10 23 27 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 23   c = 27

Area: T = 112.2549721603
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 21.19331307973° = 21°11'35″ = 0.37698899112 rad
Angle ∠ B = β = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ C = γ = 102.5565857799° = 102°33'21″ = 1.79899373858 rad

Height: ha = 22.45499443206
Height: hb = 9.76108453568
Height: hc = 8.31547941928

Median: ma = 24.57664114549
Median: mb = 16.88002976164
Median: mc = 11.5

Inradius: r = 3.74216573868
Circumradius: R = 13.8310769269

Vertex coordinates: A[27; 0] B[0; 0] C[5.55655555556; 8.31547941928]
Centroid: CG[10.85218518519; 2.77215980643]
Coordinates of the circumscribed circle: U[13.5; -3.00766889715]
Coordinates of the inscribed circle: I[7; 3.74216573868]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.8076869203° = 158°48'25″ = 0.37698899112 rad
∠ B' = β' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ C' = γ' = 77.44441422014° = 77°26'39″ = 1.79899373858 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 = 10 ; ; b = 23 ; ; c = 27 ; ;

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

p = a+b+c = 10+23+27 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-10)(30-23)(30-27) } ; ; T = sqrt{ 12600 } = 112.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 112.25 }{ 10 } = 22.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 112.25 }{ 23 } = 9.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 112.25 }{ 27 } = 8.31 ; ;

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{ 23**2+27**2-10**2 }{ 2 * 23 * 27 } ) = 21° 11'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+27**2-23**2 }{ 2 * 10 * 27 } ) = 56° 15'4" ; ; gamma = 180° - alpha - beta = 180° - 21° 11'35" - 56° 15'4" = 102° 33'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 112.25 }{ 30 } = 3.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 21° 11'35" } = 13.83 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 27**2 - 10**2 } }{ 2 } = 24.576 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 10**2 - 23**2 } }{ 2 } = 16.8 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 10**2 - 27**2 } }{ 2 } = 11.5 ; ;
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