18 19 20 triangle

Acute scalene triangle.

Sides: a = 18   b = 19   c = 20

Area: T = 155.4499147634
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 54.99003678046° = 54°54'1″ = 0.95881921787 rad
Angle ∠ B = β = 59.72439505476° = 59°43'26″ = 1.04223795794 rad
Angle ∠ C = γ = 65.37656816478° = 65°22'32″ = 1.14110208955 rad

Height: ha = 17.27221275148
Height: hb = 16.3633068172
Height: hc = 15.54549147634

Median: ma = 17.30660682999
Median: mb = 16.48548415218
Median: mc = 15.57224115024

Inradius: r = 5.45443560573
Circumradius: R = 111.0003819643

Vertex coordinates: A[20; 0] B[0; 0] C[9.075; 15.54549147634]
Centroid: CG[9.69216666667; 5.18216382545]
Coordinates of the circumscribed circle: U[10; 4.58334924851]
Coordinates of the inscribed circle: I[9.5; 5.45443560573]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.1099632195° = 125°5'59″ = 0.95881921787 rad
∠ B' = β' = 120.2766049452° = 120°16'34″ = 1.04223795794 rad
∠ C' = γ' = 114.6244318352° = 114°37'28″ = 1.14110208955 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 18 ; ; b = 19 ; ; c = 20 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 18+19+20 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-18)(28.5-19)(28.5-20) } ; ; T = sqrt{ 24164.44 } = 155.45 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 155.45 }{ 18 } = 17.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 155.45 }{ 19 } = 16.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 155.45 }{ 20 } = 15.54 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 18**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 54° 54'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-18**2-20**2 }{ 2 * 18 * 20 } ) = 59° 43'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-18**2-19**2 }{ 2 * 19 * 18 } ) = 65° 22'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 155.45 }{ 28.5 } = 5.45 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 54° 54'1" } = 11 ; ;




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