Every Circle Question Hides a Right-Angled Triangle
Learn the parts of a circle, use the radius and diameter relationship, apply the right angle in a semicircle, find tangent lengths by Pythagoras, and draw chords, sectors and segments.
Why does Pythagoras keep appearing in circle problems?
Because two of the circle's most useful properties produce right angles, and a right angle is an invitation to use Pythagoras.
The first: a tangent is always perpendicular to the radius at the point where it touches. So the centre, the touching point and any external point form a right-angled triangle without you drawing anything extra.
The second: the perpendicular from the centre to a chord bisects it. That splits the chord into two equal halves and creates another right-angled triangle, with the radius as its hypotenuse.
And a third: the angle in a semicircle is a right angle, which puts a right angle on the circle itself.
So the working method for almost every numerical question in this chapter is the same — find the right-angled triangle, label the hypotenuse, apply . This page covers the ICSE Class 8 Mathematics chapter on circles.
The first: a tangent is always perpendicular to the radius at the point where it touches. So the centre, the touching point and any external point form a right-angled triangle without you drawing anything extra.
The second: the perpendicular from the centre to a chord bisects it. That splits the chord into two equal halves and creates another right-angled triangle, with the radius as its hypotenuse.
And a third: the angle in a semicircle is a right angle, which puts a right angle on the circle itself.
So the working method for almost every numerical question in this chapter is the same — find the right-angled triangle, label the hypotenuse, apply . This page covers the ICSE Class 8 Mathematics chapter on circles.
What are the parts of a circle called?
A circle is the set of all points at a fixed distance from a fixed point, and every term in this chapter describes some piece of that picture.
- Centre — the fixed point, usually labelled .
- Radius — a line segment from the centre to any point on the circle. Every radius of a circle has the same length.
- Diameter — a chord passing through the centre. It is the longest chord the circle has.
- Chord — a segment joining any two points on the circle.
- Arc — a part of the circle itself, the curved edge. The shorter one is the minor arc, the longer the major arc.
- Sector — the region enclosed by two radii and an arc, shaped like a slice of cake.
- Segment — the region enclosed by a chord and an arc, the shape left when a slice is cut straight across.
- Tangent — a line touching the circle at exactly one point.
- Secant — a line cutting the circle at two points.
- Circumference — the total distance around the circle.
Sector against segment is the distinction most often confused, and the difference is one line: a sector is cut by two straight radii from the centre, while a segment is cut by one straight chord that need not pass anywhere near the centre. A slice of a round roti cut from the middle outwards is a sector; the piece you get by slicing straight across one side is a segment.
Tangent against secant, and the boundary between them. Slide a line towards a circle from outside. At first it misses entirely — no points in common. Then it touches at exactly one point, and there it is a tangent. Push it further in and it cuts at two points, becoming a secant. A line and a circle can share , or points and never more, and the tangent is precisely the borderline case.
The chord that cannot exist. Since the diameter is the longest chord, a circle of radius cm has a maximum chord length of cm. A question describing a chord of cm in that circle describes nothing — and noticing that is worth as much as any calculation.
- Centre — the fixed point, usually labelled .
- Radius — a line segment from the centre to any point on the circle. Every radius of a circle has the same length.
- Diameter — a chord passing through the centre. It is the longest chord the circle has.
- Chord — a segment joining any two points on the circle.
- Arc — a part of the circle itself, the curved edge. The shorter one is the minor arc, the longer the major arc.
- Sector — the region enclosed by two radii and an arc, shaped like a slice of cake.
- Segment — the region enclosed by a chord and an arc, the shape left when a slice is cut straight across.
- Tangent — a line touching the circle at exactly one point.
- Secant — a line cutting the circle at two points.
- Circumference — the total distance around the circle.
Sector against segment is the distinction most often confused, and the difference is one line: a sector is cut by two straight radii from the centre, while a segment is cut by one straight chord that need not pass anywhere near the centre. A slice of a round roti cut from the middle outwards is a sector; the piece you get by slicing straight across one side is a segment.
Tangent against secant, and the boundary between them. Slide a line towards a circle from outside. At first it misses entirely — no points in common. Then it touches at exactly one point, and there it is a tangent. Push it further in and it cuts at two points, becoming a secant. A line and a circle can share , or points and never more, and the tangent is precisely the borderline case.
