Angle and its Measurement
Easy Overview
Ever wondered why we use 360° for a full circle? Or why your calculator has that RAD button nobody touches? Turns out, angles are everywhere — from the slope of a rooftop to the spin of a fan. This chapter is all about understanding how we measure turns and twists, and why radians aren't as scary as they sound.
Degrees and the 360° convention
A degree is 1/360th of a full rotation. Why 360? Probably because ancient astronomers noticed the Sun moves about 1° per day across the sky. Conveniently, 360 also has a ton of divisors, so it's easy to split into halves, thirds, quarters — you name it. So 1° is tiny, but stack 360 of 'em and you've done a full loop.
Radians — the natural angle
Radians feel weird at first, but they're actually the 'natural' way to measure angles. Here's the trick: take a circle of radius 1, walk along the circumference by a distance equal to the radius, and the angle you've covered is 1 radian. That's it. A full circle is about 6.28 radians (that's 2π). So when you see π/2, that's a quarter turn — 90°.
Converting between degrees and radians
Simple formula: radians = degrees × (π / 180). Or the other way: degrees = radians × (180 / π). Memorise that π radians = 180° and you're golden. For example, 60° = π/3, 45° = π/4, 30° = π/6. These show up so often they're basically free marks.
Arc length and area of a sector
If you know the angle in radians, arc length is just radius × angle. Easy. For area of a sector (think pizza slice), it's (1/2) × r² × angle. Both formulas are dead simple because radians make the math clean. Degrees would give you ugly fractions — another reason radians exist.
Key Points
- •1° = 1/360th of a full rotation
- •1 radian ≈ 57.3° — the angle where arc length equals radius
- •π radians = 180° — the golden conversion rule
- •Arc length = r × θ (θ in radians)
- •Area of sector = ½ r²θ
- •Degree is convenient, radian is natural — use radian for calculus and formulas
Practice Questions
- Convert 150° to radians and 3π/5 to degrees.
- A pendulum swings through an angle of 30°. If the string is 40 cm long, find the distance travelled by the tip.
- Find the area of a sector with radius 7 cm and central angle 60°.
- The sum of two angles is 120° and their difference is 30°. Find the angles in radians.
- A horse is tied to a pole with a 14 m rope. If it grazes an arc of length 22 m, what angle does the rope cover?