13 21 23 triangle

Acute scalene triangle.

Sides: a = 13   b = 21   c = 23

Area: T = 134.9989582931
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 33.9844117855° = 33°59'3″ = 0.593313475 rad
Angle ∠ B = β = 64.54772963821° = 64°32'50″ = 1.12765628451 rad
Angle ∠ C = γ = 81.46985857629° = 81°28'7″ = 1.42218950585 rad

Height: ha = 20.76876281433
Height: hb = 12.85661507554
Height: hc = 11.73882246027

Median: ma = 21.04216254125
Median: mb = 15.45215371404
Median: mc = 13.14334394281

Inradius: r = 4.73664765941
Circumradius: R = 11.62986750867

Vertex coordinates: A[23; 0] B[0; 0] C[5.58769565217; 11.73882246027]
Centroid: CG[9.52989855072; 3.91327415342]
Coordinates of the circumscribed circle: U[11.5; 1.72551331173]
Coordinates of the inscribed circle: I[7.5; 4.73664765941]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.0165882145° = 146°57″ = 0.593313475 rad
∠ B' = β' = 115.4532703618° = 115°27'10″ = 1.12765628451 rad
∠ C' = γ' = 98.53114142371° = 98°31'53″ = 1.42218950585 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 = 21 ; ; c = 23 ; ;

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

p = a+b+c = 13+21+23 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-13)(28.5-21)(28.5-23) } ; ; T = sqrt{ 18222.19 } = 134.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.99 }{ 13 } = 20.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.99 }{ 21 } = 12.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.99 }{ 23 } = 11.74 ; ;

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**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 33° 59'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 64° 32'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 81° 28'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.99 }{ 28.5 } = 4.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 33° 59'3" } = 11.63 ; ;




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