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Geometry and Trigonometry - Arc length and sector area

Grade 11IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The arc length is the distance along the curved edge of a sector. In radians, it is calculated as l=rθl = r\theta, while in degrees it is a fraction of the circumference: l=θ360×2πrl = \frac{\theta}{360} \times 2\pi r.

Diagram of a circular sector showing radius r, central angle theta, and arc length l.
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The sector area represents the 'pie slice' region of a circle. It is calculated as A=12r2θA = \frac{1}{2}r^2\theta for radians or A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2 for degrees.

A circle with a shaded sector highlighting the concept of sector area.
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The perimeter of a sector is the sum of the arc length and the two radii that bound the sector: P=l+2rP = l + 2r.

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To convert degrees to radians, multiply by π180\frac{\pi}{180}. This is essential when using formulae intended for radian measure.

📐Formulae

Angle conversion: θrad=θdeg×π180\text{Angle conversion: } \theta_{rad} = \theta_{deg} \times \frac{\pi}{180}

Arc length (radians): l=rθ\text{Arc length (radians): } l = r\theta

Arc length (degrees): l=θ360×2πr\text{Arc length (degrees): } l = \frac{\theta}{360} \times 2\pi r

Sector area (radians): A=12r2θ\text{Sector area (radians): } A = \frac{1}{2}r^2\theta

Sector area (degrees): A=θ360×πr2\text{Sector area (degrees): } A = \frac{\theta}{360} \times \pi r^2

💡Examples

Problem 1:

A circular pizza has a radius of 15 cm15\text{ cm}. A slice is cut with a central angle of 40∘40^{\circ}. Calculate the arc length of the crust of this slice and the area of the slice.

Solution:

l=40360×2×π×15l = \frac{40}{360} \times 2 \times \pi \times 15 l=19×30π=10π3≈10.5 cml = \frac{1}{9} \times 30\pi = \frac{10\pi}{3} \approx 10.5\text{ cm} A=40360×π×152A = \frac{40}{360} \times \pi \times 15^2 A=19×225π=25π≈78.5 cm2A = \frac{1}{9} \times 225\pi = 25\pi \approx 78.5\text{ cm}^2

Explanation:

Since the angle is given in degrees, we use the degree-based formulas. The arc length represents the crust, and the area represents the size of the slice. Final answers are rounded to 3 significant figures.

Problem 2:

A sector of a circle has a radius of 6 cm6\text{ cm} and an area of 27 cm227\text{ cm}^2. Find the central angle θ\theta in radians.

Solution:

A=12r2θA = \frac{1}{2}r^2\theta 27=12(62)θ27 = \frac{1}{2}(6^2)\theta 27=18θ27 = 18\theta θ=2718=1.5 rad\theta = \frac{27}{18} = 1.5\text{ rad}

Explanation:

We use the radian formula for sector area. By substituting the known values for Area (AA) and radius (rr), we solve for the unknown angle θ\theta.

Problem 3:

Find the perimeter of a sector where the radius is 10 cm10\text{ cm} and the central angle is 2.12.1 radians.

Solution:

l=rθ=10×2.1=21 cml = r\theta = 10 \times 2.1 = 21\text{ cm} P=l+2rP = l + 2r P=21+2(10)=21+20=41 cmP = 21 + 2(10) = 21 + 20 = 41\text{ cm}

Explanation:

First, calculate the arc length (ll) using the radian formula. Then, add two lengths of the radius to find the total boundary (perimeter) of the sector.

Problem 4:

A windshield wiper on a car has a blade length of 40 cm40\text{ cm}. It rotates through an angle of 150∘150^{\circ}. Calculate the area of the windshield cleaned by the blade.

A sector representing a windshield wiper path with a 150 degree angle and 40 cm radius.

Solution:

  1. Identify variables: r=40r = 40, θ=150∘\theta = 150^{\circ}.
  2. Since the angle is in degrees, use the formula: A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2
  3. Substitute values: A=150360×π×402A = \frac{150}{360} \times \pi \times 40^2
  4. Simplify the fraction: A=512×1600πA = \frac{5}{12} \times 1600\pi
  5. Calculate: A=8000π12≈2094.40 cm2A = \frac{8000\pi}{12} \approx 2094.40\text{ cm}^2

Explanation:

The area swept by a wiper is a sector of a circle. We use the degree-based area formula and plug in the radius (blade length) and the angle of rotation.

Problem 5:

A pendulum of length 80 cm80\text{ cm} swings such that its tip traces an arc of length 20 cm20\text{ cm}. Find the angle of the swing in radians.

A pendulum diagram showing a length of 80 cm and an arc at the bottom of 20 cm.

Solution:

  1. Identify variables: l=20l = 20, r=80r = 80.
  2. Use the radian arc length formula: l=rθl = r\theta
  3. Rearrange to solve for θ\theta: θ=lr\theta = \frac{l}{r}
  4. Substitute values: θ=2080\theta = \frac{20}{80}
  5. Simplify: θ=0.25 radians\theta = 0.25\text{ radians}

Explanation:

Because the arc length and radius are known, the simplest way to find the angle is using the radian formula l=rθl = r\theta. The result is automatically in radians.