13 24 28 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 24   c = 28

Area: T = 155.6955014371
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 27.6055300635° = 27°36'19″ = 0.48218033871 rad
Angle ∠ B = β = 58.81113776665° = 58°48'41″ = 1.02664521779 rad
Angle ∠ C = γ = 93.58333216985° = 93°35' = 1.63333370886 rad

Height: ha = 23.9533079134
Height: hb = 12.97545845309
Height: hc = 11.12110724551

Median: ma = 25.25437125983
Median: mb = 18.23545825288
Median: mc = 13.28553302556

Inradius: r = 4.79106158268
Circumradius: R = 14.02774241203

Vertex coordinates: A[28; 0] B[0; 0] C[6.73221428571; 11.12110724551]
Centroid: CG[11.57773809524; 3.70770241517]
Coordinates of the circumscribed circle: U[14; -0.87767140075]
Coordinates of the inscribed circle: I[8.5; 4.79106158268]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.3954699365° = 152°23'41″ = 0.48218033871 rad
∠ B' = β' = 121.1898622333° = 121°11'19″ = 1.02664521779 rad
∠ C' = γ' = 86.41766783015° = 86°25' = 1.63333370886 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 = 24 ; ; c = 28 ; ;

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

p = a+b+c = 13+24+28 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-13)(32.5-24)(32.5-28) } ; ; T = sqrt{ 24240.94 } = 155.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 155.7 }{ 13 } = 23.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 155.7 }{ 24 } = 12.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 155.7 }{ 28 } = 11.12 ; ;

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**2-24**2-28**2 }{ 2 * 24 * 28 } ) = 27° 36'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-13**2-28**2 }{ 2 * 13 * 28 } ) = 58° 48'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-13**2-24**2 }{ 2 * 24 * 13 } ) = 93° 35' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 155.7 }{ 32.5 } = 4.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 27° 36'19" } = 14.03 ; ;




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