Arc Length

Worked example

1. 1

   $\frac{40}{360}=\frac{1}{9}$ — Put the angle over 360. 40 goes into 360 nine times, so the arc is a ninth of the circle.

2. 2

   $2\pi \times 9=18\pi$ — The whole circumference is $2\pi r$. With r = 9 that is 18π.

3. 3

   $\frac{1}{9}\times 18\pi =2\pi$ — Take a ninth of the whole edge: 18 ÷ 9 = 2. This is the exact answer.

4. 4

   $2\pi =6.28\text{ cm}$ — Only now press π: 2 × 3.14159… = 6.283…, which is 6.28 cm to 3 s.f. If the question says “in terms of π”, stop at 2π.

Worked example

1. 1

   $\frac{75}{360}\times 2\pi \times 6.4$ — The arc first: the angle over 360, times the whole circumference.

2. 2

   $=\mathrm{8.377...}$ — Keep every digit in the calculator. Rounding now can push the final answer out.

3. 3

   $\mathrm{8.377...}+6.4+6.4$ — Add the two straight edges. Each one is a radius, 6.4 cm.

4. 4

   $=21.2\text{ cm}$ — 21.177… is 21.2 cm to 3 s.f. A perimeter is a length, so the unit is cm.

Worked example

1. 1

   $360-100={260}^{\circ }$ — The small angle is 100°. The angles round the centre add to 360°, so the big angle is 260°.

2. 2

   $\frac{260}{360}=\frac{13}{18}$ — Put the big angle over 360 and simplify: divide both by 20.

3. 3

   $2\pi \times 9=18\pi$ — The radius is 9 cm, so the whole circumference is 18π.

4. 4

   $\frac{13}{18}\times 18\pi =13\pi$ — The 18s cancel. 13π cm is the answer “in terms of π”.

5. 5

   $13\pi =40.8\text{ cm}$ — Or as a decimal: 13 × 3.14159… = 40.84…, which is 40.8 cm to 3 s.f.

Worked example

1. 1

   $\frac{50}{360}\times 2\pi r=7$ — The angle is 50° and the arc is 7 cm. Only r is unknown.

2. 2

   $\frac{100\pi }{360}r=7$ — Tidy the left side: 50 × 2π = 100π.

3. 3

   $100\pi r=2520$ — Multiply both sides by 360 to clear the fraction: 7 × 360 = 2520.

4. 4

   $r=\frac{2520}{100\pi }$ — Divide both sides by 100π, so r is alone.

5. 5

   $r=\mathrm{8.021...}$ — Work it out, and keep the radius in your calculator.

6. 6

   $\pi \times {\mathrm{8.021...}}^2=202{\text{ cm}}^2$ — The question asks for the area of the whole circle, so use $\pi r^2$ with the full radius.

Worked example

1. 1

   $30-9-9=12$ — A sector has perimeter 30 cm and radius 9 cm. Take off the two straight radii: the arc is 12 cm.

2. 2

   $\frac{x}{360}\times 2\pi \times 9=12$ — Put the arc and the radius into the formula. Only the angle x is unknown.

3. 3

   $\frac{x}{360}\times 18\pi =12$ — Tidy: 2π × 9 = 18π.

4. 4

   $x=\frac{12\times 360}{18\pi }$ — Multiply both sides by 360, then divide both sides by 18π, so x is alone.

5. 5

   $x=76.4^{\circ }$ — 12 × 360 ÷ (18π) = 76.39…, so 76.4° to 3 s.f. Check: $\frac{76.39}{360}\times 18\pi \approx 12$.

Worked example

1. 1

   $\frac{90}{360}=\frac{1}{4}$ — The angle is 90°, which is a quarter of 360°.

2. 2

   $\frac{1}{4}\times 2\pi r=\frac{\pi r}{2}$ — Take a quarter of $2\pi r$: 2 ÷ 4 = ½. Leave π and r as they are. This is the arc.

3. 3

   $P=\frac{\pi r}{2}+r+r$ — Add the two straight sides. Each one is a radius, r.

4. 4

   $P=\frac{\pi r}{2}+2r$ — Collect the like terms: r + r = 2r.

5. 5

   $P=r\left(\frac{\pi }{2}+2\right)$ — Both terms have an r, so take it out as a common factor. Expand back to check.

Worked example

1. 1

   $\frac{x}{360-x}=\frac{2}{7}$ — The minor arc is $\frac{2}{7}$ of the major arc. The minor angle is x, so the major angle is 360 − x.

2. 2

   $7x=2(360-x)$ — Cross-multiply to clear both fractions.

3. 3

   $7x=720-2x$ — Expand the bracket: 2 × 360 = 720 and 2 × x = 2x.

4. 4

   $9x=720$ — Add 2x to both sides, so every x is on the left.

5. 5

   $x=80^{\circ }$ — Divide both sides by 9. Check: the major angle is 280°, and 80 ÷ 280 = 2/7.

Worked example

1. 1

   $\frac{70}{360}\times 2\pi \times 8=\mathrm{9.773...}$ — The radius is 8 cm and the angle is 70°. Find the arc first, and keep every digit.

2. 2

   $\frac{1}{2}\times 70^{\circ }=35^{\circ }$ — A line from the centre to the middle of the chord cuts the triangle into two right-angled halves. Each has angle 35° and hypotenuse 8.

3. 3

   $8\sin 35^{\circ }=\mathrm{4.588...}$ — In one half, the half-chord is opposite the 35°, so use sin: opposite = hypotenuse × sin.

4. 4

   $2\times \mathrm{4.588...}=\mathrm{9.177...}$ — Double it to get the whole chord.

5. 5

   $\mathrm{9.773...}+\mathrm{9.177...}=19.0\text{ cm}$ — Add the arc and the chord: 18.95…, which is 19.0 cm to 3 s.f.