Law of Sines in Geometry

Key idea

Sides proportional to sines of opposite angles.

In depth

In any triangle, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. It connects sides, angles and the circumscribed circle.

Worked example

A = 30°, a = 5: 2R = 5/0.5 = 10, so R = 5.

Formula

Real-world uses

• Finding unknown sides of a triangular field from two angles and one side in surveying

Common mistake

Forgetting the relation to 2R, which links to circle geometry.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Geometry

• Heron’s Formula – Triangle area from three sides.
• Regular Polygon Area – Use apothem and perimeter.
• Arc Length of a Circle – Part of the circumference.
• Sector Area – Part of the circle’s area.
• Central and Inscribed Angles – An inscribed angle is half the central angle on the same arc.
• Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.