20 24 30 triangle

Acute scalene triangle.

Sides: a = 20   b = 24   c = 30

Area: T = 239.2476734565
Perimeter: p = 74
Semiperimeter: s = 37

Angle ∠ A = α = 41.65496722739° = 41°38'59″ = 0.72769239136 rad
Angle ∠ B = β = 52.89109950542° = 52°53'28″ = 0.92331220084 rad
Angle ∠ C = γ = 85.45993326719° = 85°27'34″ = 1.49215467317 rad

Height: ha = 23.92546734565
Height: hb = 19.93772278804
Height: hc = 15.95497823043

Median: ma = 25.25986618806
Median: mb = 22.49444437584
Median: mc = 16.21772747402

Inradius: r = 6.46661279612
Circumradius: R = 15.04772273176

Vertex coordinates: A[30; 0] B[0; 0] C[12.06766666667; 15.95497823043]
Centroid: CG[14.02222222222; 5.31765941014]
Coordinates of the circumscribed circle: U[15; 1.19112388293]
Coordinates of the inscribed circle: I[13; 6.46661279612]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.3550327726° = 138°21'1″ = 0.72769239136 rad
∠ B' = β' = 127.1099004946° = 127°6'32″ = 0.92331220084 rad
∠ C' = γ' = 94.54106673281° = 94°32'26″ = 1.49215467317 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 = 20 ; ; b = 24 ; ; c = 30 ; ;

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

p = a+b+c = 20+24+30 = 74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74 }{ 2 } = 37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37 * (37-20)(37-24)(37-30) } ; ; T = sqrt{ 57239 } = 239.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 239.25 }{ 20 } = 23.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 239.25 }{ 24 } = 19.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 239.25 }{ 30 } = 15.95 ; ;

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{ 20**2-24**2-30**2 }{ 2 * 24 * 30 } ) = 41° 38'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-20**2-30**2 }{ 2 * 20 * 30 } ) = 52° 53'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-20**2-24**2 }{ 2 * 24 * 20 } ) = 85° 27'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 239.25 }{ 37 } = 6.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 41° 38'59" } = 15.05 ; ;




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