mathematics

Subtended angle

Subtended angle is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Subtended angle rather than just read about it. In short: In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. For example, a side of a triangle subtends the opposite angle.

Key takeaways

Subtended angle belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Subtended angle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Subtended angle from memory before moving on to harder problems.

Reference excerpt

In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. For example, a side of a triangle subtends the opposite angle. More generally, an angle subtended by an arc of a curve is the angle subtended by the corresponding chord of the arc. For example, a circular arc subtends the central angle formed by the two radii through the arc endpoints. If an angle is subtended by a straight or curved segment, the segment is said to subtend the angle. Sometimes the term "subtend" is applied in the opposite sense, and the angle is said to subtend the segment. Alternately, the angle can be said to intercept or enclose the segment. The above definition of a subtended plane angle remains valid in three-dimensional space (3D), as one vertex and two endpoints (assumed non-collinear) define an Euclidean plane in 3D. For example, an arc of a great circle on a sphere subtends a central plane angle, formed by the two radii between the center of the sphere and each of the two arc endpoints. More generally, a surface subtends a solid angle if its boundary defines the cone of the angle. Many theorems in geometry relate to subtended angles. If two sides of a triangle are congruent, then the angles they subtend are also congruent, and conversely if two angles are congruent then they are subtended by congruent sides (propositions I.5–6 in Euclid's Elements), forming an isosceles triangle. More generally, the law of sines states that the sine of each angle of a triangle is proportional to the side subtending it. The inscribed angle theorem states that when the vertex of an angle inscribed in a circle lies on the same side of the chord subtending it as the center of the circle, then the central angle subtended by the same chord is twice the inscribed angle. By extension, an angle subtended by a more complex geometric figure may be defined in terms of the figure's convex hull and its diameter; for example, the angle subtended by a tree as viewed in a camera (see angular size). A subtended plane angle can also be defined for any two arbitrary isolated points and a vertex, as in two lines of sight from a particular viewer; for example, the angle subtended by two stars as seen from Earth (see angular separation).

References

Illustrations

Worked examples

Example 1 — a first encounter with Subtended angle

Start with the simplest possible case. Write down what Subtended angle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Subtended angle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Subtended angle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Subtended angle

In research: Subtended angle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Subtended angle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Subtended angle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Angle, Elementary geometry stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Subtended angle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Subtended angle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Subtended angle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Subtended angle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Subtended angle in simple terms?

Why does Subtended angle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Subtended angle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Subtended angle.