SAT circles: arcs, sectors, and radians as fractions of a whole
Arc length, sector area, and radians all come from one idea: what fraction of the circle are you working with? Drag an interactive sector and watch every measurement update together.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- A central angle of θ degrees cuts off θ/360 of the circle: the same fraction of its circumference and its area.
- Convert degrees to radians by multiplying by π/180. 180° is π radians.
- An inscribed angle is half the central angle that intercepts the same arc.
In this article
Everything is a fraction of the circle
A full circle has 360°, circumference \(\displaystyle 2\pi r\), and area \(\displaystyle \pi r^2\). A sector with a central angle of θ degrees is \(\displaystyle \frac{\theta}{360}\) of the whole circle. Its arc is that fraction of the circumference, and its area is that fraction of the circle’s area.
Move the slider to change the central angle of a circle with radius 6. Every measurement uses the same fraction.
Try it yourself · interactive
Drag the central angle
- Fraction of circle
- 90/360 = 1/4
- Radians
- π/2
- Arc length
- 3Ï€
- Sector area
- 9Ï€
arc = 1/4 × 2π(6)  · area = 1/4 × π(6)²
Radians measure angles with the radius
One radian is the angle whose arc is exactly one radius long. A full circle has \(\displaystyle 2\pi\) radians, so \(\displaystyle 180^\circ = \pi\) radians. To convert degrees to radians, multiply by \(\displaystyle \frac{\pi}{180}\). To convert radians to degrees, multiply by \(\displaystyle \frac{180}{\pi}\).
Radians make arc length simple: \(\displaystyle s = r\theta\), with θ in radians. A radius of 9 and an angle of \(\displaystyle \frac{2\pi}{3}\) give an arc of \(\displaystyle 9 \times \frac{2\pi}{3} = 6\pi\).
See the skill
Common conversions
30°
Ï€/6
1/12 of a circle.
45°
Ï€/4
1/8 of a circle.
90°
Ï€/2
A right angle, 1/4 of a circle.
180°
Ï€
A straight angle, half a circle.
Inscribed angles are half
An inscribed angle has its vertex on the circle. It is half the central angle that intercepts the same arc. If a central angle measures 100°, any inscribed angle intercepting that arc measures 50°. An inscribed angle that intercepts a semicircle is always 90°.
Your turn
Quick practice
A circle has a radius of 10. What is the length of an arc intercepted by a central angle of 72°?
Hint: What fraction of 360 is 72?
Practice: try these before checking the answers
Find the fraction of the circle first.
- 1. Convert 60° to radians.
- 2. Convert 3Ï€/4 radians to degrees.
- 3. Find the area of a sector with radius 4 and central angle 45°.
- 4. Find the arc length for radius 9 and central angle 2Ï€/3 radians.
- 5. An inscribed angle intercepts a 100° arc. What is its measure?
Sources & further reading
One step from here
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