20 27 28 triangle

Acute scalene triangle.

Sides: a = 20   b = 27   c = 28

Area: T = 255.8533351551
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 42.59988128925° = 42°35'56″ = 0.74334895424 rad
Angle ∠ B = β = 66.03105176822° = 66°1'50″ = 1.15224499404 rad
Angle ∠ C = γ = 71.37106694253° = 71°22'14″ = 1.24656531708 rad

Height: ha = 25.58553351551
Height: hb = 18.95221001149
Height: hc = 18.27552393965

Median: ma = 25.62222559506
Median: mb = 20.242228248
Median: mc = 19.19663538205

Inradius: r = 6.82327560414
Circumradius: R = 14.774408827

Vertex coordinates: A[28; 0] B[0; 0] C[8.125; 18.27552393965]
Centroid: CG[12.04216666667; 6.09217464655]
Coordinates of the circumscribed circle: U[14; 4.72195004196]
Coordinates of the inscribed circle: I[10.5; 6.82327560414]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.4011187108° = 137°24'4″ = 0.74334895424 rad
∠ B' = β' = 113.9699482318° = 113°58'10″ = 1.15224499404 rad
∠ C' = γ' = 108.6299330575° = 108°37'46″ = 1.24656531708 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 = 27 ; ; c = 28 ; ;

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

p = a+b+c = 20+27+28 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-20)(37.5-27)(37.5-28) } ; ; T = sqrt{ 65460.94 } = 255.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 255.85 }{ 20 } = 25.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 255.85 }{ 27 } = 18.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 255.85 }{ 28 } = 18.28 ; ;

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**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 42° 35'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 66° 1'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-20**2-27**2 }{ 2 * 27 * 20 } ) = 71° 22'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 255.85 }{ 37.5 } = 6.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 42° 35'56" } = 14.77 ; ;




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