13 15 23 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 15   c = 23

Area: T = 91.47223318824
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 32.02439959746° = 32°1'26″ = 0.55989241694 rad
Angle ∠ B = β = 37.72437572832° = 37°43'26″ = 0.65884037708 rad
Angle ∠ C = γ = 110.2522246742° = 110°15'8″ = 1.92442647134 rad

Height: ha = 14.07326664434
Height: hb = 12.19663109177
Height: hc = 7.95441158159

Median: ma = 18.29661744635
Median: mb = 17.11099386323
Median: mc = 8.04767384697

Inradius: r = 3.58771502699
Circumradius: R = 12.25878049223

Vertex coordinates: A[23; 0] B[0; 0] C[10.28326086957; 7.95441158159]
Centroid: CG[11.09442028986; 2.65113719386]
Coordinates of the circumscribed circle: U[11.5; -4.24330863192]
Coordinates of the inscribed circle: I[10.5; 3.58771502699]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.9766004025° = 147°58'34″ = 0.55989241694 rad
∠ B' = β' = 142.2766242717° = 142°16'34″ = 0.65884037708 rad
∠ C' = γ' = 69.74877532578° = 69°44'52″ = 1.92442647134 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 = 23 ; ;

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

p = a+b+c = 13+15+23 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-13)(25.5-15)(25.5-23) } ; ; T = sqrt{ 8367.19 } = 91.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 91.47 }{ 13 } = 14.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 91.47 }{ 15 } = 12.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 91.47 }{ 23 } = 7.95 ; ;

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**2-23**2 }{ 2 * 15 * 23 } ) = 32° 1'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 37° 43'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 110° 15'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 91.47 }{ 25.5 } = 3.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 32° 1'26" } = 12.26 ; ;




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