Circular Segment Area

Area of a circular segment

A circular segment is the slice caught between a chord and the arc it cuts off. Given radius r and a central angle θ measured in radians, the exact area comes out to A = r²·(θ - sin θ)/2. Plug in r = 5 and θ = π/2 (90°) and you get A = 25·(π/2 - 1)/2 ≈ 7.135. If all you have is the chord c and the sagitta h, Huygens' 1657 approximation gives A ≈ (h·(c² + 4h²))/(6c), which stays within a few percent on shallow segments. Notice that the segment area is just the sector area with the triangle made by the two radii and the chord subtracted out.

Applications

The textbook example is a partially filled horizontal cylindrical tank. Once you know the liquid depth, the cross-section is a circular segment, and the formula hands you the volume per unit length. Truck tanks, railway tankers and chemical reactors all lean on this. The same math turns up when you estimate material and capacity for shipping containers with curved roofs, plano-convex lenses and arched window panes.

FAQ

Radians or degrees? The formula A = r²·(θ - sin θ)/2 wants θ in radians. If you have degrees, convert first with θ_rad = θ_deg · π/180.

Major vs minor segment? What the formula above returns is the minor segment, the smaller piece. For the major segment, take π·r² - A_minor.

How accurate is Huygens' formula? It's very good while h/r < 0.3, that is for shallow segments. As you approach the semicircle limit the error grows, and at that point you should switch to the exact relation.