Length of an arc

We can find the length of an arc by using the formula:

\(\frac{\texttheta}{360} \times \pi~\text{d}\)

\(\texttheta\) is the angle of the sector and \(\text{d}\) is the diameter of the circle.

This sector has a minor arc, because the angle is less than 180⁰.

We are given the radius of the sector so we need to double this to find the diameter.

Here, \(\text{d}\) = 24 and \(\texttheta\) = 80⁰.

Substituting these values into the formula, we get:

\(\text{Arc length =}~\frac{80}{360} \times \pi \times {24}\)

\(\text{= 16.755...}\)

\(\text{= 16.8 cm (to one decimal place)}\)

Question

Find the length of the minor arc AB.

This sector contains a major arc, because the angle is greater than 180⁰.

We still follow the same process for this question. Don’t forget to double the radius to get the diameter.

\(\text{Arc length =}~\frac{210}{360} \times \pi \times {120}\)

\(\text{= 219.91...}\)

\(\text{= 219.9 mm (to one decimal place)}\)

Find the length of the major arc AB.