Radian Measure and Arc Length - Pre-Calculus
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What does it mean to say arc length is proportional to radius for a fixed central angle $\theta$?
What does it mean to say arc length is proportional to radius for a fixed central angle $\theta$?
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$\frac{s}{r}$ is constant for that $\theta$. The ratio of arc length to radius remains the same for any given angle.
$\frac{s}{r}$ is constant for that $\theta$. The ratio of arc length to radius remains the same for any given angle.
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Identify the correct arc length if $\theta$ is in degrees: which formula is correct for $s$?
Identify the correct arc length if $\theta$ is in degrees: which formula is correct for $s$?
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$s=\frac{\theta}{360}\cdot 2\pi r$. Fraction $\frac{\theta}{360}$ of circumference $2\pi r$ gives arc length.
$s=\frac{\theta}{360}\cdot 2\pi r$. Fraction $\frac{\theta}{360}$ of circumference $2\pi r$ gives arc length.
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Convert $\frac{7\pi}{4}$ radians to degrees.
Convert $\frac{7\pi}{4}$ radians to degrees.
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$315^\circ$. $\frac{7\pi}{4} \cdot \frac{180}{\pi} = \frac{7 \cdot 180}{4} = 315$.
$315^\circ$. $\frac{7\pi}{4} \cdot \frac{180}{\pi} = \frac{7 \cdot 180}{4} = 315$.
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Convert $150^\circ$ to radians.
Convert $150^\circ$ to radians.
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$\frac{5\pi}{6}$. $150 \cdot \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}$.
$\frac{5\pi}{6}$. $150 \cdot \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}$.
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Find the sector area $A$ using $A=\frac{1}{2}rs$ if $r=8$ and $s=6\pi$.
Find the sector area $A$ using $A=\frac{1}{2}rs$ if $r=8$ and $s=6\pi$.
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$A=24\pi$. Direct substitution: $A = \frac{1}{2}(8)(6\pi) = 24\pi$.
$A=24\pi$. Direct substitution: $A = \frac{1}{2}(8)(6\pi) = 24\pi$.
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Find the sector area $A$ if $r=6$ and $\theta=\frac{\pi}{3}$ radians.
Find the sector area $A$ if $r=6$ and $\theta=\frac{\pi}{3}$ radians.
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$A=6\pi$. Using $A = \frac{1}{2}r^2\theta$: $A = \frac{1}{2}(36)\frac{\pi}{3} = 6\pi$.
$A=6\pi$. Using $A = \frac{1}{2}r^2\theta$: $A = \frac{1}{2}(36)\frac{\pi}{3} = 6\pi$.
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Find the radius $r$ if arc length $s=10\pi$ and central angle $\theta=5$ radians.
Find the radius $r$ if arc length $s=10\pi$ and central angle $\theta=5$ radians.
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$r=2\pi$. Using $r = \frac{s}{\theta}$: $r = \frac{10\pi}{5} = 2\pi$.
$r=2\pi$. Using $r = \frac{s}{\theta}$: $r = \frac{10\pi}{5} = 2\pi$.
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Find the central angle $\theta$ in radians if arc length $s=12$ and radius $r=3$.
Find the central angle $\theta$ in radians if arc length $s=12$ and radius $r=3$.
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$\theta=4$. Using $\theta = \frac{s}{r}$: $\theta = \frac{12}{3} = 4$.
$\theta=4$. Using $\theta = \frac{s}{r}$: $\theta = \frac{12}{3} = 4$.
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Find the arc length $s$ if $r=5$ and $\theta=\frac{3\pi}{2}$ radians.
Find the arc length $s$ if $r=5$ and $\theta=\frac{3\pi}{2}$ radians.
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$s=\frac{15\pi}{2}$. Using $s = r\theta$: $s = 5 \cdot \frac{3\pi}{2} = \frac{15\pi}{2}$.
$s=\frac{15\pi}{2}$. Using $s = r\theta$: $s = 5 \cdot \frac{3\pi}{2} = \frac{15\pi}{2}$.
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What is the area of a full circle written to match the sector area formula $A=\frac{1}{2}r^2\theta$?
What is the area of a full circle written to match the sector area formula $A=\frac{1}{2}r^2\theta$?
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$A=\pi r^2$. Full circle has $\theta = 2\pi$, so $A = \frac{1}{2}r^2(2\pi) = \pi r^2$.
$A=\pi r^2$. Full circle has $\theta = 2\pi$, so $A = \frac{1}{2}r^2(2\pi) = \pi r^2$.
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What is the sector area formula written using arc length $s$ and radius $r$?
What is the sector area formula written using arc length $s$ and radius $r$?
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$A=\frac{1}{2}rs$. Substitute $s = r\theta$ into $A = \frac{1}{2}r^2\theta$.
$A=\frac{1}{2}rs$. Substitute $s = r\theta$ into $A = \frac{1}{2}r^2\theta$.
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What is the area of a sector with radius $r$ and central angle $\theta$ measured in radians?
What is the area of a sector with radius $r$ and central angle $\theta$ measured in radians?
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$A=\frac{1}{2}r^2\theta$. Sector is fraction $\frac{\theta}{2\pi}$ of circle area $\pi r^2$.
$A=\frac{1}{2}r^2\theta$. Sector is fraction $\frac{\theta}{2\pi}$ of circle area $\pi r^2$.
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What is the radian-to-degree conversion formula for an angle measuring $x$ radians?
What is the radian-to-degree conversion formula for an angle measuring $x$ radians?
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$x\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ since $\pi$ radians $= 180°$.
