Heron's Formula
Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 33 of 44.
With three sides and no angle, there is a formula with no trig at all. Let s be half the perimeter, s = (a + b + c)/2. Then the area is:
A = √(s(s - a)(s - b)(s - c))
For sides 5, 6 and 7: s = 18/2 = 9, and A = √(9×4×3×2), which is √216 = 6√6, about 14.70. The other way round gives the same: the law of cosines for one angle, then the area with its sine. Heron skips the angle.
On the board, the formula is written for your triangle with s still a letter. Work out s first, put it into A, then simplify and round.
Challenge
The board holds Heron's formula for a triangle with sides 7 cm, 8 cm and 9 cm. Work out s, put it into A, and round A to two decimal places. Then press Check.
Try it yourself
Solve for A
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