21 27 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 27   c = 30

Area: T = 275.3477053734
Perimeter: p = 78
Semiperimeter: s = 39

Angle ∠ A = α = 42.83334280661° = 42°50' = 0.74875843497 rad
Angle ∠ B = β = 60.9410718932° = 60°56'27″ = 1.06436161939 rad
Angle ∠ C = γ = 76.2265853002° = 76°13'33″ = 1.333039211 rad

Height: ha = 26.2243528927
Height: hb = 20.39660780544
Height: hc = 18.35664702489

Median: ma = 26.53877090194
Median: mb = 22.0966379794
Median: mc = 18.9743665961

Inradius: r = 7.0660180865
Circumradius: R = 15.44441456421

Vertex coordinates: A[30; 0] B[0; 0] C[10.2; 18.35664702489]
Centroid: CG[13.4; 6.11988234163]
Coordinates of the circumscribed circle: U[15; 3.67771775338]
Coordinates of the inscribed circle: I[12; 7.0660180865]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.1676571934° = 137°10' = 0.74875843497 rad
∠ B' = β' = 119.0599281068° = 119°3'33″ = 1.06436161939 rad
∠ C' = γ' = 103.7744146998° = 103°46'27″ = 1.333039211 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 = 27 ; ; c = 30 ; ;

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

p = a+b+c = 21+27+30 = 78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78 }{ 2 } = 39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39 * (39-21)(39-27)(39-30) } ; ; T = sqrt{ 75816 } = 275.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 275.35 }{ 21 } = 26.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 275.35 }{ 27 } = 20.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 275.35 }{ 30 } = 18.36 ; ;

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**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 42° 50' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 60° 56'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-27**2 }{ 2 * 27 * 21 } ) = 76° 13'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 275.35 }{ 39 } = 7.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 42° 50' } = 15.44 ; ;




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