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 phútTEKS 12A,12B,12C,12D,12E幾何学
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
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.
Same arc on the right circle, but the angle's vertex is on the circle (not at center) → halved.
Central angle = arc measureInscribed angle = ½ · arc measureBoth 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.
Central + inscribed
In circle O, chord AB creates a central angle of 110°. Point C is on the major arc. What is the measure of inscribed angle ACB?
The inscribed angle is half the intercepted arc. Point C is on the major arc, so angle ACB intercepts the minor arc AB = 110°. ∠ACB = 110° / 2 = 55°.
Circumference, area, sector
Circumference: C = 2πrArea: A = πr²Arc length: (θ/360) · 2πrSector area: (θ/360) · πr²For 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 = r2This 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.
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?
📌 Step 1: Recall the standard form (x − h)² + (y − k)² = r² where center = (h, k) and radius = r 📌 Step 2: Match to the given equation (x + 3)² + (y − 2)² = 49 (x − (−3))² + (y − 2)² = 7² 📌 Step 3: Read the center and radius center = (−3, 2) radius = √49 = 7 💡 Common mistake: Watch the signs! (x + 3) means h = −3 (negative) (y − 2) means k = 2 (positive) The sign in the equation is OPPOSITE to the coordinate.
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
Check yourself
Quick check #1
An inscribed angle in a circle measures 40°. What is the measure of the central angle that intercepts the same arc?
Inscribed Angle Theorem: a central angle is TWICE the inscribed angle that intercepts the same arc. 40° × 2 = 80°.
Quick check #2
A tangent line to a circle is drawn at the point of tangency. The angle between the tangent and the radius drawn to that point is:
This is a foundational circle theorem: a tangent line is ALWAYS perpendicular (90°) to the radius drawn to the point of tangency.