Question

Arc CD is 2/3 of the circumference of a circle. What is the radian measure of the central angle?

Answer 1

The radian measure of the central angle is $\fbox{\begin\\\ \bf \dfrac{4\pi}{3} \text{radians}\\\end{minispace}}$.

Further explanation:

In the question it is given that the length of the arc CD is $\left(\frac{2}{3}\right)^{rd}$ of the circumference of a circle.

Consider a circle of radius $r$ so, the circumference of the circle is $2\pi r$.

Step 1: Obtain the length of the arc

The length of the arc CD is $\left(\frac{2}{3}\right)^{rd}$ of the circumference of the circle.

The length of the arc CD is calculated as follows:

$\fbox{\begin\\\ \begin{aligned}L&=\dfrac{2}{3}\times 2\pi r\\&=\dfrac{4\pi r}{3}\end{aligned}\\\end{minispace}}$

Therefore, the length of the arc CD is $\dfrac{4\pi r}{3}$ units.

Step 2: Obtain the central angle of the arc in degree

The central angle for a circle of circumference of $2\pi r$ is $360^{\circ}$.

The central angle for an arc CD is calculated as follows:

$\fbox{\begin\\\ \begin{aligned}x&=\dfrac{4\pi r}{3}\times \dfrac{360}{2\pi r}\\&=240^{\circ}\end{aligned}\\\end{minispace}}$

This implies that the central angle of the arc CD is $240^{\circ}$.

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 $360^{\circ}$ in terms of radians is $2\pi$ so, the measure of angle $1^{\circ}$ in terms of radians is $\frac{2\pi}{360}$.

The central angle for an arc CD is calculated as follows:

$\fbox{\begin\\\ \begin{aligned}x&=\dfrac{2\pi}{360}\times 240\\&=\dfrac{4\pi}{3}\end{aligned}\\\end{minispace}}$

Therefore, the radian measure of the central angle is $\fbox{\begin\\\ \bf \dfrac{4\pi}{3} \text{radians}\\\end{minispace}}$.

Learn more:

1. A problem on composite function brainly.com/question/2723982  

2. A problem to find radius and center of circle brainly.com/question/9510228  

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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.

Answer 2

⁴/₃π radian

Further explanation

Given:

Arc CD is  $\frac{2}{3}$ 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

$\boxed{ \ \theta = \frac{2}{3} \times 360^0 \ }$

We get the degree measure of the middle angle, i.e., $\boxed{ \ \theta = 240^0 \ }$

Step-2: convert degrees to radians

Because $\boxed{ \ \pi \ radian = 180^0 \ }$, then

$\boxed{ \ \theta = 240^0 \times \frac{\pi \ radians}{180^0} \ }$

Cross out 240° and 180° because they are equally divided by 60°.

$\boxed{ \ \theta = 4 \times \frac{\pi \ radians}{3} \ }$

Hence, the radian measure of the central angle is $\boxed{\boxed{ \ \theta = \frac{4}{3} \pi \ radians \ }}$

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Quick Steps

The full angle of a circle is 2π radians.

$\boxed{ \ \theta = \frac{2}{3} \times 2 \pi \ radians \ }$

Hence, the radian measure of the central angle is $\boxed{\boxed{ \ \theta = \frac{4}{3} \pi \ radians \ }}$

One radian represents the measure of a central angle of a circle that intercepts an arc whose length equals a radius of the circle.

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3. The equation represents a circle with the same radius as the circle shown but with a center at (-1, 1)?  brainly.com/question/1952668

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