20 23 27 triangle

Acute scalene triangle.

Sides: a = 20   b = 23   c = 27

Area: T = 224.4999443206
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 46.30548463945° = 46°18'17″ = 0.80881720292 rad
Angle ∠ B = β = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ C = γ = 77.44441422014° = 77°26'39″ = 1.35216552678 rad

Height: ha = 22.45499443206
Height: hb = 19.52216907136
Height: hc = 16.63295883857

Median: ma = 23
Median: mb = 20.79106228863
Median: mc = 16.88002976164

Inradius: r = 6.41442698059
Circumradius: R = 13.8310769269

Vertex coordinates: A[27; 0] B[0; 0] C[11.11111111111; 16.63295883857]
Centroid: CG[12.70437037037; 5.54331961286]
Coordinates of the circumscribed circle: U[13.5; 3.00766889715]
Coordinates of the inscribed circle: I[12; 6.41442698059]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.6955153606° = 133°41'43″ = 0.80881720292 rad
∠ B' = β' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ C' = γ' = 102.5565857799° = 102°33'21″ = 1.35216552678 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 = 23 ; ; c = 27 ; ;

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

p = a+b+c = 20+23+27 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-20)(35-23)(35-27) } ; ; T = sqrt{ 50400 } = 224.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 224.5 }{ 20 } = 22.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 224.5 }{ 23 } = 19.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 224.5 }{ 27 } = 16.63 ; ;

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**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 46° 18'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 56° 15'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-20**2-23**2 }{ 2 * 23 * 20 } ) = 77° 26'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 224.5 }{ 35 } = 6.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 46° 18'17" } = 13.83 ; ;




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