15 19 21 triangle

Acute scalene triangle.

Sides: a = 15   b = 19   c = 21

Area: T = 137.8122145691
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 43.69224943517° = 43°41'33″ = 0.76325778848 rad
Angle ∠ B = β = 61.04547093922° = 61°2'41″ = 1.06554311698 rad
Angle ∠ C = γ = 75.2632796256° = 75°15'46″ = 1.31435835989 rad

Height: ha = 18.37549527588
Height: hb = 14.50765416517
Height: hc = 13.12549662563

Median: ma = 18.56774446276
Median: mb = 15.58804364509
Median: mc = 13.51985058346

Inradius: r = 5.01113507524
Circumradius: R = 10.85771707704

Vertex coordinates: A[21; 0] B[0; 0] C[7.26219047619; 13.12549662563]
Centroid: CG[9.42106349206; 4.37549887521]
Coordinates of the circumscribed circle: U[10.5; 2.76219118626]
Coordinates of the inscribed circle: I[8.5; 5.01113507524]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.3087505648° = 136°18'27″ = 0.76325778848 rad
∠ B' = β' = 118.9555290608° = 118°57'19″ = 1.06554311698 rad
∠ C' = γ' = 104.7377203744° = 104°44'14″ = 1.31435835989 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 = 19 ; ; c = 21 ; ;

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

p = a+b+c = 15+19+21 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-15)(27.5-19)(27.5-21) } ; ; T = sqrt{ 18992.19 } = 137.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 137.81 }{ 15 } = 18.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 137.81 }{ 19 } = 14.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 137.81 }{ 21 } = 13.12 ; ;

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**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 43° 41'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 61° 2'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-15**2-19**2 }{ 2 * 19 * 15 } ) = 75° 15'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 137.81 }{ 27.5 } = 5.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 43° 41'33" } = 10.86 ; ;




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