6 18 21 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 18   c = 21

Area: T = 50.05993397879
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 15.35988855808° = 15°21'32″ = 0.26880631228 rad
Angle ∠ B = β = 52.61768015821° = 52°37' = 0.91883364295 rad
Angle ∠ C = γ = 112.0244312837° = 112°1'28″ = 1.95551931013 rad

Height: ha = 16.6866446596
Height: hb = 5.56221488653
Height: hc = 4.76875561703

Median: ma = 19.32661480901
Median: mb = 12.5549900398
Median: mc = 8.35216465442

Inradius: r = 2.22548595461
Circumradius: R = 11.32765576894

Vertex coordinates: A[21; 0] B[0; 0] C[3.64328571429; 4.76875561703]
Centroid: CG[8.21442857143; 1.58991853901]
Coordinates of the circumscribed circle: U[10.5; -4.24774591335]
Coordinates of the inscribed circle: I[4.5; 2.22548595461]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.6411114419° = 164°38'28″ = 0.26880631228 rad
∠ B' = β' = 127.3833198418° = 127°23' = 0.91883364295 rad
∠ C' = γ' = 67.9765687163° = 67°58'32″ = 1.95551931013 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 = 6 ; ; b = 18 ; ; c = 21 ; ;

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

p = a+b+c = 6+18+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-6)(22.5-18)(22.5-21) } ; ; T = sqrt{ 2505.94 } = 50.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50.06 }{ 6 } = 16.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50.06 }{ 18 } = 5.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50.06 }{ 21 } = 4.77 ; ;

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{ 6**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 15° 21'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-6**2-21**2 }{ 2 * 6 * 21 } ) = 52° 37' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-6**2-18**2 }{ 2 * 18 * 6 } ) = 112° 1'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50.06 }{ 22.5 } = 2.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 15° 21'32" } = 11.33 ; ;




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