Todo círculo tem o mesmo punhado de partes nomeadas — e quase toda fórmula de círculo é apenas um relacionamento entre duas delas. Uma vez que você possa rotular o raio, diâmetro, corda, arco, setor, segmento, tangente e secante em uma figura, o resto da geometria do círculo segue naturalmente. Este guia percorre cada parte uma a uma com a fórmula que depende dela.
The center is the defining point of a circle — every point on the circle is exactly the radius distance away from it. The radius (plural: radii) is the most-used measurement in circle formulas because it's the simplest. Anything else you can compute (area, circumference, diameter, sector, chord length) ultimately reduces to a formula involving r.
Formulas that use the radius: Area A = πr², Circumference C = 2πr, Diameter d = 2r, Equation of a circle (x − h)² + (y − k)² = r².
The diameter is the longest chord in any circle — a straight line through the center, ending on the circle on both sides. Its length is always exactly twice the radius: d = 2r. If you only know the diameter, you can still compute everything: r = d/2, A = πd²/4, C = πd.
A common student trap: confusing diameter with radius in the area formula. If you mistakenly plug d into A = πr², you'll get an answer 4× too large. Always halve first if the figure labels the diameter.
A chord is any segment whose endpoints lie on the circle. The diameter is the special chord that happens to pass through the center; every other chord is shorter than the diameter.
Chord length formula: c = 2r × sin(θ/2), where θ is the central angle subtending the chord (the angle between the two radii drawn to the chord's endpoints).
Example: In a circle of radius 10, a chord subtended by a 60° central angle has length c = 2 × 10 × sin(30°) = 20 × 0.5 = 10. (When θ = 60°, the chord equals the radius — that's the equilateral-triangle case.)
An arc is a piece of the circumference. There are two kinds:
Arc length formula: L = r × θ (radians), or L = (θ°/360) × 2πr (degrees).
Example: In a circle of radius 6, a 90° arc has length (90/360) × 2π × 6 = (1/4) × 12π = 3π ≈ 9.42.
A sector is the pie-slice region between two radii — bounded by the radii on two sides and an arc on the curved side. Think pizza slice.
Sector area formula: A_s = ½ × r² × θ (radians), or A_s = (θ°/360) × πr² (degrees).
Example: A 45° sector in a circle of radius 8 has area (45/360) × π × 64 = (1/8) × 64π = 8π ≈ 25.13.
A segment is easily confused with a sector but it's a different region. Imagine drawing a single chord across a circle — the chord divides the circle into two regions, each bounded by the chord and an arc. Each region is a segment. (A sector, in contrast, is bounded by two radii plus an arc.)
Segment area formula: A_seg = ½ × r² × (θ − sin θ), with θ in radians.
Mnemonic: a sector is what you'd cut with two straight knife strokes from the center; a segment is what you'd cut with one straight stroke across.
A tangent is a line that just barely touches the circle — meeting it at exactly one point (the "point of tangency") without crossing into the interior. The key property:
A tangent line is always perpendicular to the radius drawn to the point of tangency.
This is the foundation of dozens of geometry-proof problems and shows up in calculus when you find the tangent line to a curve. If a problem mentions "the tangent at point P", instantly draw the radius OP — the angle there is 90°.
A secant is a line that cuts the circle at exactly two points. Visualize it as a chord whose endpoints have been extended into a full line on both sides.
The relationship that examiners love: if two secants are drawn from an external point P, the product of the two segments (external × full) is the same for both secants. This is the Power of a Point theorem.
The circumference is the total distance around the circle — its perimeter. In school textbooks the word "perimeter" is used; in geometry the technical word is "circumference". Both refer to the same length:
C = 2π × r = π × d
That ratio C/d = π (≈ 3.14159) is identical for every circle. It's the most famous constant in mathematics.
Picture a circle with center O. Inside:
Re-drawing this figure from memory is the single best study exercise for circle vocabulary. Once the labels become automatic, all the formulas are just relationships between these parts.
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