14 19 20 triangle

Acute scalene triangle.

Sides: a = 14   b = 19   c = 20

Area: T = 127.0776502549
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 41.97663420173° = 41°58'35″ = 0.73326253761 rad
Angle ∠ B = β = 65.18879583752° = 65°11'17″ = 1.13877445063 rad
Angle ∠ C = γ = 72.83656996075° = 72°50'9″ = 1.27112227711 rad

Height: ha = 18.15437860784
Height: hb = 13.37664739525
Height: hc = 12.70876502549

Median: ma = 18.2077141456
Median: mb = 14.41435353054
Median: mc = 13.36603892159

Inradius: r = 4.79553397188
Circumradius: R = 10.46661363299

Vertex coordinates: A[20; 0] B[0; 0] C[5.875; 12.70876502549]
Centroid: CG[8.625; 4.23658834183]
Coordinates of the circumscribed circle: U[10; 3.08986906086]
Coordinates of the inscribed circle: I[7.5; 4.79553397188]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.0243657983° = 138°1'25″ = 0.73326253761 rad
∠ B' = β' = 114.8122041625° = 114°48'43″ = 1.13877445063 rad
∠ C' = γ' = 107.1644300392° = 107°9'51″ = 1.27112227711 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 = 19 ; ; c = 20 ; ;

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

p = a+b+c = 14+19+20 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-14)(26.5-19)(26.5-20) } ; ; T = sqrt{ 16148.44 } = 127.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127.08 }{ 14 } = 18.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127.08 }{ 19 } = 13.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127.08 }{ 20 } = 12.71 ; ;

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{ 14**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 41° 58'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-14**2-20**2 }{ 2 * 14 * 20 } ) = 65° 11'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-14**2-19**2 }{ 2 * 19 * 14 } ) = 72° 50'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127.08 }{ 26.5 } = 4.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 41° 58'35" } = 10.47 ; ;




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