16 26 27 triangle

Acute scalene triangle.

Sides: a = 16   b = 26   c = 27

Area: T = 201.714375139
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 35.07774391825° = 35°4'39″ = 0.61222168069 rad
Angle ∠ B = β = 69.04547372744° = 69°2'41″ = 1.20550579966 rad
Angle ∠ C = γ = 75.87878235431° = 75°52'40″ = 1.32443178501 rad

Height: ha = 25.21442189238
Height: hb = 15.51664424146
Height: hc = 14.94217593623

Median: ma = 25.2698557537
Median: mb = 17.98661057486
Median: mc = 16.84548805279

Inradius: r = 5.84767754026
Circumradius: R = 13.92107167615

Vertex coordinates: A[27; 0] B[0; 0] C[5.72222222222; 14.94217593623]
Centroid: CG[10.90774074074; 4.98105864541]
Coordinates of the circumscribed circle: U[13.5; 3.39765210368]
Coordinates of the inscribed circle: I[8.5; 5.84767754026]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.9232560818° = 144°55'21″ = 0.61222168069 rad
∠ B' = β' = 110.9555262726° = 110°57'19″ = 1.20550579966 rad
∠ C' = γ' = 104.1222176457° = 104°7'20″ = 1.32443178501 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 = 26 ; ; c = 27 ; ;

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

p = a+b+c = 16+26+27 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-16)(34.5-26)(34.5-27) } ; ; T = sqrt{ 40688.44 } = 201.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 201.71 }{ 16 } = 25.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 201.71 }{ 26 } = 15.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 201.71 }{ 27 } = 14.94 ; ;

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{ 26**2+27**2-16**2 }{ 2 * 26 * 27 } ) = 35° 4'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16**2+27**2-26**2 }{ 2 * 16 * 27 } ) = 69° 2'41" ; ; gamma = 180° - alpha - beta = 180° - 35° 4'39" - 69° 2'41" = 75° 52'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 201.71 }{ 34.5 } = 5.85 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16 }{ 2 * sin 35° 4'39" } = 13.92 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 26**2+2 * 27**2 - 16**2 } }{ 2 } = 25.269 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 16**2 - 26**2 } }{ 2 } = 17.986 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 26**2+2 * 16**2 - 27**2 } }{ 2 } = 16.845 ; ;
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