Measuring Space: Perimeter and Area - Length of an Arc of a Circle

Find the length of an arc of a circle with radius $21\text{ cm}$ and a central angle of $60^\circ$. (Use $\pi = \frac{22}{7}$)

Given: radius $r = 21\text{ cm}$ and angle $\theta = 60^\circ$. Using the formula for arc length: $l = \frac{\theta}{360^\circ} \times 2 \pi r$ $l = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21$ $l = \frac{1}{6} \times 2 \times 22 \times 3$ $l = \frac{1}{6} \times 132$ $l = 22\text{ cm}$

We identify the given radius and central angle, then substitute them into the arc length formula. Simplification leads to the final length of $22\text{ cm}$.

The length of an arc of a circle is $11\text{ cm}$ and its radius is $7\text{ cm}$. Find the angle subtended by the arc at the center.

Given: $l = 11\text{ cm}$, $r = 7\text{ cm}$. We use the formula: $l = \frac{\theta}{360^\circ} \times 2 \pi r$ $11 = \frac{\theta}{360^\circ} \times 2 \times \frac{22}{7} \times 7$ $11 = \frac{\theta}{360^\circ} \times 44$ $\theta = \frac{11 \times 360^\circ}{44}$ $\theta = \frac{360^\circ}{4}$ $\theta = 90^\circ$

By rearranging the arc length formula to solve for the unknown angle $\theta$, we substitute the known values of arc length and radius to find that the angle is $90^\circ$.

Find the perimeter of a sector of a circle with radius $10.5\text{ cm}$ and central angle $60^\circ$.

Given: $r = 10.5\text{ cm}$, $\theta = 60^\circ$. First, find the arc length $l$: $l = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 10.5$ $l = \frac{1}{6} \times 2 \times 22 \times 1.5$ $l = \frac{1}{6} \times 66 = 11\text{ cm}$ Now, Perimeter of Sector $= l + 2r$: $\text{Perimeter} = 11 + 2(10.5)$ $\text{Perimeter} = 11 + 21$ $\text{Perimeter} = 32\text{ cm}$

The perimeter of a sector includes the curved arc and the two straight radii. We first calculate the arc length and then add twice the radius to it.