Arc measure and arc length
Key Notes :
The formula for the length of an arc is.
𝓁 = m / 360 . c
where 𝓁 is the arc length, C is the circumference, and m is the measure of the arc in degrees.
Learn with an example
🔔 The radius of a circle is 3 metres. What is the length of a 90° arc?
Give the exact answer in simplest form.
________ metres
The arc’s length depends on the arc’s measure and the circle’s circumference. You already know that the arc’s measure is 90°, so find the circle’s circumference.
First, find the circumference.
C = 2𝜋r
= 2𝜋(3) Plug in r=3
= 6𝜋 Multiply
The circumference is 6𝜋 metres.
Now, find the length of the arc.
𝓁 = C . m / 360
= 6𝜋 . 90 / 360 Plug in C=6𝜋 and m=90
= 3𝜋 / 2 Multiply and simplify
The length of the arc is 3𝜋 / 2 metres
🔔 The radius of a circle is 5 kilometres. What is the length of a 135° arc?
Give the exact answer in simplest form.
________ kilometres
The arc’s length depends on the arc’s measure and the circle’s circumference. You already know that the arc’s measure is 135°, so find the circle’s circumference.
First, find the circumference.
C = 2𝜋r
= 2𝜋(5) Plug in r=5
= 10𝜋 Multiply
The circumference is 10𝜋 kilometres.
Now, find the length of the arc.
𝓁=C . m / 360
= 10𝜋 . 135 / 360 Plug in C=10𝜋 and m=135
= 15𝜋 / 4 Multiply and simplify
The length of the arc is 15𝜋 / 4 kilometres.
🔔 The radius of a circle is 10 centimetres. What is the length of a 135° arc?
Give the exact answer in simplest form.
________ centimetres
The arc’s length depends on the arc’s measure and the circle’s circumference. You already know that the arc’s measure is 135°, so find the circle’s circumference.
First, find the circumference.
C = 2𝜋r
= 2𝜋(10) Plug in r=10
= 20𝜋 Multiply
The circumference is 20𝜋 centimetres.
Now, find the length of the arc.
𝓁 = C . m / 360
=20𝜋 . 135 / 360 Plug in C=20𝜋 and m=135
= 15𝜋 / 2 Multiply and simplify
The length of the arc is 15𝜋 / 2 centimetres.
Let’s practice!
