Subtend — Definition, Formula & Examples
Subtend means to 'stretch across' or 'hold under.' An arc or chord subtends an angle when the two lines forming that angle have their endpoints on the arc or chord.
A line segment or arc is said to subtend an angle at a given point if the two rays from that point pass through the endpoints of the segment or arc. In circle geometry, an arc subtends an angle at the center (central angle) or at any point on the remaining part of the circle (inscribed angle).
How It Works
When you see 'arc AB subtends angle C,' it means point C is the vertex of the angle and the two sides of the angle pass through points A and B. The same arc can subtend different angles depending on where the vertex is located. At the center of the circle, the subtended angle is the central angle. At a point on the circumference (on the major arc), the subtended angle is the inscribed angle. A key circle theorem states that the angle subtended at the center is always twice the angle subtended at the circumference by the same arc.
Worked Example
Problem: Arc PQ subtends an angle of 80° at the center O of a circle. What angle does arc PQ subtend at a point R on the major arc?
Identify the relationship: The angle subtended at the center is twice the angle subtended at the circumference by the same arc.
Solve for the inscribed angle: Substitute the known central angle and solve.
Answer: Arc PQ subtends an angle of 40° at point R on the major arc.
Why It Matters
The word 'subtend' appears throughout GCSE and IB circle theorem proofs. Misreading it can cause you to mix up which angle a problem is asking about. Recognizing what subtends what lets you correctly apply theorems like the inscribed angle theorem and the angle in a semicircle.
Common Mistakes
Mistake: Confusing which angle is subtended — placing the vertex on the wrong part of the circle.
Correction: The vertex of the subtended angle is always at the specified point (center or circumference), not on the arc itself. The arc's endpoints define where the two rays go, not where the angle sits.