The chord that cannot exist. Since the diameter is the longest chord, a circle of radius cm has a maximum chord length of cm. A question describing a chord of cm in that circle describes nothing — and noticing that is worth as much as any calculation.
Formula
How do you use the radius, diameter and the angle in a semicircle?
The relationship between radius and diameter is
and the key angle property is that the angle in a semicircle is a right angle: if is a diameter and is any other point on the circle, then
Worked example 1 — converting. A circle of radius cm has diameter cm. A circle of diameter cm has radius cm. Simple, and yet mixing the two up ruins more circle answers than any other slip — so read whether the question gave you or before substituting.
Worked example 2 — the semicircle in action. is a diameter of length cm, and lies on the circle with cm. Find and the area of triangle .
Since , triangle is right-angled at with as hypotenuse:
The two legs are cm and cm, so
Worked example 3 — a larger case. is a diameter of cm and cm.
Worked example 4 — using the radius instead. A circle has radius cm, is a diameter, and cm. Then cm and
The hypotenuse is always the diameter, never a chord. In triangle the right angle sits at , so the side opposite — which is , the diameter — is the hypotenuse. Treating or as the hypotenuse and adding instead of subtracting is the error this question type is built to catch, and the check is simple: the diameter must come out as the largest of the three sides. Here , as required.
Where the property does not apply. must be a diameter. If is merely a chord, the angle at is not and none of this works. Confirm that the segment passes through the centre before claiming the right angle.
and the key angle property is that the angle in a semicircle is a right angle: if is a diameter and is any other point on the circle, then
Worked example 1 — converting. A circle of radius cm has diameter cm. A circle of diameter cm has radius cm. Simple, and yet mixing the two up ruins more circle answers than any other slip — so read whether the question gave you or before substituting.
Worked example 2 — the semicircle in action. is a diameter of length cm, and lies on the circle with cm. Find and the area of triangle .
Since , triangle is right-angled at with as hypotenuse:
The two legs are cm and cm, so
Worked example 3 — a larger case. is a diameter of cm and cm.
Worked example 4 — using the radius instead. A circle has radius cm, is a diameter, and cm. Then cm and
The hypotenuse is always the diameter, never a chord. In triangle the right angle sits at , so the side opposite — which is , the diameter — is the hypotenuse. Treating or as the hypotenuse and adding instead of subtracting is the error this question type is built to catch, and the check is simple: the diameter must come out as the largest of the three sides. Here , as required.
Where the property does not apply. must be a diameter. If is merely a chord, the angle at is not and none of this works. Confirm that the segment passes through the centre before claiming the right angle.
How do you find the length of a tangent from an external point?
Use the right angle between the tangent and the radius, then apply Pythagoras with as the hypotenuse.
If is outside the circle and touches the circle at , then , so
Worked example 1. The radius is cm and is cm from the centre. Find the tangent length.
Worked example 2 — finding the distance to the centre. The radius is cm and the tangent length is cm.
Here the two known sides are the legs, so they are added. In the first example one was the hypotenuse, so they were subtracted. Deciding which side is the hypotenuse before writing anything is the whole skill.
Worked example 3 — finding the radius. cm and cm.
The check that catches a reversed subtraction. is the hypotenuse, so ** must be the largest of the three lengths**. In every example above it is: , then , then . An answer where the tangent came out longer than is wrong before you check anything else.
Two tangents from the same point are equal. From there are exactly two tangents to the circle, touching at and , and
Both are cm in the first example. The reason is that triangles and share the hypotenuse and have equal radii, so their third sides must match.
Worked example 4 — a chord instead of a tangent. The perpendicular from the centre bisects a chord, giving the same kind of triangle. In a circle of radius cm, how far from the centre is a chord of cm?
Half the chord is cm, and the radius is the hypotenuse:
Worked example 5 — the other direction. In a circle of radius cm, find the chord that is cm from the centre.
Remember to double the half-chord. Answering cm instead of cm is the classic loss of a mark, because the triangle only ever contains half the chord.
Worked example 6 — two circles with the same centre. Two concentric circles have radii cm and cm. A chord of the larger circle touches the smaller one. How long is it?
