8 19 21 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 19   c = 21

Area: T = 75.8954663844
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 22.36599722153° = 22°21'36″ = 0.39902551358 rad
Angle ∠ B = β = 64.62330664748° = 64°37'23″ = 1.12878852827 rad
Angle ∠ C = γ = 93.01769613098° = 93°1'1″ = 1.62334522351 rad

Height: ha = 18.9743665961
Height: hb = 7.98989119836
Height: hc = 7.22880632232

Median: ma = 19.62114168703
Median: mb = 12.73877392029
Median: mc = 10.11218742081

Inradius: r = 3.16222776602
Circumradius: R = 10.51545732201

Vertex coordinates: A[21; 0] B[0; 0] C[3.42985714286; 7.22880632232]
Centroid: CG[8.14328571429; 2.40993544077]
Coordinates of the circumscribed circle: U[10.5; -0.55333985905]
Coordinates of the inscribed circle: I[5; 3.16222776602]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.6440027785° = 157°38'24″ = 0.39902551358 rad
∠ B' = β' = 115.3776933525° = 115°22'37″ = 1.12878852827 rad
∠ C' = γ' = 86.98330386902° = 86°58'59″ = 1.62334522351 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 = 8 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 8+19+21 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-8)(24-19)(24-21) } ; ; T = sqrt{ 5760 } = 75.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 75.89 }{ 8 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 75.89 }{ 19 } = 7.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 75.89 }{ 21 } = 7.23 ; ;

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{ 8**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 22° 21'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-8**2-21**2 }{ 2 * 8 * 21 } ) = 64° 37'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-8**2-19**2 }{ 2 * 19 * 8 } ) = 93° 1'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 75.89 }{ 24 } = 3.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 22° 21'36" } = 10.51 ; ;




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