15 16 21 triangle

Acute scalene triangle.

Sides: a = 15   b = 16   c = 21

Area: T = 119.5832607431
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 45.38216583472° = 45°22'54″ = 0.79220593582 rad
Angle ∠ B = β = 49.39985335° = 49°23'55″ = 0.86221670552 rad
Angle ∠ C = γ = 85.22198081528° = 85°13'11″ = 1.48773662402 rad

Height: ha = 15.94443476575
Height: hb = 14.94878259289
Height: hc = 11.38988197553

Median: ma = 17.09553209973
Median: mb = 16.40112194669
Median: mc = 11.41327122105

Inradius: r = 4.5999331055
Circumradius: R = 10.53766493261

Vertex coordinates: A[21; 0] B[0; 0] C[9.76219047619; 11.38988197553]
Centroid: CG[10.2543968254; 3.79662732518]
Coordinates of the circumscribed circle: U[10.5; 0.87880541105]
Coordinates of the inscribed circle: I[10; 4.5999331055]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.6188341653° = 134°37'6″ = 0.79220593582 rad
∠ B' = β' = 130.60114665° = 130°36'5″ = 0.86221670552 rad
∠ C' = γ' = 94.78801918472° = 94°46'49″ = 1.48773662402 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 = 15 ; ; b = 16 ; ; c = 21 ; ;

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

p = a+b+c = 15+16+21 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-15)(26-16)(26-21) } ; ; T = sqrt{ 14300 } = 119.58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 119.58 }{ 15 } = 15.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 119.58 }{ 16 } = 14.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 119.58 }{ 21 } = 11.39 ; ;

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{ 15**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 45° 22'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 49° 23'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-15**2-16**2 }{ 2 * 16 * 15 } ) = 85° 13'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 119.58 }{ 26 } = 4.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 45° 22'54" } = 10.54 ; ;




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