21 22 28 triangle

Acute scalene triangle.

Sides: a = 21   b = 22   c = 28

Area: T = 228.2954628715
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 47.83551312921° = 47°50'6″ = 0.83548805392 rad
Angle ∠ B = β = 50.94223486763° = 50°56'32″ = 0.88991117131 rad
Angle ∠ C = γ = 81.22325200316° = 81°13'21″ = 1.41876004013 rad

Height: ha = 21.74223455919
Height: hb = 20.75440571559
Height: hc = 16.30767591939

Median: ma = 22.88655849827
Median: mb = 22.17697992774
Median: mc = 16.32548277173

Inradius: r = 6.43108346117
Circumradius: R = 14.1665904902

Vertex coordinates: A[28; 0] B[0; 0] C[13.23221428571; 16.30767591939]
Centroid: CG[13.7444047619; 5.4365586398]
Coordinates of the circumscribed circle: U[14; 2.16216802935]
Coordinates of the inscribed circle: I[13.5; 6.43108346117]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.1654868708° = 132°9'54″ = 0.83548805392 rad
∠ B' = β' = 129.0587651324° = 129°3'28″ = 0.88991117131 rad
∠ C' = γ' = 98.77774799684° = 98°46'39″ = 1.41876004013 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 = 21 ; ; b = 22 ; ; c = 28 ; ;

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

p = a+b+c = 21+22+28 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-21)(35.5-22)(35.5-28) } ; ; T = sqrt{ 52118.44 } = 228.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 228.29 }{ 21 } = 21.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 228.29 }{ 22 } = 20.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 228.29 }{ 28 } = 16.31 ; ;

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{ 21**2-22**2-28**2 }{ 2 * 22 * 28 } ) = 47° 50'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 50° 56'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-21**2-22**2 }{ 2 * 22 * 21 } ) = 81° 13'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 228.29 }{ 35.5 } = 6.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 47° 50'6" } = 14.17 ; ;




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