Friday
Friday: Arcs, Sectors and Radians
Comet swings the spotlight across the stage. "It sweeps 60 degrees. How much rim does that light up?"
"A sixth of the circle," Wren says. "The turntable's rim is 18.84 meters, so a sixth is 3.14 meters."
"And on the spotlight pool, a sixth of its rim," Comet says. "Smaller circle, smaller arc, same angle."
Nova projects three circles of different sizes, each with a 90° arc glowing. "Would you like a hint? Divide each arc by its radius."
Wren does it. "1.57. 1.57. 1.57. Always the same number for 90 degrees."
"So arc over radius measures the angle," Comet says. "Any circle. What can we make of that?"
"A new unit," Wren says. "Our designer, meet the radian. Then review the week."
Arc length and sectors
An arc is the same fraction of the circumference as its central angle is of 360°.
Arc length = (angle ÷ 360) × 2 × π × radius. For the turntable and 60°: (60 ÷ 360) × 2 × 3.14 × 3 = 3.14 meters.
A sector is the pie-slice region between two radii. Its area is the same fraction of the circle's area.
Sector area = (angle ÷ 360) × π × radius². For 60° on the turntable: (60 ÷ 360) × 3.14 × 3² = 4.71 square meters.
Why arc ÷ radius measures the angle
All circles are similar. So a 90° arc on a bigger circle is a scaled copy of a 90° arc on a smaller one.
Scale the radius by k and the arc scales by k too. So arc ÷ radius comes out the same on every circle.
That ratio depends only on the angle. It is the radian measure of the angle. The crew's table shows it for 90°.
| Radius (m) | 90° arc (m) | Arc ÷ radius |
|---|---|---|
| 1 | 1.57 | 1.57 |
| 2 | 3.14 | 1.57 |
| 3 | 4.71 | 1.57 |
Arc ÷ radius for 90° is 1.57 every time, which is π ÷ 2. So 90° = π/2 radians.
A full turn's arc is the whole circumference, 2πr, so 360° = 2π radians and 180° = π radians.
To convert, use 180° = π: divide the degrees by 180 and keep the π. 60° is 60/180 of π, which is π/3.
| Degrees | Radians |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Arcs in a day
- The minute hand of a clock sweeps 90° in 15 minutes; its tip travels a quarter of a circle.
- A door swinging open sweeps a sector of floor. A wider swing is a bigger angle and a bigger sector.
- A sprinkler head turning back and forth waters a sector. Its radius is how far the water reaches.
- A windshield wiper sweeps a ring-shaped piece of a sector, the difference of two sectors.
- Read the question.
- Tap your answer.
The spotlight sweeps 60° across the turntable, radius 3 meters. How long is the lit arc of rim, in meters? Use 3.14 for π.
The spotlight lights a 60° sector of the turntable, radius 3 meters. What is its area in square meters? Use 3.14 for π.
Two spokes apart is 120°. How long is the rim between them, in meters, on the 3 meter turntable? Use 3.14 for π.
Half the turntable is a 180° sector. What is its area in square meters, with radius 3 meters? Use 3.14 for π.- Read the question.
- Tap your answer.
| Week review | True or false? |
|---|---|
| Any two circles are similar. | ? |
| An inscribed angle equals its arc. | ? |
| An angle in a semicircle is a right angle. | ? |
| Arc length divided by radius depends on the size of the circle. | ? |
| 180° equals π radians. | ? |
| 3.14 ÷ 3 = 1.05 | ? |
Then compute the sector it sweeps. Use 3.14 for π.
Great review, designer. Tomorrow is Shop Day: a string circle for your family and the square-corner trick.