Degrees, Radians, Arc Length and Sector Area Made Easy

Imagine tying a ribbon to the centre of a circular garden and walking around its edge while keeping the ribbon stretched. Your turn has an angle, your footsteps trace an arc, and the ribbon sweeps across a sector.

These are three different measurements of one movement. Once you see the connection, the formulas stop looking like separate things to memorise.

The most important habit is simple: check the angle unit before substituting a number. A correct formula with the wrong unit can produce a very confident wrong answer.

First, name what the question is asking for

In this guide, r means radius, s means arc length, A means sector area, and θ, pronounced theta, means the angle at the centre in radians. We will use d for the numerical angle in degrees.

Quantity	What you are measuring	Typical unit
Central angle	The amount of turn between two radii	Degrees or radians
Arc length	Distance along the curved circumference	cm or m
Sector area	The region enclosed by two radii and their arc	cm² or m²
Sector perimeter	The complete boundary of that region	cm or m

A chord is the straight line joining the arc’s endpoints. It is not the arc. A segment is the region between a chord and its arc. It is not the whole sector.

Before calculating, trace the part requested with your pencil. If you are tracing a curve, you need length. If you are shading a region, you need area. If you are tracing the complete outside edge, you need perimeter.

What does one radian actually mean?

Take a flexible piece of string whose length equals the radius. Lay that string along the circle’s edge. The angle it subtends at the centre is one radian.

More generally:

θ = s/r, with s and r in the same length unit.

For example, an arc of 18 cm on a circle of radius 12 cm gives an angle of 18/12 = 1.5 radians. No π is needed in that answer. A radian measure can be a fraction, decimal, integer, or expression containing π.

Radians measure a ratio of lengths, so the length units cancel. We still write “rad” when it helps make the angle unit clear. The OpenStax explanation of angles develops this radius-to-arc interpretation.

Now use the entire circumference as the arc:

θ = 2πr/r = 2π radians.

Therefore:

360° = 2π radians, and 180° = π radians.

Notice what this says: π radians equals 180 degrees. It does not mean the number π equals the number 180.

Converting degrees to radians and back

Use the half-turn relationship to choose the multiplier:

• Degrees to radians: multiply by π/180.
• Radians to degrees: multiply by 180/π.

For instance:

150° = 150 × π/180 = 5π/6 radians.

In the other direction:

7π/12 radians = (7π/12) × (180/π) = 105°.

A useful check is to compare with a familiar turn. Since 150° is less than 180°, its radian measure must be less than π. Since 7π/12 is greater than π/2, its degree measure must exceed 90°.

Common conversions worth knowing

Degrees	Radians	Fraction of one turn
30°	π/6	1/12
45°	π/4	1/8
60°	π/3	1/6
90°	π/2	1/4
120°	2π/3	1/3
180°	π	1/2
270°	3π/2	3/4
360°	2π	1

What if degrees contain minutes and seconds?

Angular minutes and seconds divide a degree into smaller parts:

1° = 60 minutes; 1 minute = 60 seconds.

So 22°30′ means 22 + 30/60 = 22.5°, not 22.30°.

Its radian measure is:

22.5 × π/180 = π/8 radians.

For an angle written as D degrees, M minutes and S seconds, first calculate D + M/60 + S/3600 degrees. Then convert to radians. These are subdivisions of an angle, not a measurement of elapsed time.

The formula chooser: degrees or radians?

For a sector with a positive angle up to one complete turn:

Required quantity	Angle d in degrees	Angle θ in radians
Arc length s	(d/360) × 2πr	rθ
Sector area A	(d/360) × πr²	½r²θ
Perimeter of a sector with two distinct radial sides	2r + (d/360) × 2πr	2r + rθ

Both columns give the same answers. The degree column uses the fraction d/360 of a complete circle. The radian column uses θ/(2π).

For example, sector area in radians follows directly:

A = [θ/(2π)] × πr² = ½r²θ.

Similarly:

s = [θ/(2π)] × 2πr = rθ.

The CLP text on radians, arcs and sectors gives a geometric treatment of these connections.

You do not need to convert every degree question to radians. If an angle is 72°, then 72/360 = 1/5 may be the quickest route. If it is already 0.8 radians, leave it in radians.

Worked example 1: one sector, three different answers

A circular display has radius 12 cm and central angle 75°. Find its arc length, sector area and perimeter.

First convert the angle:

θ = 75 × π/180 = 5π/12 radians.

Arc length:

s = 12 × 5π/12 = 5π cm.

Sector area:

A = ½ × 12² × 5π/12 = 30π cm².

Sector perimeter:

P = 12 + 12 + 5π = 24 + 5π cm.

