13 21 28 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 21   c = 28

Area: T = 129.3833151917
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 26.10989056776° = 26°6'32″ = 0.45656863682 rad
Angle ∠ B = β = 45.30878989585° = 45°18'28″ = 0.7910772014 rad
Angle ∠ C = γ = 108.5833195364° = 108°35' = 1.89551342714 rad

Height: ha = 19.90551002949
Height: hb = 12.32222049445
Height: hc = 9.24216537084

Median: ma = 23.88799078725
Median: mb = 19.1387659209
Median: mc = 10.44403065089

Inradius: r = 4.17436500618
Circumradius: R = 14.777008383

Vertex coordinates: A[28; 0] B[0; 0] C[9.14328571429; 9.24216537084]
Centroid: CG[12.3810952381; 3.08105512361]
Coordinates of the circumscribed circle: U[14; -4.7076949792]
Coordinates of the inscribed circle: I[10; 4.17436500618]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.8911094322° = 153°53'28″ = 0.45656863682 rad
∠ B' = β' = 134.6922101041° = 134°41'32″ = 0.7910772014 rad
∠ C' = γ' = 71.41768046361° = 71°25' = 1.89551342714 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 = 28 ; ;

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

p = a+b+c = 13+21+28 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-13)(31-21)(31-28) } ; ; T = sqrt{ 16740 } = 129.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.38 }{ 13 } = 19.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.38 }{ 21 } = 12.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.38 }{ 28 } = 9.24 ; ;

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**2-28**2 }{ 2 * 21 * 28 } ) = 26° 6'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-28**2 }{ 2 * 13 * 28 } ) = 45° 18'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 108° 35' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.38 }{ 31 } = 4.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 26° 6'32" } = 14.77 ; ;




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