13. The arc and sector of a circle

The arc and sector of a circle

A circle's sector is bounded by two radii [[$ OA $]] and [[$ OB $]] and an arc [[$ AB $]]. The vertex of the sector's central angle [[$ \alpha $]] is located in the center of the circle.

Example 1

The diameter of a family-size pizza is [[$ 42,0 $]] cm. This means that its area is

[[$ A = \pi r^2 ≈ 3,14 \cdot (21 $]] cm[[$ )^2 ≈ 1385 $]] cm[[$ ^2 $]].

The circumferential length of the pizza is [[$ p = \pi d ≈ 3,14 \cdot 42 $]] cm [[$ ≈ 132 $]] cm.

Divide the pizza evenly among six people, and calculate the size of the slices received by each person.

Because the pizza is divided evenly among six people, the area of one slice of pizza is one sixth of the whole pizza ([[$ 231 $]]) cm². This means that the arc of each slice is [[$ 22 $]] cm.

You will quickly notice that the magnitude of the sector's central angle, the sector's area and the length of the sector's arc all change in the same proportion.

The length of a sector's arc

In sector calculation formulas, the size of the sector is usually expressed as the ratio of the magnitude of the central angle to the magnitude of the entire circle.

[[$$ b = \displaystyle\frac {\alpha} {360°} 2\pi r $$]]

Example 2

The radius of the circle is [[$ 4,0 $]] cm. Calculate the following features of a sector with a central angle of [[$ 26 ° $]].

a) the length of the sector's arc.
[[$ b = \displaystyle\frac {\alpha} {360°} \pi r ≈ 3,14 \cdot $]] ([[$ 4,0 $]] cm)[[$ ≈ 1,8 $]] cm.

b) the sector's area.

[[$ A = \displaystyle\frac {\alpha} {360°} \pi r^2 ≈ 3,14 \cdot ( $]][[$ 4.0 $]] cm[[$ )^2 ≈ 3,6 $]] cm[[$ ^2 $]].

Answer: The length of the sector's arc is [[$ 1,8 $]] cm, whereas the sector's area is [[$ 3,6 $]] cm[[$ ^2 $]].