15 21 27 triangle

Obtuse scalene triangle.

Sides: a = 15   b = 21   c = 27

Area: T = 156.7110521344
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 33.55773097619° = 33°33'26″ = 0.58656855435 rad
Angle ∠ B = β = 50.70435197608° = 50°42'13″ = 0.88549433622 rad
Angle ∠ C = γ = 95.73991704773° = 95°44'21″ = 1.6710963748 rad

Height: ha = 20.89547361792
Height: hb = 14.92548115566
Height: hc = 11.60881867662

Median: ma = 22.99545645751
Median: mb = 19.15107180022
Median: mc = 12.27880291578

Inradius: r = 4.97549371855
Circumradius: R = 13.5688010506

Vertex coordinates: A[27; 0] B[0; 0] C[9.5; 11.60881867662]
Centroid: CG[12.16766666667; 3.86993955887]
Coordinates of the circumscribed circle: U[13.5; -1.35768010506]
Coordinates of the inscribed circle: I[10.5; 4.97549371855]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad
∠ B' = β' = 129.2966480239° = 129°17'47″ = 0.88549433622 rad
∠ C' = γ' = 84.26108295227° = 84°15'39″ = 1.6710963748 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 = 21 ; ; c = 27 ; ;

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

p = a+b+c = 15+21+27 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-15)(31.5-21)(31.5-27) } ; ; T = sqrt{ 24558.19 } = 156.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 156.71 }{ 15 } = 20.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 156.71 }{ 21 } = 14.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 156.71 }{ 27 } = 11.61 ; ;

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**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 33° 33'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-15**2-27**2 }{ 2 * 15 * 27 } ) = 50° 42'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-15**2-21**2 }{ 2 * 21 * 15 } ) = 95° 44'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 156.71 }{ 31.5 } = 4.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 33° 33'26" } = 13.57 ; ;




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