Overview
A circle has one fixed radius, but questions describe that radius indirectly through chords, sectors, arcs or an equation. The center organizes these descriptions. A perpendicular to a chord creates a right triangle, an arc uses a fraction of the circumference, and a coordinate equation locates the center and radius. We will move between those representations while keeping linear distances, areas and angles separate.
Bisect chords to reveal equal radius squares
Two parallel chords lie on opposite sides of a circle’s center, have lengths 16 and 12, and are 14 units apart. Draw a perpendicular from the center to each chord. It bisects their lengths, creating half-chords of 8 and 6. Let u be the distance to the longer chord and v the distance to the shorter. Because the chords are on opposite sides, u + v = 14.
Both right triangles have the circle’s radius r as hypotenuse, so r² = u² + 8² = v² + 6². Subtracting gives v² − u² = 28, or (v − u)(v + u) = 28. The known sum 14 gives v − u = 2. Hence u = 6 and v = 8, and r = √(6² + 8²) = 10. This agrees with the geometric expectation that the longer chord is nearer the center.
Take the angle fraction of the correct whole
A sector with radius 3 and central angle 120° occupies 120/360 = 1/3 of its disk. The full disk area is π·3² = 9π, so the sector area is 3π. The angle fraction applies to area because sectors with the same radius divide the disk in proportion to their central angles.
The distractor 9π ignores the fraction and returns the whole disk. The value 6π is the circumference, a linear measurement rather than an area. Keeping units in mind prevents these formulas from becoming interchangeable. The radius is squared for area but appears only to the first power in circumference. Before substituting, name the whole quantity and the fraction of it that the sector represents; this makes the chosen formula follow from the picture rather than from whichever symbols happen to be familiar.
Recover an angle from an arc fraction
An arc of radius 9 has length 6π. The full circumference is 2π·9 = 18π, so the arc is 6π/18π = 1/3 of the circle. Its central angle is therefore one third of a full revolution, or 120°. This uses the same fraction principle as the sector problem, but starts with a length instead of an angle.
The answer 240° describes the remaining major arc rather than the given arc. An arc length smaller than half the circumference must correspond to a central angle smaller than 180°, which is a quick reason to reject that option. If a question uses radians, the linear relationship is arc length divided by radius; if it asks for degrees as here, the full-revolution fraction is especially transparent. Keep the angle unit explicit instead of treating a bare numerical angle as interchangeable between the two systems.
Read an equation as a center and radius
The circle (x − 2)² + (y + 1)² = 25 has center (2, −1) and radius 5. Its topmost point lies one radius above the center, so the greatest y-coordinate is −1 + 5 = 4. The option 5 reports a distance from the center rather than a coordinate in the plane. The sign inside y + 1 is another common trap: the center’s y-coordinate is −1 because the expression is y − (−1).
For transfer, identify whether your unknown is a radius, chord distance, arc, sector or coordinate. Use the center to connect the givens, and do not measure a schematic figure. When multiple chords are involved, check whether their center distances add or subtract. When fractions of a circle are involved, identify the correct whole and units. These decisions produce the equation; the remaining arithmetic should then answer the exact measurement the prompt requests.
Apply the method to a new case
A chord is 12 centimeters long and is 4 centimeters from the center of its circle. Find the radius and the length of a second chord 6 centimeters from the center. A perpendicular from the center bisects the first chord, making a right triangle with legs 6 and 4. Hence r² = 6² + 4² = 52, and r = 2√13 centimeters. For the second chord, its half-length h satisfies h² + 6² = 52, so h = 4 and its full length is 8 centimeters. Reporting 4 forgets that the right triangle uses a half-chord. Check that the more distant chord is shorter: 8 is less than 12. Both distances are smaller than the radius, so both chords exist. This existence check prevents applying a square root to a negative squared length when a proposed line lies outside the circle.