Heron from Trigonometry
Derive Heron’s formula using the law of cosines.
Key idea
Derive Heron’s formula using the law of cosines.
In depth
Starting from A = ½ab sin C and the law of cosines, eliminate the angle: sin C = √(1 − cos²C). After algebra you obtain Heron’s formula A = √(s(s − a)(s − b)(s − c)).
Worked example
Sides 5, 6, 7: s = 9, A = √(9·4·3·2) = √216 ≈ 14.7.
Real-world uses
- Deriving the area of a triangle from its sides in exam and surveying work
Common mistake
Skipping to the formula without knowing why it works.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Heron from Trigonometry?
Derive Heron’s formula using the law of cosines. It is a topic at the expert level in Trigonometry.
Where is Heron from Trigonometry used in real life?
Real-world examples: Deriving the area of a triangle from its sides in exam and surveying work.
What should I learn before Heron from Trigonometry?
Make sure you are comfortable with Mollweide and Napier Formulas, the previous topic in Trigonometry.
What is a common mistake with Heron from Trigonometry?
Skipping to the formula without knowing why it works.
How can I practise Heron from Trigonometry?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Heron from Trigonometry?
Move on to Circumradius and Inradius next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).