16 21 27 triangle

Obtuse scalene triangle.

Sides: a = 16   b = 21   c = 27

Area: T = 167.8099415707
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 36.29333999285° = 36°17'36″ = 0.63334393255 rad
Angle ∠ B = β = 50.97771974348° = 50°58'38″ = 0.89897199387 rad
Angle ∠ C = γ = 92.72994026368° = 92°43'46″ = 1.61884333894 rad

Height: ha = 20.97661769634
Height: hb = 15.9821849115
Height: hc = 12.43303270894

Median: ma = 22.8255424421
Median: mb = 19.5511214796
Median: mc = 12.89437969582

Inradius: r = 5.24440442409
Circumradius: R = 13.51553322026

Vertex coordinates: A[27; 0] B[0; 0] C[10.07440740741; 12.43303270894]
Centroid: CG[12.35880246914; 4.14334423631]
Coordinates of the circumscribed circle: U[13.5; -0.64435872477]
Coordinates of the inscribed circle: I[11; 5.24440442409]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.7076600072° = 143°42'24″ = 0.63334393255 rad
∠ B' = β' = 129.0232802565° = 129°1'22″ = 0.89897199387 rad
∠ C' = γ' = 87.27105973632° = 87°16'14″ = 1.61884333894 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 = 16 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 16+21+27 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-16)(32-21)(32-27) } ; ; T = sqrt{ 28160 } = 167.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 167.81 }{ 16 } = 20.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 167.81 }{ 21 } = 15.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 167.81 }{ 27 } = 12.43 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 21**2+27**2-16**2 }{ 2 * 21 * 27 } ) = 36° 17'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16**2+27**2-21**2 }{ 2 * 16 * 27 } ) = 50° 58'38" ; ; gamma = 180° - alpha - beta = 180° - 36° 17'36" - 50° 58'38" = 92° 43'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 167.81 }{ 32 } = 5.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16 }{ 2 * sin 36° 17'36" } = 13.52 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 27**2 - 16**2 } }{ 2 } = 22.825 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 16**2 - 21**2 } }{ 2 } = 19.551 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 16**2 - 27**2 } }{ 2 } = 12.894 ; ;
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