8.5 6.5 13.5 triangle

Obtuse scalene triangle.

Sides: a = 8.5   b = 6.5   c = 13.5

Area: T = 21.82334213347
Perimeter: p = 28.5
Semiperimeter: s = 14.25

Angle ∠ A = α = 29.82881284312° = 29°49'41″ = 0.52105990508 rad
Angle ∠ B = β = 22.35662712474° = 22°21'23″ = 0.39901905417 rad
Angle ∠ C = γ = 127.8165600321° = 127°48'56″ = 2.2310803061 rad

Height: ha = 5.1354922667
Height: hb = 6.71548988722
Height: hc = 3.2333099457

Median: ma = 9.70550244719
Median: mb = 10.80221988502
Median: mc = 3.41986985828

Inradius: r = 1.53114681638
Circumradius: R = 8.54444324765

Vertex coordinates: A[13.5; 0] B[0; 0] C[7.86111111111; 3.2333099457]
Centroid: CG[7.12203703704; 1.0787699819]
Coordinates of the circumscribed circle: U[6.75; -5.2398780998]
Coordinates of the inscribed circle: I[7.75; 1.53114681638]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.1721871569° = 150°10'19″ = 0.52105990508 rad
∠ B' = β' = 157.6443728753° = 157°38'37″ = 0.39901905417 rad
∠ C' = γ' = 52.18443996786° = 52°11'4″ = 2.2310803061 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 = 8.5 ; ; b = 6.5 ; ; c = 13.5 ; ;

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

p = a+b+c = 8.5+6.5+13.5 = 28.5 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28.5 }{ 2 } = 14.25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.25 * (14.25-8.5)(14.25-6.5)(14.25-13.5) } ; ; T = sqrt{ 476.26 } = 21.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.82 }{ 8.5 } = 5.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.82 }{ 6.5 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.82 }{ 13.5 } = 3.23 ; ;

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{ 6.5**2+13.5**2-8.5**2 }{ 2 * 6.5 * 13.5 } ) = 29° 49'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.5**2+13.5**2-6.5**2 }{ 2 * 8.5 * 13.5 } ) = 22° 21'23" ; ; gamma = 180° - alpha - beta = 180° - 29° 49'41" - 22° 21'23" = 127° 48'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.82 }{ 14.25 } = 1.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.5 }{ 2 * sin 29° 49'41" } = 8.54 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.5**2+2 * 13.5**2 - 8.5**2 } }{ 2 } = 9.705 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.5**2+2 * 8.5**2 - 6.5**2 } }{ 2 } = 10.802 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.5**2+2 * 8.5**2 - 13.5**2 } }{ 2 } = 3.419 ; ;
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