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Circle
Properties

Master the key relationships between radii, diameters, and chords to solve circle problems with confidence.

SSAT Middle Level

What Is a Circle?

A circle is the set of all points in a plane that are the same distance from a fixed point called the center. That fixed distance is called the radius.

\(\textit{O}\) \(r\) \(\textit{A}\) Every point on the circle is exactly \(r\) away from \(\textit{O}\)

Key Vocabulary

Radius
A segment from the center to any point on the circle.
Diameter
A segment through the center connecting two points on the circle. It equals \(2r\).
Chord
Any segment whose endpoints both lie on the circle.

Anatomy of a Circle

\(\textit{O}\) \(\textit{A}\) radius \(\textit{B}\) \(\textit{C}\) diameter \(\textit{D}\) \(\textit{E}\) chord

The radius (solid green), diameter (solid purple), and chord (dashed gold) are the three fundamental segments of every circle.

Core Rule: All Radii Are Equal

\(\textit{O}\) \(\textit{A}\) \(r\) \(\textit{B}\) \(r\) \(\textit{C}\) \(r\) \(\textit{D}\) \(r\) \(\overline{OA} = \overline{OB} = \overline{OC} = \overline{OD} = r\)

Diameter = \(2r\)

\(\textit{O}\) \(\textit{P}\) \(\textit{Q}\) \(r\) \(r\) \(d = 2r\)

This means if a problem tells you the diameter is 14 cm, you instantly know the radius is 7 cm.

Chords and the Center

\(\textit{O}\) \(\textit{A}\) \(\textit{B}\) \(\textit{M}\) \(\overline{AM}\) \(\overline{MB}\) A perpendicular from the center bisects the chord

Radius vs Chord vs Diameter

Property Radius Chord Diameter
Endpoints Center + circle Circle + circle Circle + circle (through center)
Passes through center? Always Not necessarily Always
Length \(r\) \(\leq 2r\) \(2r\)
Is it a chord? No Yes Yes

Equal Chords, Equal Distance

\(\textit{O}\) \(\textit{A}\) \(\textit{B}\) \(d_1\) \(\textit{C}\) \(\textit{D}\) \(d_2\) 220 220 Equal chords \(\rightarrow\) Equal distances from center

How to Solve Circle Problems

1
Identify the given info. Do you know the radius, diameter, or a chord length?
2
Draw radii to key points. Any segment from the center to the circle is a radius: use \(r\) everywhere.
3
Look for right triangles. A perpendicular from the center to a chord creates two right triangles.
4
Apply key relationships. Use \(d = 2r\), equal radii, or the perpendicular bisector property to find unknowns.
5
Check your answer. Does the chord length make sense? (It must be \(\leq 2r\).)

Example 1: Finding the Radius

A circle has a diameter of 26 cm. What is the radius?

26 cm \(\textit{P}\) \(\textit{Q}\) \(\textit{O}\)

Example 2: Equal Radii in Action

In circle \(O\), points \(A\) and \(B\) are on the circle. If \(\overline{OA} = 10\), what is \(\overline{OB}\)?

\(\textit{O}\) \(\textit{A}\) 10 \(\textit{B}\) ? All radii are equal: \(\overline{OB} = \overline{OA} = 10\)

Example 3: Bisecting a Chord

In circle \(O\) with radius 13, chord \(\overline{AB} = \) 24. A perpendicular from \(O\) meets the chord at \(M\). Find \(\overline{OM}\).

\(\textit{O}\) \(\textit{A}\) \(\textit{B}\) 24 \(\textit{M}\) ? 13 12

Longer Chord = Closer to Center

In a circle, chord \(\overline{AB} = 16\) and chord \(\overline{CD} = 10\). Which chord is closer to the center?

\(\textit{O}\) \(\textit{A}\) \(\textit{B}\) 16 \(\textit{C}\) \(\textit{D}\) 10 Longer chord = shorter distance to center

Common Mistakes

✕ Confusing Radius and Diameter
Using the diameter where the radius is needed (or vice versa). Always check: \(d = 2r\).
✕ Calling a Radius a Chord
A radius starts at the center, not on the circle. A chord must have both endpoints on the circle.
✕ Forgetting the Bisect Rule
When using the perpendicular from the center to a chord, remember to halve the chord before applying the Pythagorean theorem.

Summary of Circle Properties

Property Key Fact
All radii Are equal in length
Diameter \(d = 2r\) : the longest chord
Perpendicular from center Bisects the chord
Equal chords Are equidistant from the center
Longer chord Is closer to the center

Question 1 of 5

A circle has a radius of 9 cm. What is the diameter?

Question 2 of 5

In circle \(O\), \(\overline{OA} = 7\) and \(\overline{OB} = 7\). Both points \(A\) and \(B\) are on the circle. What is \(\overline{AB}\) called?

Question 3 of 5

A perpendicular from the center of a circle to a chord of length 20 meets it at point \(M\). What is \(\overline{AM}\)?

Question 4 of 5

In a circle with radius 5, the perpendicular distance from the center to chord \(\overline{AB}\) is 3. What is the length of \(\overline{AB}\)?

Question 5 of 5

Which of the following is always true about a diameter of a circle?