Segment Area Calculator

Circles, Angle Measures, Arcs, Central \u0026 Inscribed Angles, Tangents, Secants \u0026 Chords – Geometry
Circles, Angle Measures, Arcs, Central \u0026 Inscribed Angles, Tangents, Secants \u0026 Chords – Geometry

Segment Area Calculator

Calculate the area and height of a segment by entering the central angle and radius in the calculator below.

Results:

Segment Area (A)

Segment Height (h)

Arc Length (s)

Chord Length (a)

On this page:

How to Calculate the Area of a Segment

A segment is a part of a circle separated by a chord connecting two points on the circle’s edge. Think of a segment as a part of a circle that has been sliced off using a straight line.

A segment is often represented as part of a sector, as seen in the image below. The segment is the shaded part of the sector.

diagram of a circular segment showing the radius, central angle, chord length, height, and arc length

In addition to area, a segment is defined by its height (also called the sagitta), chord length, and arc length. The area of a segment can be found using a simple formula.

Segment Area Formula

The area can be found using the segment area formula:

segment area = r² × (θ – sin(θ)) / 2

Thus, a segment’s area is equal to the radius r squared, times the central angle θ in radians minus the sine of θ, divided by 2.

Note that the central angle must be in radians for this equation to work. If you have the central angle in degrees, use our degrees to radians converter to convert to radians, or use the equation in the next section.

Graphic showing the segment area formula where the area is equal to the radius squared, times the central angle minus the sine of central angle, divided by 2.

For example, let’s find the area of a segment for a circle with a radius of 17 and a central angle of 1.25 radians.

segment area = 17² × (1.25 – sin(1.25)) / 2
segment area = 289 × (1.25 – 0.094898) / 2
segment area = 289 × 0.301021) / 2
segment area = 86.995 / 2
segment area = 43.497

Thus, the segment area is 43.497.

Segment Area Formula for a Central Angle in Degrees

If you want to find the area of a segment, but your central angle is in degrees instead of radians, you can find the area using the following formula:

segment area = r² × ((θ × π)/360 – sin(θ))/2

This is essentially the same formula as above, but the (θ × π)/360 part converts the angle to radians.

How to Find the Height of a Segment

The following formula defines the height, or sagitta, of a segment.

segment height = r × (1 – cos(θ / 2))

The height of a segment is equal to the radius r times 1 minus the cosine of the central angle θ in radians divided by 2. Note here that again, the central angle must be in radians, not degrees.

How to Find the Arc Length of a Segment

The arc of a segment is the rounded outer portion of the circle that connects the two radii of the circle. The arc length can be found using the formula:

arc length (s) = radius (r) × central angle (θ)

The arc length s of a segment is equal to the radius r times the central angle θ in radians. Again, as with most of the above equations, the central angle must be in radians, not degrees.

You might also be interested in our sector area calculator for solving the parts of a sector.

You are watching: Segment Area Calculator. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post

Related Posts