13 25 27 triangle

Acute scalene triangle.

Sides: a = 13   b = 25   c = 27

Area: T = 161.685545853
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 28.62545509534° = 28°37'28″ = 0.5499592661 rad
Angle ∠ B = β = 67.11546195238° = 67°6'53″ = 1.17113710869 rad
Angle ∠ C = γ = 84.26108295227° = 84°15'39″ = 1.47106289056 rad

Height: ha = 24.87546859277
Height: hb = 12.93548366824
Height: hc = 11.97767006318

Median: ma = 25.19442453747
Median: mb = 17.11099386323
Median: mc = 14.65443508898

Inradius: r = 4.97549371855
Circumradius: R = 13.5688010506

Vertex coordinates: A[27; 0] B[0; 0] C[5.05655555556; 11.97767006318]
Centroid: CG[10.68551851852; 3.99222335439]
Coordinates of the circumscribed circle: U[13.5; 1.35768010506]
Coordinates of the inscribed circle: I[7.5; 4.97549371855]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.3755449047° = 151°22'32″ = 0.5499592661 rad
∠ B' = β' = 112.8855380476° = 112°53'7″ = 1.17113710869 rad
∠ C' = γ' = 95.73991704773° = 95°44'21″ = 1.47106289056 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 = 25 ; ; c = 27 ; ;

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

p = a+b+c = 13+25+27 = 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)(32.5-25)(32.5-27) } ; ; T = sqrt{ 26142.19 } = 161.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 161.69 }{ 13 } = 24.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 161.69 }{ 25 } = 12.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 161.69 }{ 27 } = 11.98 ; ;

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**2-25**2-27**2 }{ 2 * 25 * 27 } ) = 28° 37'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 67° 6'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-25**2 }{ 2 * 25 * 13 } ) = 84° 15'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 161.69 }{ 32.5 } = 4.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 28° 37'28" } = 13.57 ; ;




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