Touching means the chord is a tangent to the small circle, so the radius of cm meets it at right angles — and being perpendicular from the common centre, it also bisects the chord:
This problem needed both properties at once, which is exactly why they are taught together.
If is outside the circle and touches the circle at , then , so
Worked example 1. The radius is cm and is cm from the centre. Find the tangent length.
Worked example 2 — finding the distance to the centre. The radius is cm and the tangent length is cm.
Here the two known sides are the legs, so they are added. In the first example one was the hypotenuse, so they were subtracted. Deciding which side is the hypotenuse before writing anything is the whole skill.
Worked example 3 — finding the radius. cm and cm.
The check that catches a reversed subtraction. is the hypotenuse, so ** must be the largest of the three lengths**. In every example above it is: , then , then . An answer where the tangent came out longer than is wrong before you check anything else.
Two tangents from the same point are equal. From there are exactly two tangents to the circle, touching at and , and
Both are cm in the first example. The reason is that triangles and share the hypotenuse and have equal radii, so their third sides must match.
Worked example 4 — a chord instead of a tangent. The perpendicular from the centre bisects a chord, giving the same kind of triangle. In a circle of radius cm, how far from the centre is a chord of cm?
Half the chord is cm, and the radius is the hypotenuse:
Worked example 5 — the other direction. In a circle of radius cm, find the chord that is cm from the centre.
Remember to double the half-chord. Answering cm instead of cm is the classic loss of a mark, because the triangle only ever contains half the chord.
Worked example 6 — two circles with the same centre. Two concentric circles have radii cm and cm. A chord of the larger circle touches the smaller one. How long is it?
Touching means the chord is a tangent to the small circle, so the radius of cm meets it at right angles — and being perpendicular from the common centre, it also bisects the chord:
This problem needed both properties at once, which is exactly why they are taught together.
How do you construct and shade chords, arcs, sectors and segments?
Set the compass to the radius, not the diameter — then every part of the figure is built from that one circle.
Constructing a circle of given radius. To draw a circle of radius cm: mark a point , open the compass to cm against a ruler, place the point at and turn the pencil through a full revolution. For a circle of diameter cm, set the compass to cm instead — this is where the against confusion costs a whole diagram.
Drawing and shading each part.
- A chord: mark two points on the circle and join them with a ruler. It need not pass through .
- A diameter: draw a chord through , extending to the circle on both sides.
- A sector: draw two radii, then shade the region between them and the arc they cut off. The boundary is two straight lines plus a curve.
- A segment: draw one chord, then shade the region between the chord and the arc on one side. The boundary is one straight line plus a curve.
- **A tangent at a point **: join , then construct a perpendicular to at . The perpendicular is the tangent, and constructing it with compass and ruler is what makes it exact.
Worked example — a construction with a calculation. Draw a circle of radius cm and a chord cm long, then mark and measure the distance from the centre to the chord.
Before drawing, predict the answer. Half the chord is cm, so
Now draw it: circle of radius cm, chord of cm set with the compass, then the perpendicular from to the chord. Measuring that perpendicular should give cm.
A prediction measured afterwards is a genuine check on your drawing. If the measurement comes out as cm that is pencil width; if it comes out as cm, either the chord or the radius was set wrongly.
Which regions add up to what. A minor sector and a major sector together make the whole circle, and so do a minor segment and a major segment. Shading the wrong one of a pair is the commonest diagram error — so label the region you have shaded in words, minor segment, rather than relying on the shading alone to be read correctly.
The semicircle is both at once. Cut along a diameter and the region you get is a sector (the two radii lie in a straight line) and a segment (the diameter is a chord). It is the single case where the two definitions describe the same region, which is a neat way to remember that a diameter counts as a chord.
Constructing a circle of given radius. To draw a circle of radius cm: mark a point , open the compass to cm against a ruler, place the point at and turn the pencil through a full revolution. For a circle of diameter cm, set the compass to cm instead — this is where the against confusion costs a whole diagram.
Drawing and shading each part.
- A chord: mark two points on the circle and join them with a ruler. It need not pass through .
- A diameter: draw a chord through , extending to the circle on both sides.
- A sector: draw two radii, then shade the region between them and the arc they cut off. The boundary is two straight lines plus a curve.
- A segment: draw one chord, then shade the region between the chord and the arc on one side. The boundary is one straight line plus a curve.
