Circle Theorems Calculator
Our free circle calculator solves circle theorems problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Circle Theorems Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: Inscribed Angle = Central Angle / 2
Worked example — Inscribed angle: 40 degrees | Arc: 13.963 | Chord: 12.856
Formula
Inscribed Angle = Central Angle / 2
The inscribed angle theorem states that an inscribed angle is half the central angle subtending the same arc. Related theorems cover tangent-radius perpendicularity, equal tangent lengths, chord-chord products, and cyclic quadrilateral angle sums.
Worked Examples
Example 1: Inscribed Angle from Central Angle
Problem:A circle has radius 10 and a central angle of 80 degrees. Find the inscribed angle, arc length, and chord length.
Solution:Radius r = 10, central angle = 80 degrees = 1.3963 radians Inscribed angle = 80 / 2 = 40 degrees Arc length = r * theta = 10 * 1.3963 = 13.963 Chord = 2r * sin(theta/2) = 20 * sin(40) = 20 * 0.6428 = 12.856 Perpendicular distance from center to chord = sqrt(100 - 41.28) = 7.660
Result:Inscribed angle: 40 degrees | Arc: 13.963 | Chord: 12.856
Example 2: Tangent Length from External Point
Problem:Find the tangent length from a point 15 units from the center of a circle with radius 10.
Solution:Distance d = 15, radius r = 10 Tangent length = sqrt(d^2 - r^2) = sqrt(225 - 100) = sqrt(125) = 11.180 Angle between tangent and line to center = arccos(r/d) = arccos(10/15) = 48.19 degrees Power of point = d^2 - r^2 = 225 - 100 = 125 Angle between two tangents = 2 * arcsin(r/d) = 2 * arcsin(0.667) = 83.62 degrees
Result:Tangent length: 11.180 | External angle: 48.19 degrees | Power: 125
Frequently Asked Questions
What is the inscribed angle theorem?
The inscribed angle theorem states that an inscribed angle (an angle formed by two chords that share an endpoint on the circle) is exactly half the central angle that subtends the same arc. For example, if a central angle is 80 degrees, any inscribed angle subtending the same arc will be 40 degrees, regardless of where on the circle the vertex of the inscribed angle is placed. This theorem has a powerful corollary: all inscribed angles subtending the same arc are equal. This means if you pick any point on the major arc and draw lines to the endpoints of a chord, the angle formed is always the same. This property is fundamental in circle geometry and appears in proofs involving cyclic quadrilaterals and tangent-chord angles.
What is the angle in a semicircle theorem?
The angle in a semicircle theorem (also called Thales' theorem) states that any angle inscribed in a semicircle is a right angle (90 degrees). More precisely, if AB is a diameter of a circle and C is any point on the circle other than A or B, then angle ACB equals exactly 90 degrees. This follows directly from the inscribed angle theorem: the central angle for a diameter is 180 degrees, and the inscribed angle is half of that, giving 90 degrees. This theorem is one of the oldest results in mathematics, attributed to Thales of Miletus around 600 BCE. It has practical applications in construction for creating perfect right angles and in navigation for determining positions relative to landmarks.
What is the tangent-radius theorem?
The tangent-radius theorem states that a tangent line to a circle is always perpendicular to the radius drawn to the point of tangency. This means the angle between the tangent and the radius at the point where they meet is exactly 90 degrees. This theorem is crucial for calculating tangent lengths from external points. If a point is at distance d from the center and the circle has radius r, the tangent length is the square root of (d squared minus r squared), derived from the Pythagorean theorem applied to the right triangle formed by the radius, tangent, and line from center to external point. The tangent-radius relationship is fundamental in engineering, optics, and computer graphics where tangent lines to circles determine reflection angles and contact points.
What is the power of a point theorem?
The power of a point theorem relates the products of lengths along any line through a fixed point that intersects a circle. For a point P outside the circle, if a line through P intersects the circle at points A and B, then PA times PB is constant for all such lines. This constant value is called the power of the point and equals d squared minus r squared, where d is the distance from P to the center and r is the radius. For a tangent from P, the power equals the square of the tangent length (since both intersection points coincide). For a point inside the circle, the power is negative: PA times PB = r squared minus d squared. This theorem unifies many circle relationships and is used extensively in advanced geometry proofs and in constructing radical axes of circle pairs.
What is the cyclic quadrilateral theorem?
