8 13 16 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 13   c = 16

Area: T = 51.68111135716
Perimeter: p = 37
Semiperimeter: s = 18.5

Angle ∠ A = α = 29.7977347297° = 29°47'50″ = 0.52200618187 rad
Angle ∠ B = β = 53.85440789884° = 53°51'15″ = 0.9439930994 rad
Angle ∠ C = γ = 96.34985737146° = 96°20'55″ = 1.68215998409 rad

Height: ha = 12.92202783929
Height: hb = 7.95109405495
Height: hc = 6.46601391964

Median: ma = 14.01878457689
Median: mb = 10.85112672071
Median: mc = 7.24656883731

Inradius: r = 2.79435737066
Circumradius: R = 8.04993621606

Vertex coordinates: A[16; 0] B[0; 0] C[4.719875; 6.46601391964]
Centroid: CG[6.906625; 2.15333797321]
Coordinates of the circumscribed circle: U[8; -0.89900737004]
Coordinates of the inscribed circle: I[5.5; 2.79435737066]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.2032652703° = 150°12'10″ = 0.52200618187 rad
∠ B' = β' = 126.1465921012° = 126°8'45″ = 0.9439930994 rad
∠ C' = γ' = 83.65114262854° = 83°39'5″ = 1.68215998409 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 ; ; b = 13 ; ; c = 16 ; ;

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

p = a+b+c = 8+13+16 = 37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-8)(18.5-13)(18.5-16) } ; ; T = sqrt{ 2670.94 } = 51.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 51.68 }{ 8 } = 12.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 51.68 }{ 13 } = 7.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 51.68 }{ 16 } = 6.46 ; ;

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{ 8**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 29° 47'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-8**2-16**2 }{ 2 * 8 * 16 } ) = 53° 51'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-8**2-13**2 }{ 2 * 13 * 8 } ) = 96° 20'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 51.68 }{ 18.5 } = 2.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 29° 47'50" } = 8.05 ; ;




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