- **A tangent at a point **: join , then construct a perpendicular to at . The perpendicular is the tangent, and constructing it with compass and ruler is what makes it exact.
Worked example — a construction with a calculation. Draw a circle of radius cm and a chord cm long, then mark and measure the distance from the centre to the chord.
Before drawing, predict the answer. Half the chord is cm, so
Now draw it: circle of radius cm, chord of cm set with the compass, then the perpendicular from to the chord. Measuring that perpendicular should give cm.
A prediction measured afterwards is a genuine check on your drawing. If the measurement comes out as cm that is pencil width; if it comes out as cm, either the chord or the radius was set wrongly.
Which regions add up to what. A minor sector and a major sector together make the whole circle, and so do a minor segment and a major segment. Shading the wrong one of a pair is the commonest diagram error — so label the region you have shaded in words, minor segment, rather than relying on the shading alone to be read correctly.
The semicircle is both at once. Cut along a diameter and the region you get is a sector (the two radii lie in a straight line) and a segment (the diameter is a chord). It is the single case where the two definitions describe the same region, which is a neat way to remember that a diameter counts as a chord.
Exam tip
Exam tip: decide which side is the hypotenuse first
Find the right-angled triangle, then name its hypotenuse before substituting. Adding when you should subtract is the single biggest source of lost marks here.
For a tangent from an external point, is the hypotenuse: . So with gives cm.
For a chord, the radius is the hypotenuse and the triangle contains only half the chord. Radius cm with the chord cm from the centre gives a half-chord of cm, so the chord is ** cm — remember to double.
For the angle in a semicircle, the diameter** is the hypotenuse, and it must come out as the longest side. Diameter cm with one leg cm gives the other as cm.
Check the largest side. If your tangent is longer than , or a leg is longer than the diameter, the subtraction went the wrong way.
**Read against carefully — a compass is always set to the radius.
Two tangents from one external point are equal.
The angle in a semicircle** needs to be a diameter, not just a chord.
Sector is bounded by two radii; segment by one chord. Label which region you shaded in words.
And remember the diameter is the longest chord — a chord longer than is impossible.
For a tangent from an external point, is the hypotenuse: . So with gives cm.
For a chord, the radius is the hypotenuse and the triangle contains only half the chord. Radius cm with the chord cm from the centre gives a half-chord of cm, so the chord is ** cm — remember to double.
For the angle in a semicircle, the diameter** is the hypotenuse, and it must come out as the longest side. Diameter cm with one leg cm gives the other as cm.
Check the largest side. If your tangent is longer than , or a leg is longer than the diameter, the subtraction went the wrong way.
**Read against carefully — a compass is always set to the radius.
Two tangents from one external point are equal.
The angle in a semicircle** needs to be a diameter, not just a chord.
Sector is bounded by two radii; segment by one chord. Label which region you shaded in words.
And remember the diameter is the longest chord — a chord longer than is impossible.
Did you know
Why a wheel has spokes of exactly equal length
A bicycle wheel is a circle held together by spokes, and every spoke is cut to the same length. That is not for tidiness — it is the definition of a circle, turned into engineering.
A circle is all the points at a fixed distance from a fixed point. Make every spoke that fixed distance and the rim they support has no choice but to be circular. Shorten one spoke by two millimetres and the rim is pulled inwards there, giving the flat spot a cyclist feels as a bump once per revolution.
The tangent property shows up in the same wheel. At the exact point where the tyre meets the road, the road is a tangent to the circle — and the radius to that point is perpendicular to the road. That is why the load of the bicycle travels straight down the vertical spoke rather than at an angle, and it is the same right angle used in every tangent calculation on this page.
The chord property is visible too. Two spokes and the piece of rim between them bound a sector; the straight line joining their tips would be a chord, and the perpendicular from the hub to that chord bisects it. On a wheel with an even number of spokes, opposite pairs line up into a single diameter through the hub.
So the vocabulary in this chapter is not arbitrary labelling. Centre, radius, tangent and chord each name something a wheel actually has to get right, and the right angles the definitions force are what make the calculations possible.
A circle is all the points at a fixed distance from a fixed point. Make every spoke that fixed distance and the rim they support has no choice but to be circular. Shorten one spoke by two millimetres and the rim is pulled inwards there, giving the flat spot a cyclist feels as a bump once per revolution.
