13 21 27 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 21   c = 27

Area: T = 133.2198570402
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 28.02882451983° = 28°1'42″ = 0.48991851623 rad
Angle ∠ B = β = 49.38331749605° = 49°22'59″ = 0.86218989981 rad
Angle ∠ C = γ = 102.5898579841° = 102°35'19″ = 1.79105084932 rad

Height: ha = 20.49551646773
Height: hb = 12.68774828954
Height: hc = 9.8688042252

Median: ma = 23.29769955144
Median: mb = 18.40551623193
Median: mc = 11.07992599031

Inradius: r = 4.36878219804
Circumradius: R = 13.83325309635

Vertex coordinates: A[27; 0] B[0; 0] C[8.4632962963; 9.8688042252]
Centroid: CG[11.82109876543; 3.28993474173]
Coordinates of the circumscribed circle: U[13.5; -3.01547823895]
Coordinates of the inscribed circle: I[9.5; 4.36878219804]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.9721754802° = 151°58'18″ = 0.48991851623 rad
∠ B' = β' = 130.617682504° = 130°37'1″ = 0.86218989981 rad
∠ C' = γ' = 77.41114201588° = 77°24'41″ = 1.79105084932 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 = 27 ; ;

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

p = a+b+c = 13+21+27 = 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)(30.5-21)(30.5-27) } ; ; T = sqrt{ 17747.19 } = 133.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.22 }{ 13 } = 20.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.22 }{ 21 } = 12.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.22 }{ 27 } = 9.87 ; ;

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**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 28° 1'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 49° 22'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 102° 35'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.22 }{ 30.5 } = 4.37 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 28° 1'42" } = 13.83 ; ;




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