19 20 21 triangle

Acute scalene triangle.

Sides: a = 19   b = 20   c = 21

Area: T = 172.3376879396
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 55.1550095421° = 55°9' = 0.96325507479 rad
Angle ∠ B = β = 59.75109669494° = 59°45'3″ = 1.04328511045 rad
Angle ∠ C = γ = 65.09989376296° = 65°5'56″ = 1.13661908012 rad

Height: ha = 18.1410724147
Height: hb = 17.23436879396
Height: hc = 16.4133036133

Median: ma = 18.17327818454
Median: mb = 17.34993515729
Median: mc = 16.43992822228

Inradius: r = 5.74545626465
Circumradius: R = 11.57661641211

Vertex coordinates: A[21; 0] B[0; 0] C[9.57114285714; 16.4133036133]
Centroid: CG[10.19904761905; 5.47110120443]
Coordinates of the circumscribed circle: U[10.5; 4.87441743668]
Coordinates of the inscribed circle: I[10; 5.74545626465]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.8549904579° = 124°51' = 0.96325507479 rad
∠ B' = β' = 120.2499033051° = 120°14'57″ = 1.04328511045 rad
∠ C' = γ' = 114.901106237° = 114°54'4″ = 1.13661908012 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 = 19 ; ; b = 20 ; ; c = 21 ; ;

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

p = a+b+c = 19+20+21 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-19)(30-20)(30-21) } ; ; T = sqrt{ 29700 } = 172.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 172.34 }{ 19 } = 18.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 172.34 }{ 20 } = 17.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 172.34 }{ 21 } = 16.41 ; ;

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{ 19**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 55° 9' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 59° 45'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-19**2-20**2 }{ 2 * 20 * 19 } ) = 65° 5'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 172.34 }{ 30 } = 5.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 55° 9' } = 11.58 ; ;




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