The Angle From the Area
Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 32 of 44.
With the area and two sides known, the rule gives the sine of the angle between them. A sine below 1 belongs to an acute angle and its supplement, and both make a triangle with the same area.
Area 18 with sides 9 and 8: 18 = 1/2×9×8×sin(phi), so sin φ = 1/2 and φ is 30° or 150°. Both are real triangles: one thin and pointed, one wide open. Nothing here rules either out, so both are answers.
Challenge
A triangle has area 45 cm² and two sides of 12 cm and 10 cm. Find every possible angle θ between those sides, to two decimal places, then press Check.
Try it yourself
Use the figure to find theta
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Triangles Without Right Angles
4Law of Cosines: SAS
The Law of CosinesAn Obtuse Angle BetweenThrough the HillA Hooked DriveRecap: Two Sides, One Angle7Area of a Triangle
Area With a SineArea With an Obtuse AngleWorking BackwardsThe Angle From the AreaHeron's FormulaRecap: A Garden Bed2Law of Sines: AAS and ASA
Why the Law of Sines HoldsPair Side With AngleThe Third Angle FirstTwo Coast StationsRecap: Law of Sines8Bearings and Navigation
Reading BearingsTwo Ships From PortTurning Off CourseFinding a Course AngleRecap: Ships at Sea