96 89 116 triangle

Acute scalene triangle.

Sides: a = 96   b = 89   c = 116

Area: T = 4171.705515946
Perimeter: p = 301
Semiperimeter: s = 150.5

Angle ∠ A = α = 53.91662301718° = 53°54'58″ = 0.94110157368 rad
Angle ∠ B = β = 48.52436058471° = 48°31'25″ = 0.8476896687 rad
Angle ∠ C = γ = 77.56601639812° = 77°33'37″ = 1.35436802299 rad

Height: ha = 86.91105241555
Height: hb = 93.74661833587
Height: hc = 71.92659510252

Median: ma = 91.56769154225
Median: mb = 96.72551260015
Median: mc = 72.14222206478

Inradius: r = 27.71989711592
Circumradius: R = 59.39444179967

Vertex coordinates: A[116; 0] B[0; 0] C[63.58218965517; 71.92659510252]
Centroid: CG[59.86106321839; 23.97553170084]
Coordinates of the circumscribed circle: U[58; 12.79444085116]
Coordinates of the inscribed circle: I[61.5; 27.71989711592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.0843769828° = 126°5'2″ = 0.94110157368 rad
∠ B' = β' = 131.4766394153° = 131°28'35″ = 0.8476896687 rad
∠ C' = γ' = 102.4439836019° = 102°26'23″ = 1.35436802299 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 = 96 ; ; b = 89 ; ; c = 116 ; ;

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

p = a+b+c = 96+89+116 = 301 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 301 }{ 2 } = 150.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 150.5 * (150.5-96)(150.5-89)(150.5-116) } ; ; T = sqrt{ 17403123.94 } = 4171.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4171.71 }{ 96 } = 86.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4171.71 }{ 89 } = 93.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4171.71 }{ 116 } = 71.93 ; ;

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{ 89**2+116**2-96**2 }{ 2 * 89 * 116 } ) = 53° 54'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 96**2+116**2-89**2 }{ 2 * 96 * 116 } ) = 48° 31'25" ; ;
 gamma = 180° - alpha - beta = 180° - 53° 54'58" - 48° 31'25" = 77° 33'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4171.71 }{ 150.5 } = 27.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 96 }{ 2 * sin 53° 54'58" } = 59.39 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 89**2+2 * 116**2 - 96**2 } }{ 2 } = 91.567 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 116**2+2 * 96**2 - 89**2 } }{ 2 } = 96.725 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 89**2+2 * 96**2 - 116**2 } }{ 2 } = 72.142 ; ;
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