13 29 30 triangle

Acute scalene triangle.

Sides: a = 13   b = 29   c = 30

Area: T = 186.4833243215
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 25.38549039049° = 25°23'6″ = 0.44330501534 rad
Angle ∠ B = β = 73.00438351829° = 73°14″ = 1.27441572905 rad
Angle ∠ C = γ = 81.61112609121° = 81°36'41″ = 1.42443852096 rad

Height: ha = 28.69897297254
Height: hb = 12.86109133252
Height: hc = 12.43222162143

Median: ma = 28.77993328623
Median: mb = 18.00769431054
Median: mc = 16.73332005307

Inradius: r = 5.18800900893
Circumradius: R = 15.16222202148

Vertex coordinates: A[30; 0] B[0; 0] C[3.8; 12.43222162143]
Centroid: CG[11.26766666667; 4.14440720714]
Coordinates of the circumscribed circle: U[15; 2.21219949916]
Coordinates of the inscribed circle: I[7; 5.18800900893]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.6155096095° = 154°36'54″ = 0.44330501534 rad
∠ B' = β' = 106.9966164817° = 106°59'46″ = 1.27441572905 rad
∠ C' = γ' = 98.38987390879° = 98°23'19″ = 1.42443852096 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 = 29 ; ; c = 30 ; ;

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

p = a+b+c = 13+29+30 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-13)(36-29)(36-30) } ; ; T = sqrt{ 34776 } = 186.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 186.48 }{ 13 } = 28.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 186.48 }{ 29 } = 12.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 186.48 }{ 30 } = 12.43 ; ;

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**2-29**2-30**2 }{ 2 * 29 * 30 } ) = 25° 23'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29**2-13**2-30**2 }{ 2 * 13 * 30 } ) = 73° 14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-13**2-29**2 }{ 2 * 29 * 13 } ) = 81° 36'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 186.48 }{ 36 } = 5.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 25° 23'6" } = 15.16 ; ;




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