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Triangle area

Contents

Components1.

Formula for area of a triangle2.

Area using trigonometry3.

Heron's formula4.

5. Calculator

Components

A B

C

α β

γ

O1

ab

c

Vertices of a triangle : А, B, C;Sides of length: a, b, c;Angle: α, β, γ;Inscribed circle: кRadius of Inscribed circle : r

k

r

Formula for area of a triangle

A B

ab

c

C

hc

ha

hb

Formula for area of a triangle

A B

O1

ab

c

k

r

C

Area using trigonometry

BA

C

A1

γ

900 ab

АА1 = ha altitude ΔАВС to side ВС = а.

1) γ < 90o

From ΔАA1С => ha = b.sinγ.

Result :

2) γ = 90o => ha= b, sinγ = 1,

ha

BA

C

A1

γ

900

ab

ha

3) γ > 90o

From ΔАСА1 =>: ha = b.sin(180o – γ) = b.sinγ. =>

Analogously were prepared the following formulas:

Area using trigonometry

Area using trigonometry

3) From sinus theorem : => replace b in the formula

rezult:

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.

Calculator

Rezult

c=

S=

Clear

b=

a=

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