14 18 20 triangle

Acute scalene triangle.

Sides: a = 14   b = 18   c = 20

Area: T = 122.3766468326
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 42.83334280661° = 42°50' = 0.74875843497 rad
Angle ∠ B = β = 60.9410718932° = 60°56'27″ = 1.06436161939 rad
Angle ∠ C = γ = 76.2265853002° = 76°13'33″ = 1.333039211 rad

Height: ha = 17.4822352618
Height: hb = 13.59773853696
Height: hc = 12.23876468326

Median: ma = 17.6921806013
Median: mb = 14.73109198627
Median: mc = 12.64991106407

Inradius: r = 4.70767872433
Circumradius: R = 10.29660970948

Vertex coordinates: A[20; 0] B[0; 0] C[6.8; 12.23876468326]
Centroid: CG[8.93333333333; 4.07992156109]
Coordinates of the circumscribed circle: U[10; 2.45114516892]
Coordinates of the inscribed circle: I[8; 4.70767872433]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.1676571934° = 137°10' = 0.74875843497 rad
∠ B' = β' = 119.0599281068° = 119°3'33″ = 1.06436161939 rad
∠ C' = γ' = 103.7744146998° = 103°46'27″ = 1.333039211 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 = 14 ; ; b = 18 ; ; c = 20 ; ;

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

p = a+b+c = 14+18+20 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-14)(26-18)(26-20) } ; ; T = sqrt{ 14976 } = 122.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 122.38 }{ 14 } = 17.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 122.38 }{ 18 } = 13.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 122.38 }{ 20 } = 12.24 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 18**2+20**2-14**2 }{ 2 * 18 * 20 } ) = 42° 50' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14**2+20**2-18**2 }{ 2 * 14 * 20 } ) = 60° 56'27" ; ; gamma = 180° - alpha - beta = 180° - 42° 50' - 60° 56'27" = 76° 13'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 122.38 }{ 26 } = 4.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14 }{ 2 * sin 42° 50' } = 10.3 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 20**2 - 14**2 } }{ 2 } = 17.692 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 14**2 - 18**2 } }{ 2 } = 14.731 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 14**2 - 20**2 } }{ 2 } = 12.649 ; ;
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