19 20 24 triangle

Acute scalene triangle.

Sides: a = 19   b = 20   c = 24

Area: T = 184.285493563
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 50.1621560023° = 50°9'42″ = 0.87554843803 rad
Angle ∠ B = β = 53.92769820096° = 53°55'37″ = 0.94112033917 rad
Angle ∠ C = γ = 75.91114579674° = 75°54'41″ = 1.32549048815 rad

Height: ha = 19.39884142768
Height: hb = 18.4288493563
Height: hc = 15.35770779691

Median: ma = 19.94436706752
Median: mb = 19.19663538205
Median: mc = 15.37985564992

Inradius: r = 5.85503154168
Circumradius: R = 12.37221452989

Vertex coordinates: A[24; 0] B[0; 0] C[11.18875; 15.35770779691]
Centroid: CG[11.72991666667; 5.11990259897]
Coordinates of the circumscribed circle: U[12; 3.0121640632]
Coordinates of the inscribed circle: I[11.5; 5.85503154168]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.8388439977° = 129°50'18″ = 0.87554843803 rad
∠ B' = β' = 126.073301799° = 126°4'23″ = 0.94112033917 rad
∠ C' = γ' = 104.0898542033° = 104°5'19″ = 1.32549048815 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 = 19 ; ; b = 20 ; ; c = 24 ; ;

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

p = a+b+c = 19+20+24 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-19)(31.5-20)(31.5-24) } ; ; T = sqrt{ 33960.94 } = 184.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 184.28 }{ 19 } = 19.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 184.28 }{ 20 } = 18.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 184.28 }{ 24 } = 15.36 ; ;

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{ 19**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 50° 9'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 53° 55'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-19**2-20**2 }{ 2 * 20 * 19 } ) = 75° 54'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 184.28 }{ 31.5 } = 5.85 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 50° 9'42" } = 12.37 ; ;




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