5 13 16 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 13   c = 16

Area: T = 28.56657137142
Perimeter: p = 34
Semiperimeter: s = 17

Angle ∠ A = α = 15.94223686056° = 15°56'33″ = 0.27882468227 rad
Angle ∠ B = β = 45.57329959992° = 45°34'23″ = 0.79553988302 rad
Angle ∠ C = γ = 118.4854635395° = 118°29'5″ = 2.06879470007 rad

Height: ha = 11.42662854857
Height: hb = 4.39547251868
Height: hc = 3.57107142143

Median: ma = 14.36114066163
Median: mb = 9.91221138008
Median: mc = 5.74545626465

Inradius: r = 1.68803361008
Circumradius: R = 9.10218205462

Vertex coordinates: A[16; 0] B[0; 0] C[3.5; 3.57107142143]
Centroid: CG[6.5; 1.19902380714]
Coordinates of the circumscribed circle: U[8; -4.34108682605]
Coordinates of the inscribed circle: I[4; 1.68803361008]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.0587631394° = 164°3'27″ = 0.27882468227 rad
∠ B' = β' = 134.4277004001° = 134°25'37″ = 0.79553988302 rad
∠ C' = γ' = 61.51553646048° = 61°30'55″ = 2.06879470007 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 = 5 ; ; b = 13 ; ; c = 16 ; ;

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

p = a+b+c = 5+13+16 = 34 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34 }{ 2 } = 17 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17 * (17-5)(17-13)(17-16) } ; ; T = sqrt{ 816 } = 28.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.57 }{ 5 } = 11.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.57 }{ 13 } = 4.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.57 }{ 16 } = 3.57 ; ;

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{ 5**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 15° 56'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-5**2-16**2 }{ 2 * 5 * 16 } ) = 45° 34'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-5**2-13**2 }{ 2 * 13 * 5 } ) = 118° 29'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.57 }{ 17 } = 1.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 15° 56'33" } = 9.1 ; ;




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