SAT-77: Radians vs. Degrees and Arc Length

SAT-77: Radians vs. Degrees and Arc Length

Description: Angles can be measured in degrees or radians. The SAT expects you to convert between them and to find an arc length — the distance along part of a circle's edge.

The conversion fact

A full circle is 360° = 2π radians, so 180° = π radians.

• Degrees → radians: multiply by π/180.
• Radians → degrees: multiply by 180/π.

(Oʻzbekcha: 180° = π radian; gradusdan radianga oʻtish uchun π/180 ga koʻpaytiramiz.)

Worked Example 1 — degrees to radians

Convert 90° to radians.

• 90 × π/180 = π/2 radians.

Worked Example 2 — radians to degrees

Convert 3π/4 radians to degrees.

• (3π/4) × (180/π) = 3 × 180 / 4 = 135°.

(Oʻzbekcha: radiandan gradusga oʻtish uchun 180/π ga koʻpaytiramiz.)

Arc length — a fraction of the way around

An arc is part of the circle's circumference. The whole circumference is 2πr. An arc that spans a central angle is that same fraction of the full circle:

arc length = (central angle / 360°) × 2πr (degrees)
or simply arc length = r × θ when θ is in radians.

(Oʻzbekcha: yoy uzunligi — aylana uzunligining markaziy burchakka mos qismi.)

Worked Example 3 — arc length in degrees

A circle has radius 6. Find the length of an arc with central angle 60°.

• Fraction = 60/360 = 1/6 of the circle.
• Circumference = 2π(6) = 12π. Arc = (1/6)(12π) = 2π.

Shortcut: in radians, arc length is just r × θ. For r = 6 and θ = π/3 (which is 60°): 6 × π/3 = 2π — same answer. (Oʻzbekcha: radianlarda yoy uzunligi = r × θ, juda qulay.)

What a radian actually is

A radian is the angle you get when the arc length equals the radius. Because the full circumference is 2πr, wrapping the radius around the circle fits exactly 2π times — that is why a full circle is 2π radians. This is also why the radian formula arc = rθ is so clean: θ literally counts how many "radius-lengths" of arc you have. Understanding this makes the conversions feel natural instead of memorized. (Oʻzbekcha: bir radian — yoy uzunligi radiusga teng boʻlgan burchak; shuning uchun toʻliq aylana 2π radian.)

Worked Example 4 — find the radius from an arc

An arc of length 10π comes from a central angle of 120° in some circle. Find the radius.

• Fraction = 120/360 = 1/3, so arc = (1/3)(2πr) = (2πr)/3.
• Set (2πr)/3 = 10π → 2πr = 30π → r = 15.

(Oʻzbekcha: yoy formulasidan radiusni topish uchun teskari ishlaymiz.)

Common mistake — mixing the two formulas

Use arc = (angle/360)·2πr only when the angle is in degrees, and arc = rθ only when θ is in radians. Plugging a degree value into rθ gives a hugely wrong answer. When in doubt, convert to one system first. (Oʻzbekcha: rθ formulasi faqat radian uchun; gradusni avval radianga aylantiring.)

Practice 1

Convert 45° to radians.

Practice 2

A circle of radius 10 has an arc with central angle 90°. Find the arc length.

Key words — Kalit soʻzlar

• Degree — gradus
• Radian — radian
• Convert — oʻzgartirish (almashtirish)
• Arc length — yoy uzunligi
• Central angle — markaziy burchak
• Circumference — aylana uzunligi
• Radius — radius
• Fraction — qism (ulush)
• π (pi) — pi

Summary

• 180° = π radians; ×π/180 to go to radians, ×180/π to go to degrees.
• Arc length = (central angle / 360°) × 2πr in degrees.
• In radians, arc length = r × θ.