Circles

Circles

1. Basic Properties

Odr

Property Formula Relationship
Radius Half the diameter
Diameter Longest chord
Area Space inside
Circumference Perimeter of circle

On the SAT, you'll almost always need to extract from a given diameter, area, or circumference before solving.

2. Arcs and Sectors

sµr

Arc length (portion of circumference):

Sector area (pizza slice):

Using degrees instead:

Degree ↔ Radian Conversion

Degrees Radians

3. Equation of a Circle (Coordinate Geometry)

xy¡4¡3¡2¡1123456¡4¡3¡2¡1123456(h;k)r(x;y)

Standard Form

  • Center:
  • Radius:

Watch the signs: → center , radius .

General (Expanded) Form

To convert to standard form → complete the square:

Example:

Center: , Radius:

Finding Center and Radius from General Form (Quick Formula)

Given :

4. Lines and Chords

PrTangentChord

Tangent Lines

A tangent touches the circle at exactly one point.

This creates a right angle → use the Pythagorean theorem to find unknown lengths.

Chords

A chord is a segment connecting two points on the circle. The diameter is the longest chord.

Key properties:

  • A radius perpendicular to a chord bisects the chord
  • Equal chords are equidistant from the center
  • The perpendicular from the center to a chord bisects the chord

5. Inscribed Angles and Central Angles

O2®A®BC

Inscribed Angle Theorem

Both angles intercept the same arc.

Special Cases

Configuration Result
Inscribed angle on a semicircle Always
Two inscribed angles on the same arc They are equal
Central angle Equals the intercepted arc

6. Intersecting Chords, Secants, and Tangents

Two Chords Intersecting Inside

Two Secants from External Point

(where and are the near and far intersections of one secant, and , for the other)

Tangent-Secant from External Point

SAT Tips

  1. Always identify first — most problems give you diameter, area, or circumference and expect you to work with the radius
  2. Completing the square appears frequently — practice converting general form to standard form quickly
  3. Tangent = right angle — when you see a tangent line, set up Pythagorean theorem
  4. Arc/sector as a fraction — think of it as (degrees) or (radians) of the full circle
  5. Inscribed angle in semicircle = 90° — this shortcut appears on nearly every SAT
  6. On Desmos — type to graph a circle and visually verify intersections or tangent points
  7. Coordinate problems — if asked for a point on the circle, substitute into the equation and solve; if two points, use distance formula for the diameter
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