13 15 15 triangle

Acute isosceles triangle.

Sides: a = 13   b = 15   c = 15

Area: T = 87.87702879249
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 51.35985772389° = 51°21'31″ = 0.8966376272 rad
Angle ∠ B = β = 64.32107113805° = 64°19'15″ = 1.12326081908 rad
Angle ∠ C = γ = 64.32107113805° = 64°19'15″ = 1.12326081908 rad

Height: ha = 13.51985058346
Height: hb = 11.716603839
Height: hc = 11.716603839

Median: ma = 13.51985058346
Median: mb = 11.86438105177
Median: mc = 11.86438105177

Inradius: r = 4.0876990136
Circumradius: R = 8.32219256164

Vertex coordinates: A[15; 0] B[0; 0] C[5.63333333333; 11.716603839]
Centroid: CG[6.87877777778; 3.905534613]
Coordinates of the circumscribed circle: U[7.5; 3.60661677671]
Coordinates of the inscribed circle: I[6.5; 4.0876990136]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.6411422761° = 128°38'29″ = 0.8966376272 rad
∠ B' = β' = 115.6799288619° = 115°40'45″ = 1.12326081908 rad
∠ C' = γ' = 115.6799288619° = 115°40'45″ = 1.12326081908 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 = 15 ; ; c = 15 ; ;

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

p = a+b+c = 13+15+15 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-13)(21.5-15)(21.5-15) } ; ; T = sqrt{ 7721.19 } = 87.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 87.87 }{ 13 } = 13.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 87.87 }{ 15 } = 11.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 87.87 }{ 15 } = 11.72 ; ;

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**2-15**2-15**2 }{ 2 * 15 * 15 } ) = 51° 21'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 64° 19'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 64° 19'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 87.87 }{ 21.5 } = 4.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 51° 21'31" } = 8.32 ; ;




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