18 19 21 triangle

Acute scalene triangle.

Sides: a = 18   b = 19   c = 21

Area: T = 159.7549804382
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 53.20218678868° = 53°12'7″ = 0.92985477628 rad
Angle ∠ B = β = 57.69773438194° = 57°41'50″ = 1.00770086193 rad
Angle ∠ C = γ = 69.10107882938° = 69°6'3″ = 1.20660362715 rad

Height: ha = 17.75499782646
Height: hb = 16.81657688823
Height: hc = 15.2144267084

Median: ma = 17.889854382
Median: mb = 17.09553209973
Median: mc = 15.24397506541

Inradius: r = 5.50986139442
Circumradius: R = 11.23994503827

Vertex coordinates: A[21; 0] B[0; 0] C[9.6199047619; 15.2144267084]
Centroid: CG[10.20663492063; 5.07114223613]
Coordinates of the circumscribed circle: U[10.5; 4.0099394581]
Coordinates of the inscribed circle: I[10; 5.50986139442]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.7988132113° = 126°47'53″ = 0.92985477628 rad
∠ B' = β' = 122.3032656181° = 122°18'10″ = 1.00770086193 rad
∠ C' = γ' = 110.8999211706° = 110°53'57″ = 1.20660362715 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 = 18 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 18+19+21 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-18)(29-19)(29-21) } ; ; T = sqrt{ 25520 } = 159.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 159.75 }{ 18 } = 17.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 159.75 }{ 19 } = 16.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 159.75 }{ 21 } = 15.21 ; ;

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{ 18**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 53° 12'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 57° 41'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-18**2-19**2 }{ 2 * 19 * 18 } ) = 69° 6'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 159.75 }{ 29 } = 5.51 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 53° 12'7" } = 11.24 ; ;




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