⁴/₃π radian
Given:
Arc CD is [tex]\frac{2}{3}[/tex] of the circumference of a circle.
We will measure the central angle of the circle in radians.
Step-1: measure the central angle (θ) in degrees
The full angle of a circle is 360°, thus
[tex]\boxed{ \ \theta = \frac{2}{3} \times 360^0 \ }[/tex]
We get the degree measure of the middle angle, i.e., [tex]\boxed{ \ \theta = 240^0 \ }[/tex]
Step-2: convert degrees to radians
Because [tex]\boxed{ \ \pi \ radian = 180^0 \ }[/tex], then
[tex]\boxed{ \ \theta = 240^0 \times \frac{\pi \ radians}{180^0} \ }[/tex]
Cross out 240° and 180° because they are equally divided by 60°.
[tex]\boxed{ \ \theta = 4 \times \frac{\pi \ radians}{3} \ }[/tex]
Hence, the radian measure of the central angle is [tex]\boxed{\boxed{ \ \theta = \frac{4}{3} \pi \ radians \ }}[/tex]
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Quick Steps
The full angle of a circle is 2π radians.
[tex]\boxed{ \ \theta = \frac{2}{3} \times 2 \pi \ radians \ }[/tex]
Hence, the radian measure of the central angle is [tex]\boxed{\boxed{ \ \theta = \frac{4}{3} \pi \ radians \ }}[/tex]
One radian represents the measure of a central angle of a circle that intercepts an arc whose length equals a radius of the circle.
Keywords: arc CD is 2/3 of the circumference of a circle, what is the radian measure of the central angle, converts, pi, π, the full angle
The radian measure of the central angle is [tex]\fbox{\begin\\\ \bf \dfrac{4\pi}{3} \text{radians}\\\end{minispace}}[/tex].
Further explanation:
In the question it is given that the length of the arc CD is [tex]\left(\frac{2}{3}\right)^{rd}[/tex] of the circumference of a circle.
Consider a circle of radius [tex]r[/tex] so, the circumference of the circle is [tex]2\pi r[/tex].
Step 1: Obtain the length of the arc
The length of the arc CD is [tex]\left(\frac{2}{3}\right)^{rd}[/tex] of the circumference of the circle.
The length of the arc CD is calculated as follows:
[tex]\fbox{\begin\\\ \begin{aligned}L&=\dfrac{2}{3}\times 2\pi r\\&=\dfrac{4\pi r}{3}\end{aligned}\\\end{minispace}}[/tex]
Therefore, the length of the arc CD is [tex]\dfrac{4\pi r}{3}[/tex] units.
Step 2: Obtain the central angle of the arc in degree
The central angle for a circle of circumference of [tex]2\pi r[/tex] is [tex]360^{\circ}[/tex].
The central angle for an arc CD is calculated as follows:
[tex]\fbox{\begin\\\ \begin{aligned}x&=\dfrac{4\pi r}{3}\times \dfrac{360}{2\pi r}\\&=240^{\circ}\end{aligned}\\\end{minispace}}[/tex]
This implies that the central angle of the arc CD is [tex]240^{\circ}[/tex].
Step 3: Obtain the central angle of the arc in radian
Radian is defined as an angle which is subtended by an arc at the center such that the length of an arc is equal to the radius of the circle.
The measure of angle [tex]360^{\circ}[/tex] in terms of radians is [tex]2\pi[/tex] so, the measure of angle [tex]1^{\circ}[/tex] in terms of radians is [tex]\frac{2\pi}{360}[/tex].
The central angle for an arc CD is calculated as follows:
[tex]\fbox{\begin\\\ \begin{aligned}x&=\dfrac{2\pi}{360}\times 240\\&=\dfrac{4\pi}{3}\end{aligned}\\\end{minispace}}[/tex]
Therefore, the radian measure of the central angle is [tex]\fbox{\begin\\\ \bf \dfrac{4\pi}{3} \text{radians}\\\end{minispace}}[/tex].
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Circle
Keywords: Circle, arc, radian, degree, central angle, circumference, length of arc, measure, circumference, angle, 360 degree, 2pir, radius, diameter.