$x\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ since $\pi$ radians $= 180°$.
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What is the degree-to-radian conversion formula for an angle measuring $x^\circ$?
What is the degree-to-radian conversion formula for an angle measuring $x^\circ$?
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$x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ since $180° = \pi$ radians.
$x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ since $180° = \pi$ radians.
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What is the radian measure of a right angle?
What is the radian measure of a right angle?
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$\frac{\pi}{2}$. Quarter circle is one-fourth of $2\pi$ radians.
$\frac{\pi}{2}$. Quarter circle is one-fourth of $2\pi$ radians.
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What is the radian measure of a straight angle (a semicircle)?
What is the radian measure of a straight angle (a semicircle)?
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$\pi$. Half circle is half of $2\pi$ radians.
$\pi$. Half circle is half of $2\pi$ radians.
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What is the radian measure of a full rotation (one complete circle)?
What is the radian measure of a full rotation (one complete circle)?
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$2\pi$. Full circle's arc length equals circumference: $\frac{2\pi r}{r} = 2\pi$.
$2\pi$. Full circle's arc length equals circumference: $\frac{2\pi r}{r} = 2\pi$.
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What is the circumference of a circle written using the arc length formula $s=r\theta$?
What is the circumference of a circle written using the arc length formula $s=r\theta$?
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$C=2\pi r$. Full circle has angle $2\pi$, so $s = r(2\pi) = 2\pi r$.
$C=2\pi r$. Full circle has angle $2\pi$, so $s = r(2\pi) = 2\pi r$.
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What is the arc length formula for a circle of radius $r$ with central angle $\theta$ measured in radians?
What is the arc length formula for a circle of radius $r$ with central angle $\theta$ measured in radians?
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$s=r\theta$. Multiply radius by angle in radians to get arc length.
$s=r\theta$. Multiply radius by angle in radians to get arc length.
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What is the definition of radian measure of a central angle $\theta$ in terms of arc length $s$ and radius $r$?
What is the definition of radian measure of a central angle $\theta$ in terms of arc length $s$ and radius $r$?
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$\theta=\frac{s}{r}$. Radian measure equals arc length divided by radius.
$\theta=\frac{s}{r}$. Radian measure equals arc length divided by radius.
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Find the radius $r$ when arc length $s=10\pi$ and $\theta=\frac{5\pi}{2}$ (radians).
Find the radius $r$ when arc length $s=10\pi$ and $\theta=\frac{5\pi}{2}$ (radians).
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$4$. Use $r=\frac{s}{\theta}$: $r=\frac{10\pi}{\frac{5\pi}{2}}=4$.
$4$. Use $r=\frac{s}{\theta}$: $r=\frac{10\pi}{\frac{5\pi}{2}}=4$.
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Identify the central angle $\theta$ in radians if the arc length equals the radius, $s=r$.
Identify the central angle $\theta$ in radians if the arc length equals the radius, $s=r$.
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$1$. When $s=r$, then $\theta=\frac{s}{r}=1$ radian.
$1$. When $s=r$, then $\theta=\frac{s}{r}=1$ radian.
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Find the sector area $A$ when $r=4$ and arc length $s=6$ (use $A=\frac{1}{2}rs$).
Find the sector area $A$ when $r=4$ and arc length $s=6$ (use $A=\frac{1}{2}rs$).
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$12$. Use $A=\frac{1}{2}rs$: $A=\frac{1}{2}(4)(6)=12$.
$12$. Use $A=\frac{1}{2}rs$: $A=\frac{1}{2}(4)(6)=12$.
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What is the area formula $A$ of a circle in terms of radius $r$?
What is the area formula $A$ of a circle in terms of radius $r$?
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$A=\pi r^2$. Circle area is $\pi$ times radius squared.
$A=\pi r^2$. Circle area is $\pi$ times radius squared.
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What is the sector area formula $A$ for a central angle $\theta$ in radians and radius $r$?
What is the sector area formula $A$ for a central angle $\theta$ in radians and radius $r$?
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$A=\frac{1}{2}r^2\theta$. Sector area is half the product of radius squared and angle.
$A=\frac{1}{2}r^2\theta$. Sector area is half the product of radius squared and angle.
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What is the radian measure of $180^\circ$?
What is the radian measure of $180^\circ$?
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$\pi$. 180° equals $\pi$ radians (half circle).
$\pi$. 180° equals $\pi$ radians (half circle).
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What is the radian-to-degree conversion formula for an angle of $x$ radians?
What is the radian-to-degree conversion formula for an angle of $x$ radians?
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$x\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.
$x\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.
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What is the degree-to-radian conversion formula for an angle of $x^\circ$?
What is the degree-to-radian conversion formula for an angle of $x^\circ$?
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$x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
$x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.
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What is the radian measure of a full revolution (one complete circle)?
What is the radian measure of a full revolution (one complete circle)?
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$2\pi$. A full circle spans $2\pi$ radians (360°).
$2\pi$. A full circle spans $2\pi$ radians (360°).
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Find the arc length $s$ when $r=5$ and $\theta=\frac{3\pi}{2}$ (radians).
Find the arc length $s$ when $r=5$ and $\theta=\frac{3\pi}{2}$ (radians).
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$\frac{15\pi}{2}$. Use $s=r\theta$: $s=5\cdot\frac{3\pi}{2}=\frac{15\pi}{2}$.
$\frac{15\pi}{2}$. Use $s=r\theta$: $s=5\cdot\frac{3\pi}{2}=\frac{15\pi}{2}$.
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