The three answers are approximately 15.71 cm, 94.25 cm² and 39.71 cm respectively.

Now check without repeating the same calculation. The sector is 75/360 = 5/24 of the circle. The full area is 144π cm², and 5/24 of 144π is 30π. That agrees.

The perimeter includes both straight radii. Merely writing 5π cm would measure only the curved part of the display.

Worked example 2: the angle is already in radians

A sprinkler reaches 8 m from its pivot and turns through 1.25 radians. Find the arc traced by its outermost point and the area of the sector it sweeps.

s = 8 × 1.25 = 10 m.

A = ½ × 8² × 1.25 = 40 m².

There is no reason to insert π. The number 1.25 is already a valid radian measure.

This model assumes the sprinkler covers the full radius throughout its turn. The curved distance at the outer edge and the covered area describe different aspects of the same sweep.

Worked example 3: find the angle from an arc

A wheel has radius 40 cm. A marked point travels 30 cm around its rim. Through what angle has the wheel turned?

Use θ = s/r:

θ = 30/40 = 3/4 radian.

If degrees are requested:

d = (3/4) × 180/π = 135/π degrees, approximately 42.97°.

Suppose instead the radius were given as 0.4 m. You would first write 30 cm as 0.3 m, giving 0.3/0.4 = 0.75 radians. Dividing 30 by 0.4 without converting would mix centimetres and metres.

Worked example 4: area and arc length are given

A sector has area 54 cm² and arc length 9 cm. Find its radius and central angle.

Since s = rθ, substitute into the area formula:

A = ½r × rθ = ½rs.

Therefore:

r = 2A/s = (2 × 54)/9 = 12 cm.

Then:

θ = s/r = 9/12 = 3/4 radian.

Check the answer:

½ × 12² × 3/4 = 54 cm².

This shortcut is especially useful when the question gives arc length instead of angle. It works because the arc and area belong to the same sector.

Worked example 5: the larger arc is requested

Two radii of a circle of radius 9 cm enclose a smaller angle of 80°. Find the major arc length and the corresponding major sector area.

The major angle is:

360° - 80° = 280°.

Now use that angle consistently:

s = (280/360) × 2π × 9 = 14π cm.

A = (280/360) × π × 9² = 63π cm².

For a useful check, the smaller arc is 4π cm and the smaller sector area is 18π cm². The pairs add to 18π cm and 81π cm², the full circumference and full area.

A small angle marked on a diagram does not automatically mean the question wants the small region. Read “minor” and “major” carefully.

Worked example 6: a curved strip has two arcs

A curved garden path lies between two concentric circles of radii 10 m and 6 m. It spans 60°. Find its area and complete perimeter.

This is an annular sector: a large sector with a smaller sector removed.

Area:

A = (60/360) × π × (10² - 6²)

A = (1/6) × π × 64 = 32π/3 m².

Perimeter: add the outer arc, inner arc and two straight ends.

• Outer arc = (1/6) × 2π × 10 = 10π/3 m.
• Inner arc = (1/6) × 2π × 6 = 2π m.
• Each straight end = 10 - 6 = 4 m.

Therefore:

P = 16π/3 + 8 m.

The area uses subtraction because a region was removed. The perimeter uses addition because both curved edges remain exposed. Subtracting the inner arc would not trace the boundary of the path.

Sector, triangle and segment: keep the regions separate

Consider a circle of radius 6 cm with a 90° sector. Join the arc’s endpoints with a straight chord.

The sector contains a right-angled triangle and a curved segment.

• Sector area = (90/360) × π × 6² = 9π cm².
• Triangle area = ½ × 6 × 6 = 18 cm².
• Minor segment area = 9π - 18 cm².

The arc itself has length 3π cm, whereas the chord has length 6√2 cm by Pythagoras. The curved route is longer than the straight route.

For a minor sector of angle θ between 0 and π radians, the general segment formula is ½r²(θ - sin θ). The sine must also be evaluated with θ in radians. The AMSI lesson on radian measure explains the sector-minus-triangle construction.

Do not use a segment formula simply because a diagram looks like a slice. Look for a chord: its presence tells you which boundary separates the triangle from the segment.

Negative angles, complete turns and distance travelled

In the usual convention, counterclockwise rotation is positive and clockwise rotation is negative. A clockwise turn of 120° has signed angle -2π/3 radians.

A physical length or area is not negative. For a single sweep in one direction, use the magnitude of the angle when finding distance or unsigned area.

Also distinguish where you finish from how far you travel. A point on a circle of radius 3 cm turns through 450° in one direction:

θ = 450 × π/180 = 5π/2 radians.

Distance travelled = 3 × 5π/2 = 15π/2 cm.

