Transcript Formula for area of ​​a triangle

Triangle area
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Contents
1.
Components
2.
Formula for area of ​a triangle
3.
Area using trigonometry
4.
Heron's formula
5.
Calculator
Components
C
γ
b
a
r
O1
k
α
A
β
c
Vertices of a triangle : А, B, C;
Sides of length: a, b, c;
Angle: α, β, γ;
Inscribed circle: к
Radius of Inscribed circle : r
B
Formula for area of ​a triangle
C
b
ha
A
hb
a
hc
c
B
Formula for area of ​a triangle
C
b
a
r
O1
k
A
c
B
Area using trigonometry
C
АА1 = ha altitude ΔАВС to side ВС = а.
γ
1) γ < 90o
From ΔАA1С => ha = b.sinγ.
A1
b
ha
900
a
Result :
2) γ = 90o => ha= b, sinγ = 1,
A
B
Area using trigonometry
3) γ > 90o
From ΔАСА1 =>: ha = b.sin(180o – γ) = b.sinγ.
A1
ha
900
b
γ
A
=>
C
a
B
Analogously were prepared the following
formulas:
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.
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Rezult
b=
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c=
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