Circular segment - wikipedia Circular segment In geometry, a circular segment ( symbol : ⌓ ) is a region of a circle which is `` cut off '' from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two - dimensional space that is bounded by an arc ( of less than 180 ° ) of a circle and by the chord connecting the endpoints of the arc. Contents 1 Formula 1.1 Area 1.2 Applications 2 See also 3 References 4 External links Formula A circular segment ( in green ) is enclosed between a secant / chord ( the dashed line ) and the arc whose endpoints equal the chord 's ( the arc shown above the green area ). Let R be the radius of the circle, θ the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the sagitta ( height ) of the segment, and d the height of the triangular portion. The radius is R = h + d = h 2 + c 2 8 h ( displaystyle R = h + d = ( \frac ( h ) ( 2 ) ) + ( \frac ( c ^ ( 2 ) ) ( 8h ) ) ) The radius in terms of h and c can be derived above by using the Intersecting Chords Theorem, where 2R ( the diameter ) and c are perpendicularly intersecting chords. The arc length is s = α 180 ∘ π R = θ R = arcsin ⁡ ( c h + c 2 4 h ) ( h + c 2 4 h ) ( displaystyle s = ( \frac ( alpha ) ( 180 ^ ( circ ) ) ) pi R = ( \theta ) R = arcsin left ( ( \frac ( c ) ( h+ ( \frac ( c ^ ( 2 ) ) ( 4h ) ) ) ) \right ) left ( h+ ( \frac ( c ^ ( 2 ) ) ( 4h ) ) \right ) ) The arc length in terms of arcsin can be derived above by considering an inscribed angle that subtends the same arc, and one side of the angle is a diameter. The angle thus inscribed is θ / 2 and is part of a right triangle whose hypotenuse is the diameter. This is also useful in deriving other inverse trigonometric forms below. With further aid of half - angle formulae and pythagorean identities, the chord length is c = 2 R sin ⁡ θ 2 = R 2 − 2 cos ⁡ θ = 2 R 1 − ( d / R ) 2 ( displaystyle c = 2R sin ( \frac ( \theta ) ( 2 ) ) = R ( sqrt ( 2 - 2 cos \theta ) ) = 2R ( sqrt ( 1 - ( d / R ) ^ ( 2 ) ) ) ) The sagitta is h = R ( 1 − cos ⁡ θ 2 ) = R − R 2 − c 2 4 ( displaystyle h = R left ( 1 - cos ( \frac ( \theta ) ( 2 ) ) \right ) = R - ( sqrt ( R ^ ( 2 ) - ( \frac ( c ^ ( 2 ) ) ( 4 ) ) ) ) ) The angle is θ = 2 arctan ⁡ c 2 d = 2 arccos ⁡ d R = 2 arccos ⁡ ( 1 − h R ) = 2 arcsin ⁡ c 2 R ( displaystyle \theta = 2 arctan ( \frac ( c ) ( 2d ) ) = 2 arccos ( \frac ( d ) ( R ) ) = 2 arccos left ( 1 - ( \frac ( h ) ( R ) ) \right ) = 2 arcsin ( \frac ( c ) ( 2R ) ) ) Area The area A of the circular segment is equal to the area of the circular sector minus the area of the triangular portion A = R 2 2 ( θ − sin ⁡ θ ) ( displaystyle A = ( \frac ( R ^ ( 2 ) ) ( 2 ) ) left ( \theta - sin \theta \right ) ) with the central angle in radians, or A = R 2 2 ( α π 180 ∘ − sin ⁡ α ) ( displaystyle A = ( \frac ( R ^ ( 2 ) ) ( 2 ) ) left ( ( \frac ( alpha pi ) ( 180 ^ ( circ ) ) ) - sin alpha \right ) ) with the central angle in degrees. Applications The area formula can be used in calculating the volume of a partially - filled cylindrical tank. In the design of windows or doors with rounded tops, c and h may be the only known values and can be used to calculate R for the draftsman 's compass setting. One can reconstruct the full dimensions of a complete circular object from fragments by measuring the arc length and the chord length of the fragment. See also Spherical segment Conic section Cross section References Weisstein, Eric W. `` Circular segment ''. MathWorld. External links Definition of a circular segment With interactive animation Formulae for area of a circular segment With interactive animation Retrieved from `` https://en.wikipedia.org/w/index.php?title=Circular_segment&oldid=823648963 '' Categories : Circles Talk Contents About Wikipedia Azərbaycanca Беларуская Català Čeština Dansk Deutsch Español Esperanto فارسی Français Հայերեն Italiano മലയാളം Nederlands Norsk Norsk nynorsk ភាសា ខ្មែរ Polski Português Русский Slovenščina தமிழ் ไทย Українська Tiếng Việt 中文 Edit links This page was last edited on 2 February 2018, at 15 : 10. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: arc length of a portion of a circle
Answer: circular segment