Question

A circle has the radius of 15 inches. What is the length of the arc intercepted by a central angle of 120 degrees?

Answer 1

`" "`
`"Length of the Arc"` `color(blue)(~~ 31.42 " inches"`

Explanation:

`" "`
`color(green)("Step 1":`

Construct a circle using the given Radius (r) = 15 inches.

With a Central Angle of `color(red)(120^@`, using two radii, a sector is created:

`color(blue)(Radii: OB = OA = "15 inches"`

`color(blue)("Central Angle: "/_AOB = alpha = 120^@`

`color(green)("Step 2":`

Find the Circumference (C):

Formula: Circumference (C): `color(red)(2* pi* r)` Units.

`color(red)(pi` is the ratio of the circumference of a circle to its diameter.

For many purposes you can use `color(green)(3.14159`

`rArr C = 2*pi*15`

Using a calculator:

`C~~94.24777961`

`color(green)("Step 3":`

To find the Arc Length use the formula: `color(brown)(theta^@/360^@*Circumference)`

`rArr theta^@/360^@*Circumference`

`rArr (120^@/360^@)*94.24777961`

`rArr (1/3)*94.24777961`

Using a calculator:

`rArr ~~ 31.41592654`

Hence,

Arc Length: `~~ 31.42` inches.

`color(green)("Step 4":`

Verify the final result using a construction in Geometry:

Hope it helps.