学习 · 几何

Circles: Arcs, Chords, Tangents, and Inscribed Angles

The Geometry CBE has a whole TEKS category just for circles. Master the central-angle / inscribed-angle / tangent rules and the circle-equation form (x − h)² + (y − k)² = r².

9 分钟 TEKS 12A,12B,12C,12D,12E 几何

A whole TEKS category just for circles

On the Texas Geometry CBE, TEKS 12A–12E is dedicated to circles — arcs, chords, tangents, central and inscribed angles, sector area, and the equation of a circle. About 1 in 8 Geometry questions tests one of these. Memorize the rules visually once and you have an entire test category solved.

The vocabulary

O radius diameter chord tangent 90° arc
The five parts you must name on sight. The tangent always meets the radius at 90°.
Radius (r)
From center to any point on the circle.
Diameter (d)
Through the center, edge to edge. d = 2r.
Chord
Any line segment with endpoints on the circle. The diameter is the longest chord.
Tangent
A line that touches the circle at exactly one point. Perpendicular to the radius at that point.
Arc
A piece of the circle's circumference between two points.
Sector
A “pizza slice” of the circle — bounded by two radii and an arc.

Central angle vs inscribed angle

The single biggest source of CBE circle questions: the inscribed angle is half the central angle when both subtend the same arc.

110° CENTRAL ANGLE = arc 110° 55° INSCRIBED ANGLE = half of arc
Same arc on the right circle, but the angle's vertex is on the circle (not at center) → halved.
Central angle = arc measure Inscribed angle = ½ · arc measure Both intercept the same arc, but the inscribed angle is always exactly half.
Special case

An inscribed angle that intercepts a semicircle (diameter) is always 90°. This is why right triangles inscribed in circles always have the diameter as the hypotenuse.

Practice

Central + inscribed

In circle O, chord AB creates a central angle of 110°. Point C is on the major arc. Find the inscribed angle ACB.

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Circumference, area, sector

Circumference: C = 2πr Area: A = πr² Arc length: (θ/360) · 2πr Sector area: (θ/360) · πr² For arc length and sector area, θ is the central angle in degrees.

The equation of a circle

If a circle has center (h, k) and radius r, every point (x, y) on it satisfies:

(x − h)2 + (y − k)2 = r2 This is the distance formula squared. Everything inside parentheses gets negated to find the center.
Sign flip trap

(x + 3)2 means h = −3, not +3. The formula uses (x − h), so a plus inside means a minus outside. Same for (y + 2)2 → k = −2.

Practice

Read center & radius from the equation

The equation of a circle is (x + 3)² + (y − 2)² = 49. What is the center and radius?

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3-second recap

  • Tangent ⊥ radius at the point of tangency
  • Inscribed angle = ½ central angle (same arc)
  • Inscribed angle on a diameter = 90°
  • Circle equation: (x − h)² + (y − k)² = r² — flip the signs to get the center
  • Sector / arc are just fractions (θ/360) of the whole circle