SAT-78: Area of a Sector of a Circle

SAT-78: Area of a Sector of a Circle

Description: A sector is a "pizza slice" of a circle — the region between two radii and the arc between them. Its area is just the matching fraction of the whole circle's area, exactly parallel to how arc length worked in SAT-77.

The sector area formula

sector area = (central angle / 360°) × πr² (degrees)
or sector area = ½ r² θ when θ is in radians.

The whole circle's area is πr², and the sector is the angle's fraction of it. (Oʻzbekcha: sektor yuzasi — butun doira yuzasining markaziy burchakka mos qismi.)

The parallel with arc length

Both arc length and sector area use the same fraction, central angle ÷ 360°. The only difference is what you multiply it by: circumference (2πr) for arc length, area (πr²) for sector area. Find the fraction once, then choose the right "whole." (Oʻzbekcha: yoy va sektor bir xil ulushdan foydalanadi — faqat butuni farq qiladi.)

Worked Example 1 — quarter circle

A circle has radius 4. Find the area of a sector with central angle 90°.

• Fraction = 90/360 = 1/4. Whole area = π(4)² = 16π.
• Sector = (1/4)(16π) = 4π.

Worked Example 2 — other angles

A circle has radius 6 and a sector with central angle 120°. Find the sector area.

• Fraction = 120/360 = 1/3. Whole area = π(6)² = 36π.
• Sector = (1/3)(36π) = 12π.

(Oʻzbekcha: 120° — doiraning uchdan biri, shuning uchun yuzaning uchdan biri.)

Worked Example 3 — work backward to the angle

A circle of radius 10 has a sector of area 25π. What is the central angle?

• Whole area = 100π. Fraction = 25π / 100π = 1/4.
• Central angle = (1/4)(360°) = 90°.

Tip: keep answers in terms of π unless told to use a decimal — the SAT usually wants the exact form. (Oʻzbekcha: javobni π bilan qoldiring, agar oʻnli kasr soʻralmasa.)

Worked Example 4 — perimeter of a sector

A sector of a circle with radius 9 has a central angle of 40°. Find its full perimeter (not just the area).

• The perimeter is two radii plus the arc: 9 + 9 + arc.
• Arc = (40/360)(2π·9) = (1/9)(18π) = 2π. So perimeter = 18 + 2π ≈ 24.28.

Don't forget the two straight radius edges — a sector's boundary is two radii and the arc. (Oʻzbekcha: sektor perimetri — ikkita radius va yoy uzunligi yigʻindisi.)

Sector area vs triangle area — don't confuse them

A sector is the curved pizza slice; it is not the triangle formed by the two radii and a straight chord. The sector uses πr² and the angle fraction; the triangle uses ½·base·height. SAT problems sometimes ask for the region between them (the sector minus the triangle, called a segment), so read carefully whether the boundary is curved or straight. (Oʻzbekcha: sektor — egri "tilim"; u ikki radius va vatardan iborat uchburchak emas.)

Practice 1

A circle of radius 3 has a sector with central angle 60°. Find its area.

Practice 2

A sector of a circle with radius 8 has area 8π. Find the central angle.

Key words — Kalit soʻzlar

• Sector — sektor
• Central angle — markaziy burchak
• Area — yuza
• Radius — radius
• Fraction — ulush (qism)
• Circle — doira (aylana)
• Radii (plural) — radiuslar
• Arc — yoy
• π (pi) — pi

Summary

• Sector area = (central angle / 360°) × πr².
• Same fraction as arc length — just multiply by the circle's area instead of circumference.
• Work backward by setting the sector ÷ whole area equal to the angle fraction.