The tangent property shows up in the same wheel. At the exact point where the tyre meets the road, the road is a tangent to the circle — and the radius to that point is perpendicular to the road. That is why the load of the bicycle travels straight down the vertical spoke rather than at an angle, and it is the same right angle used in every tangent calculation on this page.
The chord property is visible too. Two spokes and the piece of rim between them bound a sector; the straight line joining their tips would be a chord, and the perpendicular from the hub to that chord bisects it. On a wheel with an even number of spokes, opposite pairs line up into a single diameter through the hub.
So the vocabulary in this chapter is not arbitrary labelling. Centre, radius, tangent and chord each name something a wheel actually has to get right, and the right angles the definitions force are what make the calculations possible.
Key takeaways
Circles, chords and tangents: quick revision
- A circle is all points at a fixed distance (the radius) from a fixed point (the centre), with .
- Chord joins two points on the circle; the diameter is a chord through the centre and is the longest chord. A chord of cm is impossible in a circle of radius cm.
- Arc is part of the curve; sector is bounded by two radii and an arc; segment is bounded by one chord and an arc.
- Tangent meets the circle at exactly one point; a secant cuts it at two. A line and a circle share , or points and no more.
- A semicircle is both a sector and a segment, because a diameter counts as a chord.
- **Angle in a semicircle is , with the diameter** as hypotenuse. Diameter cm and cm give cm and area .
- Diameter cm with cm gives cm and area ; radius cm with cm gives cm.
- The diameter must be the largest of the three sides — a free check.
- Tangent is perpendicular to the radius, so with as hypotenuse.
- , gives cm. , gives cm (legs, so add). , gives cm.
- Two tangents from the same external point are equal.
- The perpendicular from the centre bisects a chord. Radius cm with a chord of cm gives a distance of cm; radius cm with a distance of cm gives a half-chord of cm, so the **chord is cm.
- Concentric circles**: radii cm and cm, with a chord of the larger touching the smaller, give a half-chord of cm and a chord of ** cm — using both properties at once.
- Constructions: set the compass to the radius**, so a circle of diameter cm needs cm. Draw a tangent by constructing a perpendicular to the radius at the point of contact.
- Predict then measure: radius cm with a chord of cm should give a perpendicular distance of cm.
- Minor and major sectors make the whole circle, as do minor and major segments — so label the region you shade.
Draw one circle and mark on it a chord, a sector, a segment and a tangent, then calculate one length in the figure and measure it — matching your own prediction is the point at which the properties stop being definitions to recite.
- Chord joins two points on the circle; the diameter is a chord through the centre and is the longest chord. A chord of cm is impossible in a circle of radius cm.
- Arc is part of the curve; sector is bounded by two radii and an arc; segment is bounded by one chord and an arc.
- Tangent meets the circle at exactly one point; a secant cuts it at two. A line and a circle share , or points and no more.
- A semicircle is both a sector and a segment, because a diameter counts as a chord.
- **Angle in a semicircle is , with the diameter** as hypotenuse. Diameter cm and cm give cm and area .
- Diameter cm with cm gives cm and area ; radius cm with cm gives cm.
- The diameter must be the largest of the three sides — a free check.
- Tangent is perpendicular to the radius, so with as hypotenuse.
- , gives cm. , gives cm (legs, so add). , gives cm.
- Two tangents from the same external point are equal.
- The perpendicular from the centre bisects a chord. Radius cm with a chord of cm gives a distance of cm; radius cm with a distance of cm gives a half-chord of cm, so the **chord is cm.
- Concentric circles**: radii cm and cm, with a chord of the larger touching the smaller, give a half-chord of cm and a chord of ** cm — using both properties at once.
- Constructions: set the compass to the radius**, so a circle of diameter cm needs cm. Draw a tangent by constructing a perpendicular to the radius at the point of contact.
- Predict then measure: radius cm with a chord of cm should give a perpendicular distance of cm.
- Minor and major sectors make the whole circle, as do minor and major segments — so label the region you shade.
Draw one circle and mark on it a chord, a sector, a segment and a tangent, then calculate one length in the figure and measure it — matching your own prediction is the point at which the properties stop being definitions to recite.