What Is a Circle Segment?

A circle segment is the region of a circle that lies between a chord and the corresponding arc. Imagine slicing a round pie with a straight cut that does not pass through the center. The small wedge you cut off is a segment. When designs call for partially filled circles or when engineers need to measure the cross-sectional area of a liquid rising inside a cylindrical tank, the formula for a segment becomes extremely handy. Whereas sectors refer to wedge-shaped pieces defined by two radii, segments are bounded by a single chord and the curved arc above or below it.

Deriving the Area Formula

The area of a circular segment can be expressed in terms of the radius $r$ and the central angle $\theta$ in radians. The equation reads

$A=\frac{r^2}{2}\left(\theta ,-,\sin ,\theta \right)$.

However, designers often know the segment height $h$ rather than the central angle. The height represents the distance from the chord to the arc, measured along a line that is perpendicular to the chord. By drawing the geometry, you can see that the angle relates to the height through

$\theta =2\arccos \left(\frac{r-h}{r}\right)$.

Substituting this relationship into the area formula yields the more practical expression used by the calculator:

$A=r^2\arccos \left(\frac{r-h}{r}\right)-\left(r-h,\surd ,\left(2rh-h^2\right)\right)$.

Because the trigonometric operations might look intimidating, the calculator handles them automatically. You simply provide the radius and height measurements, and the script computes the area directly.

Step-by-Step Example

Suppose you have a circular garden pond with a radius of 2 m, and you are designing a walkway that cuts a chord across the pond. If the segment above the walkway should contain 0.5 m of water at its deepest point, that means the height $h$ equals 0.5 m. Plugging these values into the formula gives the segment area. The calculation reveals how much surface area of water will remain on the far side of the walkway. Knowing this helps with estimating the pond volume as well as landscaping features around it.

Applications Across Disciplines

Circle segment area calculations appear in many fields beyond landscaping. Architects use them when planning curved roofs or arched openings that only partially cover a cylindrical space. Mechanical engineers need the formula when analyzing how fluids fill pipes or tanks that are tilted at an angle. In civil engineering, roadway designers consider cross-sectional segments when shaping curved embankments or tunnels. Even artists rely on the geometry of circle segments when laying out circular windows or designing curved furniture components.

Typical Angles and Heights

The table below pairs a few common central angles with corresponding height ratios, providing a quick sanity check when you perform manual calculations. Each value assumes a unit radius.

Angle (°)	Height Ratio h/r
30	0.134
60	0.268
90	0.414
120	0.586
150	0.866

If you measure a height that corresponds closely to one of these ratios, you can quickly estimate the angle without a calculator. Conversely, knowing the angle helps you estimate the height. The calculator includes all the trigonometric steps so you do not need to do them by hand.

Limitations and Assumptions

This tool assumes that the segment height is less than or equal to the circle radius. When the height equals the radius, the chord passes through the center, and the formula effectively returns half of the circle’s area. If the height exceeds the radius, the geometry no longer describes a simple segment. In those cases you might consider treating the problem as a larger segment or using a different coordinate system. The calculator also assumes measurements are in the same units. If you mix meters and centimeters, you will get inconsistent results.

Why Bother With Circle Segments?

Understanding how to compute the area of a circle segment empowers you to solve design challenges that at first glance seem complicated. Many structures include partial curves, from aqueducts and tunnels to ornate windows and arches. The ability to quantify partial circular areas also aids in tasks like estimating the volume of fluid between a fill line and the top of a cylindrical tank. In manufacturing, cutting materials from sheet stock often involves arcs and chords. By knowing the area of each piece, you can minimize waste and improve efficiency.

Beyond practical uses, there is also an elegance in the geometry itself. The relationships among radius, angle, chord length, and segment height form a network of trigonometric identities that tie together key parts of a circle. Exploring those links deepens your understanding of mathematics, especially when you see how apparently unrelated measurements lead to the same result.

Try It Yourself

Enter any radius and segment height in the form above. When you press the button, the JavaScript code calculates the area in square meters using the formula derived earlier. The result appears immediately beneath the form. If you want to reuse the value in another application, click the copy button to send the text to your clipboard. All of this runs entirely in your browser with no data transmitted elsewhere, so feel free to test multiple scenarios.

Whether you are a student checking homework, a builder planning a curved structure, or a hobbyist curious about geometry, this calculator saves time and prevents mistakes. The included explanation and table offer additional guidance so you can tackle similar problems even without the web page. Keep exploring circular segments and you’ll notice patterns that connect to everything from planetary orbits to the design of camera lenses. Geometry surrounds us—this tool lets you peer a little deeper into one of its many facets.