8 13 14 triangle

Acute scalene triangle.

Sides: a = 8   b = 13   c = 14

Area: T = 51.17106703102
Perimeter: p = 35
Semiperimeter: s = 17.5

Angle ∠ A = α = 34.21660511313° = 34°12'58″ = 0.59771827493 rad
Angle ∠ B = β = 66.03105176822° = 66°1'50″ = 1.15224499404 rad
Angle ∠ C = γ = 79.75334311865° = 79°45'12″ = 1.3921959964 rad

Height: ha = 12.79326675776
Height: hb = 7.8722410817
Height: hc = 7.31100957586

Median: ma = 12.90334879006
Median: mb = 9.36774969976
Median: mc = 8.21658383626

Inradius: r = 2.92440383034
Circumradius: R = 7.11334499078

Vertex coordinates: A[14; 0] B[0; 0] C[3.25; 7.31100957586]
Centroid: CG[5.75; 2.43766985862]
Coordinates of the circumscribed circle: U[7; 1.26553733009]
Coordinates of the inscribed circle: I[4.5; 2.92440383034]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.7843948869° = 145°47'2″ = 0.59771827493 rad
∠ B' = β' = 113.9699482318° = 113°58'10″ = 1.15224499404 rad
∠ C' = γ' = 100.2476568814° = 100°14'48″ = 1.3921959964 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 = 13 ; ; c = 14 ; ;

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

p = a+b+c = 8+13+14 = 35 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35 }{ 2 } = 17.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.5 * (17.5-8)(17.5-13)(17.5-14) } ; ; T = sqrt{ 2618.44 } = 51.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 51.17 }{ 8 } = 12.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 51.17 }{ 13 } = 7.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 51.17 }{ 14 } = 7.31 ; ;

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**2-13**2-14**2 }{ 2 * 13 * 14 } ) = 34° 12'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-8**2-14**2 }{ 2 * 8 * 14 } ) = 66° 1'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14**2-8**2-13**2 }{ 2 * 13 * 8 } ) = 79° 45'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 51.17 }{ 17.5 } = 2.92 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 34° 12'58" } = 7.11 ; ;




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