Circle Theorems Calculator

What Are Circle Theorems?

Circle theorems are fundamental geometric propositions and laws in Euclidean geometry that describe the mathematical relationships between chords, tangents, secants, radii, and angles within or touching a circle. They provide powerful rules to find unknown angles and segment lengths without measuring them directly.

These theorems are widely applied in mechanical design, structural civil engineering, computer graphics collision detection, astronomy, and architectural surveying. You can also explore our Chord Length Calculator, Circle Calculator, and Circle Equation Calculator to analyze related properties.

The Core Circle Theorems Explained

1. Inscribed Angle Theorem

The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference:

$$\theta_{\text{central}} = 2 \times \psi_{\text{inscribed}}$$

Corollary: All inscribed angles that subtend the exact same circular arc are equal ($\psi_1 = \psi_2$).

2. Angle in a Semicircle (Thales' Theorem)

Any angle inscribed in a semicircle (subtended by the circle diameter) is always a right angle ($90^\circ$ or $\pi/2$ radians):

$$\angle C = 90^\circ$$

If the diameter has length $c$, the inscribed triangle forms a right triangle where $a^2 + b^2 = c^2$.

3. Cyclic Quadrilateral Theorem

A cyclic quadrilateral is a four-sided polygon whose four vertices all lie on the circumference of a single circle. Opposite interior angles are supplementary:

$$\angle A + \angle C = 180^\circ, \quad \angle B + \angle D = 180^\circ$$

Exterior Angle Property: The exterior angle formed by extending one side of a cyclic quadrilateral is equal to the opposite interior angle.

4. Intersecting Chords Theorem (Chord Power Theorem)

When two chords $AB$ and $CD$ intersect inside a circle at point $P$, the product of the segments of one chord equals the product of the segments of the other:

$$AP \times PB = CP \times PD \implies a \times b = c \times d$$

5. Tangent-Secant and Secant-Secant Theorem

When a tangent segment and a secant line are drawn to a circle from a common external point $P$:

Tangent-Secant: The square of the tangent length $T$ equals the product of the external secant segment $a$ and the total secant length $(a + b)$: $$T^2 = a \times (a + b)$$
Secant-Secant: For two secant lines with external segments $a, c$ and internal chords $b, d$: $$a \times (a + b) = c \times (c + d)$$

6. Tangent to Radius Theorem & Tangents from an External Point

A tangent line to a circle is always perpendicular ($90^\circ$) to the radius drawn to the point of tangency. Furthermore, when two tangents are drawn from the same external point $P$ to contact points $T_1$ and $T_2$:

$$PT_1 = PT_2 = \sqrt{OP^2 - r^2}$$

The angle between the two tangents and the central angle $\angle T_1OT_2$ are supplementary ($\gamma + \theta = 180^\circ$).

7. Alternate Segment Theorem

The angle between a tangent and a chord drawn from the point of contact is equal to the inscribed angle in the alternate circular segment:

$$\angle(\text{Tangent}, \text{Chord}) = \angle(\text{Inscribed in Alternate Segment})$$

Summary of Circle Theorem Rules

Theorem	Governing Formula	Key Takeaway
Inscribed Angle	$\theta = 2\psi$	Central angle is double the inscribed angle
Thales' Theorem	$\angle = 90^\circ$	Angle inscribed in a semicircle is a right angle
Cyclic Quad	$A + C = 180^\circ$	Opposite angles are supplementary
Intersecting Chords	$a \cdot b = c \cdot d$	Segment products of intersecting chords are equal
Tangent-Secant	$T^2 = a(a + b)$	Tangent squared equals external times total secant
Alternate Segment	$\alpha = \beta$	Tangent-chord angle equals angle in opposite segment

Frequently Asked Questions

What is the inscribed angle theorem?

The inscribed angle theorem states that an angle inscribed in a circle is half the central angle that subtends the same arc on the circle ($\theta_{\text{central}} = 2 \times \psi_{\text{inscribed}}$).

Why do opposite angles in a cyclic quadrilateral add up to 180 degrees?

Because opposite vertices subtend two complementary arcs that together make up the full circle ($360^\circ$). By the inscribed angle theorem, each opposite angle equals half of its subtended central angle, so their sum is $360^\circ / 2 = 180^\circ$.

What is the intersecting chords formula?

When two chords intersect inside a circle at point $P$, the product of the lengths of the two parts of one chord is equal to the product of the lengths of the two parts of the other chord: $a \times b = c \times d$.

What is Thales' theorem?

Thales' theorem states that if $A$, $B$, and $C$ are distinct points on a circle where the line segment $AC$ is a diameter, the angle $\angle ABC$ is always a right angle ($90^\circ$).

Are tangents from an external point to a circle equal in length?

Yes. Two tangent segments drawn to a circle from the same external point have identical lengths ($PT_1 = PT_2$). The line connecting the external point to the center of the circle bisects the angle between the two tangents.

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