8 10 13 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 10   c = 13

Area: T = 39.98804639793
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 37.95880285807° = 37°57'29″ = 0.66224925763 rad
Angle ∠ B = β = 50.251118676° = 50°15'4″ = 0.8777048662 rad
Angle ∠ C = γ = 91.79107846593° = 91°47'27″ = 1.60220514153 rad

Height: ha = 9.99551159948
Height: hb = 7.99660927959
Height: hc = 6.15108406122

Median: ma = 10.88657705285
Median: mb = 9.56655632349
Median: mc = 6.30547601065

Inradius: r = 2.57993847729
Circumradius: R = 6.50331761546

Vertex coordinates: A[13; 0] B[0; 0] C[5.11553846154; 6.15108406122]
Centroid: CG[6.03884615385; 2.05502802041]
Coordinates of the circumscribed circle: U[6.5; -0.20332242548]
Coordinates of the inscribed circle: I[5.5; 2.57993847729]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.0421971419° = 142°2'31″ = 0.66224925763 rad
∠ B' = β' = 129.749881324° = 129°44'56″ = 0.8777048662 rad
∠ C' = γ' = 88.20992153407° = 88°12'33″ = 1.60220514153 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 = 8 ; ; b = 10 ; ; c = 13 ; ;

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

p = a+b+c = 8+10+13 = 31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-8)(15.5-10)(15.5-13) } ; ; T = sqrt{ 1598.44 } = 39.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 39.98 }{ 8 } = 10 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 39.98 }{ 10 } = 8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 39.98 }{ 13 } = 6.15 ; ;

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{ 8**2-10**2-13**2 }{ 2 * 10 * 13 } ) = 37° 57'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-8**2-13**2 }{ 2 * 8 * 13 } ) = 50° 15'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-8**2-10**2 }{ 2 * 10 * 8 } ) = 91° 47'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 39.98 }{ 15.5 } = 2.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 37° 57'29" } = 6.5 ; ;




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