SAT Math · Geometry and Trigonometry · Skill 4 of 4

Circles, arcs and radians

Find arc lengths and sector areas, switch between degrees and radians, use the circle angle rules, and read and write the equation of a circle, including by completing the square.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up
Start with the sector lab →Jump to the challenge
Hi, I’m Io. Every arc and sector question asks one thing: what fraction of the whole circle is this?
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Slice the circle

Explore · 5 min

This circle has radius 6, so its circumference is 12π and its area is 36π. Open the slice with the slider. The arc is part of the circumference, and the sector is part of the area. Both are the same fraction: angle ÷ 360.

Arc length and sector area

Practise · 5 min

Arc length(angle ÷ 360) × 2πr
Sector area(angle ÷ 360) × πr²
Worked example: a sector has radius 9 and a central angle of 80°. Find its arc length and area.
  1. Fraction of the circle: 80 ÷ 360 = 2/9
  2. Arc length: 2/9 × 2π(9) = 2/9 × 18π = 4π
  3. Sector area: 2/9 × π(9²) = 2/9 × 81π = 18π

Radians

Practise · 5 min

A radian is another unit for angles. One radian is the angle that makes an arc exactly one radius long. A full turn is 2π radians, so:

180° = π radians
  • Degrees to radians: multiply by π/180. So 60° = π/3
  • Radians to degrees: multiply by 180/π. So 3π/4 = 135°
  • With the angle in radians, arc length = rθ

Circle angle rules

Learn · 5 min

Inscribed angleHalf the central angle on the same arc
Angle in a semicircleStanding on a diameter, it is always 90°
TangentMeets the radius at 90°, so look for a right triangle

The equation of a circle

Explore · 5 min

Every point on a circle is the same distance r from the centre (h, k). The Pythagorean theorem turns that into an equation:

(x − h)² + (y − k)² = r²

Watch the signs: a centre at x = 2 gives (x − 2)², and a centre at y = −1 gives (y + 1)². The right side is r², not r.

Completing the square

Practise · 6 min

The SAT often hides the centre and radius in an expanded equation. Complete the square for x and for y to get back to the standard form.

Worked example: x² + y² − 6x + 4y − 12 = 0
  1. Group and move the number: (x² − 6x) + (y² + 4y) = 12
  2. Half of −6 is −3, and (−3)² = 9. Half of 4 is 2, and 2² = 4. Add both to each side: (x² − 6x + 9) + (y² + 4y + 4) = 25
  3. Factor: (x − 3)² + (y + 2)² = 25
  4. Centre (3, −2), radius √25 = 5

ShortcutFor x² + y² + Dx + Ey + F = 0, the centre is (−D/2, −E/2). Knowing the reason matters for the radius, so practise the full method too.

Circles in Desmos

Tools · 4 min

Use the built-in Desmos calculator to see a circle, test a point, or find where a line crosses it. Press play on each card.

Circles in real life

Scenarios · 6 min

Each tab is a situation you could meet on test day. Answer every question to finish the chapter.

Spot the trap

Detective · 3 min

Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.

Final challenge

8 questions · 10 min

Key ideas to carry in
FractionAngle ÷ 360 of the whole circle.
Arc and sectorFraction × 2πr, or fraction × πr².
Radians180° = π. Arc length = rθ in radians.
Angle rulesInscribed = half central. Semicircle 90°. Tangent ⊥ radius.
Equation(x − h)² + (y − k)² = r². Watch the signs.
Complete the squareHalve, square, add to both sides.
Challenge