameter} = 2r$
Half-chord, distance from the centre and radius
Chord, distance and radius
$r^2 = p^2 + \left(\frac{c}{2}\right)^2 \qquad c = 2\sqrt{r^2 - p^2}$
A chord of the larger of two concentric circles
Concentric circles
$\text{Chord} = 2\sqrt{R^2 - r^2} \qquad R^2 - r^2 = \left(\frac{C}{2}\right)^2 - \left(\frac{c}{2}\right)^2$
Two parallel chords and the radius
Two parallel chords
$\text{gap} = p_1 - p_2 \text{ (same side)} \qquad \text{gap} = p_1 + p_2 \text{ (opposite sides)}$

Watch out for (8)

Calling the radius the longest chord
→ Parts of a circle and the equal-chord facts
Mixing up a secant and a tangent
→ Parts of a circle and the equal-chord facts
Putting the whole chord in the right triangle
→ Half-chord, distance from the centre and radius
Stopping at the half-chord
→ Half-chord, distance from the centre and radius
Treating the inner radius as half the chord
→ A chord of the larger of two concentric circles
Taking the point of contact as an end of the chord
→ A chord of the larger of two concentric circles
Subtracting the distances for chords on opposite sides
→ Two parallel chords and the radius
Placing the longer chord farther from the centre
→ Two parallel chords and the radius

Test yourself on Circles

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter test Drill all 40 questions