8 19 20 triangle

Acute scalene triangle.

Sides: a = 8   b = 19   c = 20

Area: T = 75.74325738934
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 23.49435060326° = 23°29'37″ = 0.41100390331 rad
Angle ∠ B = β = 71.22436450997° = 71°13'25″ = 1.24330871123 rad
Angle ∠ C = γ = 85.28328488678° = 85°16'58″ = 1.48884665082 rad

Height: ha = 18.93656434734
Height: hb = 7.97329025151
Height: hc = 7.57442573893

Median: ma = 19.0921883092
Median: mb = 11.90658808998
Median: mc = 10.60766017178

Inradius: r = 3.22330882508
Circumradius: R = 10.03439869763

Vertex coordinates: A[20; 0] B[0; 0] C[2.575; 7.57442573893]
Centroid: CG[7.525; 2.52547524631]
Coordinates of the circumscribed circle: U[10; 0.82551634027]
Coordinates of the inscribed circle: I[4.5; 3.22330882508]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.5066493967° = 156°30'23″ = 0.41100390331 rad
∠ B' = β' = 108.77663549° = 108°46'35″ = 1.24330871123 rad
∠ C' = γ' = 94.71771511322° = 94°43'2″ = 1.48884665082 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 = 19 ; ; c = 20 ; ;

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

p = a+b+c = 8+19+20 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-8)(23.5-19)(23.5-20) } ; ; T = sqrt{ 5736.94 } = 75.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 75.74 }{ 8 } = 18.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 75.74 }{ 19 } = 7.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 75.74 }{ 20 } = 7.57 ; ;

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**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 23° 29'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-8**2-20**2 }{ 2 * 8 * 20 } ) = 71° 13'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-8**2-19**2 }{ 2 * 19 * 8 } ) = 85° 16'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 75.74 }{ 23.5 } = 3.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 23° 29'37" } = 10.03 ; ;




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