Areas Related to Circles · Class 10 Mathematics: quick revision, 34 bites

What a sector is

A slice of the circle

The region between two radii and the arc joining them.

Takeaway. Two radii and an arc, nothing else.

θ over 360

The whole chapter in one fraction

A full turn is 360°, so an angle of θ marks off $\frac{\theta}{360}$ of the circle.

Takeaway. Same slice, same fraction, every time.

A fraction of πr²

Area of a sector

Takeaway. Square units.

A fraction of 2πr

Length of the arc

Takeaway. Ordinary length units.

Add the two radii

Arc is not perimeter

Takeaway. A pizza slice has crust AND two straight edges.

Put θ = 360

A check that never fails

The fraction becomes 1 and the formulas give $\pi r^2$ and $2\pi r$.

Takeaway. Test it on the whole circle.

60° in a circle of radius 21

A first sector

Takeaway. And the arc is 22 cm.

Multiples of 7

Why 21?

$\frac{22}{7}$ only cancels neatly when the radius is 7, 14, 21 or 28.

Takeaway. Choose π to suit the radius you were given.

The quadrant

A quarter circle

A sector of 90°, so $\frac14$ of everything.

Takeaway. Its arc is a quarter of the circumference, not a quarter of the radius.

r = 14

A quadrant worked out

Takeaway. Arc 22 cm, perimeter 50 cm.

Minor plus major

They add to the whole

The two angles add to 360°, so the two areas add to $\pi r^2$.

Takeaway. Given one, subtract for the other.

Rearrange, do not guess

Finding the angle

Takeaway. The same move works from the arc length.

Area 3π, radius 6

Working backwards

Takeaway. The π cancels, which is why leaving it symbolic pays.

Getting the unit right

Square units for area

Area is cm² or m²; arc length is cm or m.

Takeaway. Check the unit before you stop.

Simplify θ/360 early

Cancel first

Turn $\frac{60}{360}$ into $\frac16$ before multiplying anything.

Takeaway. Short arithmetic is accurate arithmetic.

Read the question twice

Radius, not diameter

If a question gives the diameter, halve it before substituting.

Takeaway. Both formulas want r.

Sectors outside the classroom

The wiper's sweep

A windscreen wiper sweeps a sector; so does a radar scan and a garden sprinkler.

Takeaway. The same fraction of a circle each time.

What a segment is

Cut with a chord

The region between a CHORD and one of the arcs it cuts off.

Takeaway. One straight edge, not two.

Sector against segment

Not the same thing

Takeaway. Count the straight edges to tell them apart.

Sector minus triangle

The one formula

Takeaway. A segment has no formula of its own.

Half r-squared sine theta

The triangle underneath

Triangle $OAB$ has $OA = OB = r$, so it is isosceles.

Takeaway. Half, two sides, sine of the angle between them.

An equilateral triangle

Sixty is special

With a 60° apex, an isosceles triangle has 60° base angles too.

Takeaway. The chord equals the radius.

A right triangle with legs r

Ninety is easiest

Takeaway. No surd in the area at all.

Same area, different shape

One-twenty repeats sixty

$\sin 120° = \sin 60° = \frac{\sqrt3}{2}$, so the triangle areas match.

Takeaway. One is tall and narrow, the other short and wide.

The syllabus restriction

Why just three angles

Segment problems use only 60°, 90° and 120° — the angles whose sine is known exactly.

Takeaway. No calculator is ever needed.

r = 10, θ = 90°

A right-angled segment

Takeaway. A π term minus a plain number — the standard shape.

r = 21, θ = 60°

A sixty-degree segment

Takeaway. Keep the surd exact until the very last line.

Surds last

Do not round early

A segment is a small difference between two large numbers.

Takeaway. Substitute decimals only at the end.

Subtract, do not restart

The major segment

Takeaway. Much quicker than a 270° sector plus a triangle.

Two r sine half theta

How long is the chord?

Takeaway. The perpendicular from the centre bisects both the chord and the angle.

Arc plus chord

The boundary of a segment

No radii are involved — they are inside the segment's region, not on its edge.

Takeaway. Compare a sector, whose perimeter is arc + 2r.

Not from the circle

Subtract from the sector

The triangle comes off the SECTOR, which is the slice containing it.

Takeaway. Draw the sector first and the mistake disappears.

π minus something

Check the shape of your answer

A segment area is a $\pi$ term minus a surd or an integer.

Takeaway. The form of the answer is itself a check.

Where segments show up

Liquid in a pipe

The cross-section of a partly filled horizontal pipe is exactly a segment.

Takeaway. Engineers compute this area to measure flow.