3.3 Circles and Measurement

Why This Section Matters

Circles combine equation form, arc measurement, and angle relationships. Praxis 5165 items ask you to write circle equations from center-radius data, find arc length and sector area, and reason about inscribed and central angles. Measurement questions often embed π — know when to leave answers in terms of π versus a decimal approximation.

Circle items frequently cross domains: you may need the distance formula to find radius from diameter endpoints, then write the standard equation — linking Section 3.2 skills to this section.

Standard Form of a Circle

A circle with center (h, k) and radius r satisfies:

(x − h)² + (y − k)² = r²

Watch the signs: center (2, −1) produces (x − 2)² + (y + 1)² = r² because (y − (−1)) = y + 1.

General form x² + y² + Dx + Ey + F = 0 can be converted by completing the square, but Praxis items usually give center-radius information directly.

Worked Example: Equation from Center and Radius

Center (2, −1), radius 4:

(x − 2)² + (y + 1)² = 16

The right side is r² = 16, not 4.

Worked Example: Equation from a Diameter

Diameter endpoints (1, 2) and (7, 6).

• Center (midpoint): ((1 + 7)/2, (2 + 6)/2) = (4, 4)
• Diameter length: √[(7 − 1)² + (6 − 2)²] = √(36 + 16) = √52 = 2√13
• Radius r = √13, so r² = 13

Equation: (x − 4)² + (y − 4)² = 13

Arc Length and Sector Area

For a circle of radius r and central angle θ in degrees:

The factor θ/360 is the fraction of the full circle. Simplify the fraction before multiplying — 60° gives 1/6, 90° gives 1/4, 120° gives 1/3.

Worked Example: Sector Area

Radius 6, central angle 60°.

Full circle area = π(6²) = 36π. Sector fraction = 60/360 = 1/6.

Sector area = (1/6)(36π) = 6π square units.

Worked Example: Arc Length

Same circle (r = 6, θ = 60°):

Arc length = (1/6)(2π · 6) = (1/6)(12π) = 2π units.

Worked Example: Larger Sector

Radius 10, central angle 120°. Fraction = 120/360 = 1/3.

Sector area = (1/3)π(100) = (100π/3) square units.

Angle Relationships in Circles

• A central angle has its vertex at the center; its measure equals its intercepted arc.
• An inscribed angle has its vertex on the circle; its measure is half the intercepted arc.
• An angle inscribed in a semicircle is a right angle (Thales' theorem).

If two inscribed angles intercept the same arc, they are congruent. A central angle and an inscribed angle intercepting the same arc satisfy: central = 2 × inscribed.

Chords, Tangents, and Secants

• A radius to a tangent point is perpendicular to the tangent line.
• Two tangents from an external point to a circle have equal lengths.
• In the same circle, congruent chords are equidistant from the center.
• The perpendicular from the center to a chord bisects the chord.

These facts support calculation items and short proof fragments.

Right Triangle in a Circle

If a chord of length 8 is 3 units from the center of a circle with radius 5, half the chord is 4, forming a 3-4-5 right triangle between center, foot of perpendicular, and chord endpoint. This pattern appears when finding chord length or distance from center.

Composite Figures with Semicircles

For a rectangle topped by a semicircle sharing the rectangle's width as diameter, total area = rectangle area + (1/2)πr². Identify whether the radius equals half the width before substituting.

Common Traps

• Writing r instead of r² on the right side of the circle equation.
• Sign errors on the y-term when the center has a negative coordinate.
• Using θ/180 instead of θ/360 for sector fractions.
• Confusing diameter with radius in arc-length formulas.
• Reporting arc length when the item asks for sector area (or vice versa).

Section Checklist

• Identify center (h, k) and radius r before writing (x − h)² + (y − k)² = r².
• For arc and sector items, compute θ/360 first as a simplified fraction.
• Decide early whether the item wants an exact π form or a decimal approximation.
• Link diameter endpoints to midpoint and distance formulas when needed.

Inscribed Quadrilaterals

A quadrilateral is cyclic (inscribable in a circle) when opposite angles are supplementary. If ∠A + ∠C = 180° and ∠B + ∠D = 180°, the vertices lie on one circle. Praxis may ask which condition guarantees a cyclic quadrilateral.

Arc Measure vs. Arc Length

Arc measure is in degrees (or radians). Arc length is a linear distance along the circle. A 90° arc on a small circle has the same measure as a 90° arc on a large circle but a shorter arc length — measure depends on radius.

Link to Praxis Practice

Circle equation items frequently test sign errors on (y − k). When reviewing misses, rewrite the center in words before choosing an answer: "center two right, one down" maps to (2, −1) and (y + 1)² on the left side.

Radian Connection (Secondary)

Some calculus-adjacent items use radians: arc length L = rθ when θ is in radians. To convert, multiply degrees by π/180. A 60° angle is π/3 radians, so on a circle of radius 6, L = 6 · (π/3) = 2π — matching the degree formula result. Knowing both forms prevents confusion when a stem mixes function graphs with circular motion.

Tangent-Line Slope

A tangent to a circle is perpendicular to the radius at the point of tangency. If the radius from center (2, 3) to point (6, 3) is horizontal, the tangent at (6, 3) is vertical (undefined slope). This perpendicular-radius fact links circle geometry to the slope rules from Section 3.2.