13 21 25 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 21   c = 25

Area: T = 136.4488479288
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 31.31990870573° = 31°19'9″ = 0.54766211879 rad
Angle ∠ B = β = 57.10766549978° = 57°6'24″ = 0.99766991545 rad
Angle ∠ C = γ = 91.57442579449° = 91°34'27″ = 1.59882723112 rad

Height: ha = 20.99220737366
Height: hb = 12.99550932655
Height: hc = 10.9165878343

Median: ma = 22.15328779169
Median: mb = 16.93436942219
Median: mc = 12.19663109177

Inradius: r = 4.62553721793
Circumradius: R = 12.50547197954

Vertex coordinates: A[25; 0] B[0; 0] C[7.06; 10.9165878343]
Centroid: CG[10.68766666667; 3.63986261143]
Coordinates of the circumscribed circle: U[12.5; -0.34435362581]
Coordinates of the inscribed circle: I[8.5; 4.62553721793]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.6810912943° = 148°40'51″ = 0.54766211879 rad
∠ B' = β' = 122.8933345002° = 122°53'36″ = 0.99766991545 rad
∠ C' = γ' = 88.42657420551° = 88°25'33″ = 1.59882723112 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 = 25 ; ;

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

p = a+b+c = 13+21+25 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-13)(29.5-21)(29.5-25) } ; ; T = sqrt{ 18618.19 } = 136.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 136.45 }{ 13 } = 20.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 136.45 }{ 21 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 136.45 }{ 25 } = 10.92 ; ;

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**2-25**2 }{ 2 * 21 * 25 } ) = 31° 19'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-25**2 }{ 2 * 13 * 25 } ) = 57° 6'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 91° 34'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 136.45 }{ 29.5 } = 4.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 31° 19'9" } = 12.5 ; ;




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