13 23 25 triangle

Acute scalene triangle.

Sides: a = 13   b = 23   c = 25

Area: T = 148.3821897481
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 31.07217412281° = 31°4'18″ = 0.54223041888 rad
Angle ∠ B = β = 65.94400486139° = 65°56'24″ = 1.15108709572 rad
Angle ∠ C = γ = 82.9888210158° = 82°59'18″ = 1.44884175076 rad

Height: ha = 22.82879842278
Height: hb = 12.9032773694
Height: hc = 11.87105517985

Median: ma = 23.12546621597
Median: mb = 16.27111400953
Median: mc = 13.88334433769

Inradius: r = 4.86549802453
Circumradius: R = 12.59441912843

Vertex coordinates: A[25; 0] B[0; 0] C[5.3; 11.87105517985]
Centroid: CG[10.1; 3.95768505995]
Coordinates of the circumscribed circle: U[12.5; 1.53774179996]
Coordinates of the inscribed circle: I[7.5; 4.86549802453]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.9288258772° = 148°55'42″ = 0.54223041888 rad
∠ B' = β' = 114.0659951386° = 114°3'36″ = 1.15108709572 rad
∠ C' = γ' = 97.0121789842° = 97°42″ = 1.44884175076 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 = 23 ; ; c = 25 ; ;

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

p = a+b+c = 13+23+25 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-13)(30.5-23)(30.5-25) } ; ; T = sqrt{ 22017.19 } = 148.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 148.38 }{ 13 } = 22.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 148.38 }{ 23 } = 12.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 148.38 }{ 25 } = 11.87 ; ;

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**2-23**2-25**2 }{ 2 * 23 * 25 } ) = 31° 4'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-13**2-25**2 }{ 2 * 13 * 25 } ) = 65° 56'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-13**2-23**2 }{ 2 * 23 * 13 } ) = 82° 59'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 148.38 }{ 30.5 } = 4.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 31° 4'18" } = 12.59 ; ;




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