The cyclic quadrilateral theorem states that the opposite angles of a quadrilateral inscribed in a circle (a cyclic quadrilateral) sum to 180 degrees. If the four vertices A, B, C, D lie on a circle, then angle A plus angle C equals 180 degrees, and angle B plus angle D equals 180 degrees. The converse is also true: if a quadrilateral has opposite angles summing to 180 degrees, then it can be inscribed in a circle. This theorem follows from the inscribed angle theorem, as opposite angles subtend arcs that together make the full circle (360 degrees), and each inscribed angle is half its corresponding arc. Cyclic quadrilaterals appear in astronomy for calculating planetary positions and in computational geometry for Delaunay triangulation algorithms.
What is the alternate segment theorem?
The alternate segment theorem (also called the tangent-chord angle theorem) states that the angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle in the alternate (opposite) segment of the circle. In other words, if a tangent at point P makes an angle alpha with chord PQ, then any inscribed angle in the arc on the other side of PQ from the tangent also equals alpha. This elegant result connects tangent properties with inscribed angles. The proof uses the fact that the tangent-chord angle equals half the intercepted arc, which is the same relationship as for inscribed angles. This theorem is particularly useful in solving complex geometry problems involving both tangents and chords simultaneously.
How does the perpendicular from center to chord theorem work?
The perpendicular from the center of a circle to a chord bisects the chord, and conversely, a line from the center to the midpoint of a chord is perpendicular to the chord. If a chord has length c and the circle has radius r, the perpendicular distance from the center to the chord is d = sqrt(r^2 - (c/2)^2). This creates two right triangles, each with hypotenuse r, one leg c/2, and the other leg d. This theorem provides a practical method for finding unknown measurements: if you know the radius and chord length, you can find the distance from center to chord; if you know the radius and distance, you can find the chord length. Surveyors and engineers use this relationship to determine tunnel radii from chord and sagitta measurements.
What is the two-tangent theorem?
The two-tangent theorem states that the two tangent segments drawn from an external point to a circle are equal in length. If point P is outside a circle and tangent lines from P touch the circle at points A and B, then PA = PB. Furthermore, the line from P to the center O bisects the angle APB and also bisects the angle AOB. The angle APB can be calculated as 2 times arcsin(r/d), where r is the radius and d is the distance from P to the center. This theorem is used in engineering to design cam profiles, in surveying to calculate distances to inaccessible circular structures, and in computer vision for estimating circle positions from tangent line observations. The equal tangent property also explains why a string wrapped around a circle pulls equally on both sides.
How are intersecting chords related in a circle?
When two chords intersect inside a circle, the products of their segments are equal. If chord AB intersects chord CD at point P inside the circle, then AP times PB equals CP times PD. This is a special case of the power of a point theorem where the point lies inside the circle. For example, if one chord is divided into segments of length 3 and 8 by the intersection point, and the other chord has one segment of length 4, then the other segment must have length (3 times 8)/4 = 6. This relationship extends to secant lines from external points: if two secants from external point P intersect the circle at A, B and C, D respectively, then PA times PB equals PC times PD. These properties are useful in surveying and construction for indirect distance measurements.
How do circle theorems apply to real-world problem solving?
Circle theorems have extensive practical applications across many fields. In architecture, the inscribed angle theorem helps design arched windows and domed ceilings with precise angular relationships. In navigation, the angle in a semicircle theorem allows sailors to determine their position relative to two landmarks when the angle between them is 90 degrees. In engineering, tangent properties determine gear tooth profiles (involute gears), cam follower contact points, and pipe fitting angles. In computer graphics, circle theorems enable efficient algorithms for circle-circle and line-circle intersections used in collision detection and rendering. In optics, tangent lines to circular mirrors determine reflection paths. In sports analytics, the inscribed angle theorem helps analyze shooting angles in basketball and soccer goal scoring probabilities.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Great Circle Calculator
Calculate great circle with inputs, formulas, and instant results.
🧮Circle Sector Perimeter Calculator
Calculate circle sector perimeter with inputs, formulas, and instant results.
🧮Circle Through Three Points Calculator
Calculate circle through three points with inputs, formulas, and instant results.
🧮Circle Segment Area Calculator
Calculate circle segment area with inputs, formulas, and instant results.
🧮Unit Circle Calculator
Calculate unit circle with inputs, formulas, and instant results.
🧮Circumscribed Circle Calculator
Calculate circumscribed circle with inputs, formulas, and instant results.
🧮Area of a Circle Calculator
Calculate area of acircle with inputs, formulas, and instant results.
🧮Equation of a Circle Calculator
Calculate equation of acircle with inputs, formulas, and instant results.