Triangle area

1. Triangle area

2. 1. 2. 3. 4. Components Formula for area of ​​a triangle Area using trigonometry Heron's formula Contents 5. Calculator

3. Components C γ b a r O1 k α β A B c Vertices of a triangle : А, B, C; Sides of length: a, b, c; Angle: α, β, γ; Inscribed circle: к Radiusof Inscribed circle : r

4. Formula for area of ​​a triangle C b a hb ha hc A B c

5. Formula for area of ​​a triangle C b a r O1 k A B c

6. Area using trigonometry C АА1 = ha altitude ΔАВС to side ВС = а. 1) γ< 90o FromΔАA1С => ha = b.sinγ. Result: 2) γ= 90o => ha= b, sinγ = 1, γ A1 ha b a 900 B A

7. Area using trigonometry 3) γ> 90o FromΔАСА1=>: ha = b.sin(180o – γ) = b.sinγ. => A1 ha 900 C b a γ B A Analogously were prepared the following formulas:

8. Area using trigonometry 3) Fromsinus theorem : => replace b in the formula rezult:

9. Heron's formula Let a,b,c be the lengths of the sides of a triangle. The area is given by: Heron's formula Hero (or Heron) of Alexandria (c. 10 – 70 AD) was an ancient Greek mathematician and engineer who was active in his native city ofAlexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition. Hero published a well recognized description of a steam-powered device called an aeolipile (sometimes called a "Hero engine"). Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. He is said to have been a follower of theAtomists. Some of his ideas were derived from the works of Ctesibius. Much of Hero's original writings and designs have been lost, but some of his works were preserved in Arabic manuscripts.

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