Circle Theorems Calculator - Inscribed Angle, Arc, Chord

A circle theorems calculator is a geometry tool that applies the standard circle theorems to a single circle, so you can get every related angle and length from just the radius and a central angle. The page runs the inscribed angle theorem, the arc length formula, the chord length formula, the sector area formula, and the segment area formula, and returns the values in one pass for homework checks, exam revision, and quick classroom demonstrations.

• • Classroom practice on circle theorems: confirm the relationships between central angles, inscribed angles, arc, chord, and segment area without measuring by hand.
• • Homework and exam revision: solve a problem that names only the central angle and the radius by reading the inscribed angle, chord length, or segment area directly.
• • Quick sanity check on a sketch: plug in a labeled angle and one length and check the rest of the page's outputs against the sketch.
• • Connecting theorems to a single picture: show how the inscribed angle theorem, Thales' theorem, the tangent-chord theorem, and the segment formula sit on the same circle.

The page keeps the input set to a radius and a central angle, because the standard circle theorems are written in those terms. Every other angle or length on the page is a direct consequence of a single theorem or formula.

Angles are shown in both degrees and radians; lengths and areas use the same unit you entered for the radius. The page is informational and assumes the radius and angle are exact.

If the angle you have is in radians or gradians instead of degrees, the Angle Converter handles the unit flip without recomputing the circle.

The page runs five standard formulas on the radius and the central angle, in radians, and then re-labels each result in the units you entered. The inscribed angle theorem, the arc length formula, the chord length formula, the sector area formula, and the segment area formula are each applied once and shown together.

• r: Radius of the circle. The same unit is used for every length and area output.
• theta_c (deg): Central angle in degrees. The page caps anything above 360 at 360.
• theta_c (rad): Central angle in radians, computed as theta_c (deg) * pi / 180. The arc, chord, and area formulas use this value.
• theta_i: Inscribed angle, equal to half the central angle by the inscribed angle theorem. Reported in degrees and radians.
• s (arc length): r * theta_radians. For a full circle this is the circumference 2*pi*r.
• c (chord length): 2r * sin(theta_radians / 2). For theta = pi the chord is the diameter 2r.

The arc length formula and the chord length formula both come from the right triangle that drops the perpendicular from the center of the circle to the chord. In that triangle, the hypotenuse is the radius, the short leg is half the chord, and the angle at the center is half of the central angle.

The segment area is the sector area minus the isoceles triangle formed by the two radii and the chord, which is why the page subtracts 0.5 * r^2 * sin(theta_radians) from the sector area formula. For a 180-degree central angle the segment area is the semicircle area because sin(pi) is 0.

According to Wolfram MathWorld, the inscribed angle is half the central angle.

If you already know the arc length instead of the central angle, the Arc Length Calculator works the same problem backwards from s = r * theta.

Four ideas make the rest of the calculator work. Each is a circle theorem in its own right, and together they explain why every other output on the page is a function of just the radius and the central angle.

These four ideas are enough to set up almost every circle-theorem problem you will meet in a standard high school geometry course. The arc, chord, sector, and segment formulas are the trigonometric consequences of these angle relations on a single circle of known radius.

If the four ideas feel comfortable, the rest of the page is a speed dial: the page reuses the same radius and central angle to print every related value.

According to Wikipedia, an angle inscribed in a semicircle is always 90 degrees.

The dedicated Inscribed Angle Calculator lets you solve for a missing central or inscribed angle given an arc and a radius, which is the inverse of the relationship this page is built on.

Five short steps cover every common case, from a clean textbook example to a Thales-style semicircle.

1. 1 Enter the radius: Type the radius in the first field. The default is 5; use 0 to test the degenerate case where every length and area collapses to 0.
2. 2 Enter the central angle: Type the central angle in degrees. The page accepts 0 to 360 and caps anything larger at 360, so a 540-degree input is treated as a full rotation.
3. 3 Read the inscribed angle: The primary output is the inscribed angle, half the central angle by the inscribed angle theorem. Its radian value sits next to it.
4. 4 Read the lengths and areas: Look at the arc length, chord length, sector area, and segment area in the same unit you used for the radius.
5. 5 Reset or try Thales' case: Click Reset to return to the default example. Set the central angle to 180 to see Thales' theorem in one click.

Try radius 6 and central angle 90 degrees. The page gives an inscribed angle of 45 degrees, an arc length of about 9.4248, a chord length of about 8.4853 (which is 6 * sqrt(2)), a sector area of about 28.2743, and a segment area of about 10.2743.

If the problem gives the central angle in radians or in a different unit, the Central Angle Calculator handles the conversion and the inverse solve in one place.

These benefits matter most when you are working through a circle-theorem problem by hand and need a fast, trustworthy check.

• • One entry, five outputs: The page returns the inscribed angle, the radian equivalents, the arc length, the chord length, the sector area, and the segment area at the same time.
• • Each output is tied to a named theorem: The page labels each result with the theorem or formula that produced it.
• • Built-in Thales check: Setting the central angle to 180 returns the 90-degree inscribed angle, the diameter as the chord, and the semicircle area as the segment area.
• • Angles in degrees and radians: Every angle is shown in both units, so the page lines up with whatever unit your textbook uses.
• • Reset to a worked example: Reset restores the default radius of 5 and central angle of 60, the equilateral-triangle case where the chord length equals the radius.

The page is most useful as a check, not as a replacement for understanding the theorems. Use it to confirm a homework answer, compare your sketch against the numbers, or pre-compute a value.

Because the page is built around a single circle, the relationship between the inputs and the outputs is fixed and predictable. That makes it a natural complement to the dedicated chord, arc, sector, and segment calculators.

For a deeper look at just the chord relationship, the Chord Length Calculator returns the chord from a radius and central angle with the same formula in a focused view.

The formulas are fixed, but a few factors change how the result should be read or how the page should be used in a problem.

• • The page is for a single circle. For the angle between two secants, the tangent-chord line equation, or the angle between a tangent and a secant, use the dedicated tools in this category.
• • The page assumes the central angle is a single rotation between 0 and 360 degrees. Angles larger than one full rotation are capped at 360.
• • The outputs are exact for the formulas, not for measured circles. If you read a radius and an angle off a drawing, treat the result as a check rather than ground truth.

If the central angle is very close to 0, the small-angle approximation kicks in and the chord length is approximately r * theta_radians, the same as the arc length. The page does not switch to the approximation.

According to Wolfram MathWorld, the segment area is A = (1/2) r^2 (theta - sin theta).

For the segment area on its own, with the option to flip the inputs around, the Segment Area Calculator is a single-formula page that does the same work in fewer fields.