Inscribed — Definition, Formula & Examples

Inscribed refers to a geometric figure drawn inside another figure so that it fits snugly, touching the outer figure at specific points without crossing it. The most common examples are inscribed angles and inscribed polygons within circles.

A figure is inscribed in another figure if it is contained within the outer figure and meets it at maximal points of tangency or intersection. Specifically, a polygon is inscribed in a circle (its circumscribed circle) when every vertex of the polygon lies on the circle, and an angle is inscribed in a circle when its vertex lies on the circle and its sides are chords of that circle.

Key Formula

\theta_{\text{inscribed}} = \frac{1}{2}\, \theta_{\text{central}}

Where:

$\theta_{\text{inscribed}}$ = The inscribed angle, with vertex on the circle
$\theta_{\text{central}}$ = The central angle subtending the same arc

How It Works

When a polygon is inscribed in a circle, the circle passes through all of its vertices and is called the circumscribed circle (or circumcircle). Conversely, when a circle is inscribed in a polygon, the circle is tangent to every side of the polygon and is called the incircle. An inscribed angle in a circle is formed by two chords that share an endpoint on the circle. The Inscribed Angle Theorem states that an inscribed angle measures exactly half the central angle that subtends the same arc.

Worked Example

Problem: An inscribed angle in a circle intercepts an arc of 130°. What is the measure of the inscribed angle?

Identify the relationship: An inscribed angle equals half the central angle (or equivalently, half the intercepted arc).

\theta_{\text{inscribed}} = \frac{1}{2} \times \text{intercepted arc}

Substitute and solve: The intercepted arc is 130°, so divide by 2.

\theta_{\text{inscribed}} = \frac{1}{2} \times 130° = 65°

Answer: The inscribed angle measures 65°.

Why It Matters

Inscribed figures appear throughout high school geometry and trigonometry, particularly when proving angle relationships in circles. In engineering and architecture, inscribed polygons help determine tolerances when fitting shapes inside circular openings or bearings.

Common Mistakes

Mistake: Confusing inscribed with circumscribed and mixing up which figure is inside which.

Correction: Inscribed means the figure is inside. A triangle inscribed in a circle has its vertices on the circle. A circle circumscribed around a triangle is the same setup, described from the circle's perspective.