Circular Segment Area: Sector Minus Triangle

Distinguish a segment from a sector

A circular sector is bounded by two radii and their connecting arc. A circular segment is bounded by a chord and an arc. The minor segment can be found by subtracting the triangle formed by the two radii and chord from the corresponding sector. A question that asks for a segment therefore cannot normally be answered with the sector’s fraction of total circle area alone. Identify the chord, radius and central angle on a sketch before choosing the region whose area is required.

Use the segment formula in radians

For a minor central angle θ in radians and radius r, segment area is r²(θ − sin θ)/2. The sector contributes r²θ/2, and the isosceles triangle contributes r² sin θ/2. The sine input must use the same radian angle as the first term. If a supplied angle is in degrees, convert it using θ = degrees × π/180. Mixing a degree number such as 60 directly with a radian sine result would combine incompatible quantities inside the subtraction.

Work a sixty-degree example

With r = 10 cm and a central angle of 60°, θ is π/3 radians and sin θ is √3/2. Segment area is 50(π/3 − √3/2), approximately 9.0586 cm². The sector area alone is 100π/6, approximately 52.3599 cm², while the triangle area is approximately 43.3013 cm². Their difference confirms the segment result. The whole circle’s area is 100π cm², so the small segment’s area should be much smaller than the full disc.

Keep minor and major regions separate

If the desired region is the major segment bounded by the same chord, subtract the minor segment area from the whole-circle area. The two segments together cover the disc. If the problem instead gives chord length c and radius r, the minor angle can be obtained from θ = 2 arcsin(c/(2r)), provided 0 ≤ c ≤ 2r. An impossible chord longer than the diameter indicates inconsistent data. The local Circle Calculator supplies full-circle values; the segment formula and angle step require separate arithmetic.

Distinguish the circle’s basic measurements

The radius is the distance from the centre to the circular boundary. The diameter passes through the centre and spans the circle, so d = 2r. Circumference is the boundary length, C = 2πr = πd, while area is the enclosed surface, A = πr². These measurements are different quantities. Radius, diameter and circumference use length units; area uses square units. Identify which one is given and which one is requested before entering a value into a field labelled radius.

Keep π and intermediate values unrounded

Use the calculator’s π value or retain π symbolically until the final rounding step. For a radius of 3 units, circumference is exactly 6π units and area is exactly 9π square units, approximately 18.8496 and 28.2743 respectively. Replacing π with 3 too early can introduce a larger error than the desired answer precision. Exact unit definitions or a long decimal result do not make a measured radius exact, so report final digits appropriate to the measurement or the exercise’s stated rounding instruction.

Match the local tool to the calculation

The Circle Calculator takes a radius and returns full-circle dimensions such as diameter, circumference and area. It does not infer a radius from an uploaded picture or directly accept every possible circular measurement. If the source gives a diameter, divide by two first. If it gives circumference, divide by 2π; if it gives area, take √(A/π), using consistent units. A partial-circle problem may require additional arithmetic beyond the full-circle outputs. The calculation describes an ideal plane circle and does not by itself establish a real object’s clearance, fit or measurement accuracy.

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