12 18 20 triangle

Acute scalene triangle.

Sides: a = 12   b = 18   c = 20

Area: T = 106.6543645039
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 36.33660575146° = 36°20'10″ = 0.63441838408 rad
Angle ∠ B = β = 62.7220387264° = 62°43'13″ = 1.09546772659 rad
Angle ∠ C = γ = 80.94435552214° = 80°56'37″ = 1.41327315469 rad

Height: ha = 17.77656075064
Height: hb = 11.85504050043
Height: hc = 10.66553645039

Median: ma = 18.05554700853
Median: mb = 13.82202749611
Median: mc = 11.57658369028

Inradius: r = 4.26661458015
Circumradius: R = 10.12662361883

Vertex coordinates: A[20; 0] B[0; 0] C[5.5; 10.66553645039]
Centroid: CG[8.5; 3.55551215013]
Coordinates of the circumscribed circle: U[10; 1.59439445852]
Coordinates of the inscribed circle: I[7; 4.26661458015]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.6643942485° = 143°39'50″ = 0.63441838408 rad
∠ B' = β' = 117.2879612736° = 117°16'47″ = 1.09546772659 rad
∠ C' = γ' = 99.05664447786° = 99°3'23″ = 1.41327315469 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 = 12 ; ; b = 18 ; ; c = 20 ; ;

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

p = a+b+c = 12+18+20 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-12)(25-18)(25-20) } ; ; T = sqrt{ 11375 } = 106.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.65 }{ 12 } = 17.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.65 }{ 18 } = 11.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.65 }{ 20 } = 10.67 ; ;

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{ 12**2-18**2-20**2 }{ 2 * 18 * 20 } ) = 36° 20'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 62° 43'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-12**2-18**2 }{ 2 * 18 * 12 } ) = 80° 56'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.65 }{ 25 } = 4.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 36° 20'10" } = 10.13 ; ;




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