13 16 23 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 16   c = 23

Area: T = 100.6987567001
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 33.18798823619° = 33°10'48″ = 0.57990981926 rad
Angle ∠ B = β = 42.34326054522° = 42°20'33″ = 0.7399017879 rad
Angle ∠ C = γ = 104.4787512186° = 104°28'39″ = 1.82334765819 rad

Height: ha = 15.49219333848
Height: hb = 12.58771958752
Height: hc = 8.7566310174

Median: ma = 18.71549672722
Median: mb = 16.88219430161
Median: mc = 8.95882364336

Inradius: r = 3.87329833462
Circumradius: R = 11.87771489284

Vertex coordinates: A[23; 0] B[0; 0] C[9.60986956522; 8.7566310174]
Centroid: CG[10.87695652174; 2.9198770058]
Coordinates of the circumscribed circle: U[11.5; -2.96992872321]
Coordinates of the inscribed circle: I[10; 3.87329833462]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.8220117638° = 146°49'12″ = 0.57990981926 rad
∠ B' = β' = 137.6577394548° = 137°39'27″ = 0.7399017879 rad
∠ C' = γ' = 75.52224878141° = 75°31'21″ = 1.82334765819 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 = 13 ; ; b = 16 ; ; c = 23 ; ;

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

p = a+b+c = 13+16+23 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-13)(26-16)(26-23) } ; ; T = sqrt{ 10140 } = 100.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 100.7 }{ 13 } = 15.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 100.7 }{ 16 } = 12.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 100.7 }{ 23 } = 8.76 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13**2-16**2-23**2 }{ 2 * 16 * 23 } ) = 33° 10'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 42° 20'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-16**2 }{ 2 * 16 * 13 } ) = 104° 28'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 100.7 }{ 26 } = 3.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 33° 10'48" } = 11.88 ; ;




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