It finishes at the same position as a 90° turn, but replacing 450° with 90° would discard one whole revolution of travel.

For area, be equally careful. After one complete sweep, the whole disc has already been covered. Using ½r²θ for a rotation beyond 2π counts repeated coverage; it is not the area of a larger non-overlapping sector inside that circle.

How to catch an answer that cannot be right

Use these quick checks before submitting your work:

1. Angle check: Is your formula expecting degrees or radians?
2. Radius check: If the question gives a diameter, did you halve it?
3. Size check: For a sector smaller than a full circle, is its area smaller than πr² and its arc shorter than 2πr?
4. Boundary check: For perimeter, have you counted every exposed edge exactly once?
5. Unit check: Length uses cm or m; area uses cm² or m².

For example, with r = 5 cm and angle 60°, the incorrect substitution s = 5 × 60 gives 300 cm. But the entire circumference is only 10π cm, about 31.42 cm. The answer cannot represent a sixth of that circle.

The correct result is s = 5 × π/3 = 5π/3 cm.

Keep π in an exact answer unless a decimal or a particular approximation is requested. If you use a calculator for trigonometric functions, check its angle mode. Ordinary multiplication by π does not change with DEG or RAD mode, but values such as sin(θ) do.

For more practice interpreting angles around a full circle, see our guide to trigonometric signs, quadrants and the unit circle.

Try these eight questions yourself

Cover the answers while you work. Write the formula before substituting.

1. Convert 225° to radians.
2. Convert 11π/18 radians to degrees.
3. Convert 15°45′ to radians.
4. Find the arc length and area of a sector with radius 7 cm and angle 120°.
5. A sector has area 32 cm² and radius 8 cm. Find its angle in radians and its arc length.
6. A sector has arc length 14 cm and radius 10 cm. Find its area and perimeter.
7. A sector has radius 4 cm and angle 270°. Find its area and perimeter.
8. A point moves clockwise through 300° on a circle of radius 6 cm. Find the signed radian angle and the distance travelled.

Answers with short working

1. 225 × π/180 = 5π/4 radians.
2. (11π/18) × 180/π = 110°.
3. 15 + 45/60 = 63/4 degrees; multiplying by π/180 gives 7π/80 radians.
4. θ = 2π/3. Arc = 14π/3 cm; area = 49π/3 cm².
5. θ = 2A/r² = 64/64 = 1 radian; s = 8 × 1 = 8 cm.
6. A = ½rs = 70 cm²; P = 14 + 20 = 34 cm.
7. A = (3/4) × 16π = 12π cm²; P = 8 + 6π = 8 + 6π cm.
8. Signed angle = -5π/3 radians; distance = 6 × 5π/3 = 10π cm.

Frequently asked questions

1. What is the difference between a degree and a radian?

Both measure an angle. A degree is 1/360 of a full turn. One radian is the central angle formed when the intercepted arc has the same length as the radius. A half-turn is 180° or π radians.

2. Can I use s = rθ when the angle is in degrees?

Not directly. Convert the angle to radians first, or use s = (d/360) × 2πr with d in degrees. Putting the degree number into rθ gives the wrong length.

3. Does every radian answer contain π?

No. An arc of 20 cm on a circle of radius 10 cm subtends 2 radians. Expressions containing π are common because familiar degree angles are often simple fractions of a half-turn.

4. When should I use A = ½rs?

Use it when you know the radius and the arc length of the same sector. It follows from A = ½r²θ and s = rθ. Both lengths must use the same unit.

5. Is sector perimeter the same as arc length?

No. For a sector with two distinct straight sides, perimeter equals arc length plus twice the radius. Arc length measures only the curved edge.

6. How do I choose between the minor and major sector?

Read which region the question requests. If the smaller angle is d degrees, the larger angle is 360 - d degrees. Use the angle belonging to the requested region in both its area and arc calculations.

7. What happens if the radius doubles but the angle stays fixed?

Arc length doubles because s is proportional to r. Sector area becomes four times as large because A is proportional to r². The angle itself does not change.

8. What happens if the angle doubles but the radius stays fixed?

Arc length and sector area both double, provided you are describing the corresponding sweep. For an ordinary non-overlapping sector, the resulting angle must not exceed one full turn.

9. Can arc length or sector area be negative?

Ordinary geometric lengths and areas are non-negative. A negative angle indicates clockwise direction under the usual convention. Use its magnitude for the unsigned size of a single sweep, and keep direction separate.

10. Should I leave my answer in terms of π?

Yes, unless the question requests a decimal or specifies an approximation. An answer such as 5π cm is exact. If rounding is required, keep the exact expression until your final calculation.