Data Source & Methodology
All calculations performed by this tool are based on the **Inscribed Angle Theorem**, a fundamental proposition in Euclidean geometry.
- Authoritative Source: Euclid's *Elements*, Book III, Proposition 20.
- Methodology: This tool strictly applies the theorem's formulas. All inputs and outputs are in degrees. The theorem provides a direct relationship between an angle inscribed in a circle and the arc it subtends.
The Formula Explained
The Inscribed Angle Theorem states that an angle $\theta$ (theta) inscribed in a circle is **half** of the central angle $\alpha$ (alpha) that subtends the same arc $A$ on the circle.
Crucially, the measure of the central angle $\alpha$ is **equal** to the measure of its intercepted arc $A$ (when measured in degrees). This common-sense rule gives us the primary formulas.
Formula 1: Angle from Intercepted Arc
The inscribed angle is half the measure of its intercepted arc.
Formula 2: Angle from Central Angle
The inscribed angle is half the measure of the central angle.
Derived Formulas
From the above, we can also solve for the arc or central angle:
Glossary of Variables
- Inscribed Angle ($\theta$): The angle formed by two chords in a circle that have a common endpoint on the circle's circumference.
- Central Angle ($\alpha$): An angle whose vertex is the center of the circle and whose sides are radii intersecting the circle at two points.
- Intercepted Arc ($A$): The portion of the circle's circumference that lies in the interior of the inscribed angle (or central angle).
How It Works: A Step-by-Step Example
Let's walk through a common problem.
Problem: A circle has an inscribed angle of 70°. What is the measure of the intercepted arc?
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Identify the Goal and Given Value:
- Goal: Find the Intercepted Arc ($A$).
- Given: Inscribed Angle ($\theta$) = 70°.
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Select the Correct Formula:
We need the formula that solves for the arc $A$ using the angle $\theta$. This is the reverse of the primary theorem.
$$ A = 2 \times \theta $$ -
Perform the Calculation:
Substitute the known value into the formula.
$$ A = 2 \times 70^\circ = 140^\circ $$ -
Find the Central Angle (Bonus):
The central angle $\alpha$ is always equal to the intercepted arc $A$.
$$ \alpha = A = 140^\circ $$
Result: The intercepted arc is 140° and the central angle is also 140°.
Frequently Asked Questions (FAQ)
What is a "Thales's Theorem" (angle in a semicircle)?
Thales's Theorem is a special case of the Inscribed Angle Theorem. It states that any angle inscribed in a semicircle is a right angle (90°). This is because the intercepted arc is the other half of the circle (180°). Using the formula: $\theta = \frac{1}{2} \times 180^\circ = 90^\circ$.
What about angles in a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided figure where all four vertices lie on the circle. The Inscribed Angle Theorem helps prove that the opposite angles of a cyclic quadrilateral are supplementary (add up to 180°). This is because the two opposite angles together subtend the entire circle (360°), and their sum is half of that: $\frac{1}{2} \times 360^\circ = 180^\circ$.
Does this calculator work in radians?
No, this calculator is designed for simplicity and common educational use, so it exclusively uses **degrees**. To convert, remember that $180^\circ = \pi$ radians.
Can the intercepted arc be greater than 180°?
Yes. If the inscribed angle is a "reflex angle" (greater than 180°, measured the long way around), its corresponding intercepted arc will also be a reflex arc (greater than 180°). The formula $\theta = A/2$ still holds.
What is the difference between a central angle and an inscribed angle?
The key difference is the **vertex** (the point) of the angle.
- A **Central Angle** has its vertex at the **center** of the circle.
- An **Inscribed Angle** has its vertex on the **circumference** of the circle.
Tool developed by Ugo Candido.
Mathematics content reviewed by the CalcDomain Editorial Board for accuracy and clarity.
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