15 27 30 triangle

Acute scalene triangle.

Sides: a = 15   b = 27   c = 30

Area: T = 202.0499498886
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 29.92664348666° = 29°55'35″ = 0.52223148218 rad
Angle ∠ B = β = 63.89661188627° = 63°53'46″ = 1.11551976534 rad
Angle ∠ C = γ = 86.17774462707° = 86°10'39″ = 1.50440801784 rad

Height: ha = 26.94399331848
Height: hb = 14.96766295471
Height: hc = 13.47699665924

Median: ma = 27.5366339626
Median: mb = 19.5
Median: mc = 15.87545078664

Inradius: r = 5.61224860802
Circumradius: R = 15.03334448576

Vertex coordinates: A[30; 0] B[0; 0] C[6.6; 13.47699665924]
Centroid: CG[12.2; 4.49899888641]
Coordinates of the circumscribed circle: U[15; 1.00222296572]
Coordinates of the inscribed circle: I[9; 5.61224860802]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0743565133° = 150°4'25″ = 0.52223148218 rad
∠ B' = β' = 116.1043881137° = 116°6'14″ = 1.11551976534 rad
∠ C' = γ' = 93.82325537293° = 93°49'21″ = 1.50440801784 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 = 27 ; ; c = 30 ; ;

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

p = a+b+c = 15+27+30 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-15)(36-27)(36-30) } ; ; T = sqrt{ 40824 } = 202.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 202.05 }{ 15 } = 26.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 202.05 }{ 27 } = 14.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 202.05 }{ 30 } = 13.47 ; ;

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**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 29° 55'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-15**2-30**2 }{ 2 * 15 * 30 } ) = 63° 53'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-15**2-27**2 }{ 2 * 27 * 15 } ) = 86° 10'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 202.05 }{ 36 } = 5.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 29° 55'35" } = 15.03 ; ;




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