20 21 30 triangle

Obtuse scalene triangle.

Sides: a = 20   b = 21   c = 30

Area: T = 209.4811353585
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 41.68438728078° = 41°41'2″ = 0.72875208255 rad
Angle ∠ B = β = 44.28884644613° = 44°17'18″ = 0.77329795255 rad
Angle ∠ C = γ = 94.02876627308° = 94°1'40″ = 1.64110923026 rad

Height: ha = 20.94881353585
Height: hb = 19.95106051034
Height: hc = 13.96554235724

Median: ma = 23.8855141825
Median: mb = 23.2332520311
Median: mc = 13.98221314541

Inradius: r = 5.90108831996
Circumradius: R = 15.03771378936

Vertex coordinates: A[30; 0] B[0; 0] C[14.31766666667; 13.96554235724]
Centroid: CG[14.77222222222; 4.65551411908]
Coordinates of the circumscribed circle: U[15; -1.05661799235]
Coordinates of the inscribed circle: I[14.5; 5.90108831996]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.3166127192° = 138°18'58″ = 0.72875208255 rad
∠ B' = β' = 135.7121535539° = 135°42'42″ = 0.77329795255 rad
∠ C' = γ' = 85.97223372692° = 85°58'20″ = 1.64110923026 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 = 21 ; ; c = 30 ; ;

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

p = a+b+c = 20+21+30 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-20)(35.5-21)(35.5-30) } ; ; T = sqrt{ 43882.44 } = 209.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 209.48 }{ 20 } = 20.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 209.48 }{ 21 } = 19.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 209.48 }{ 30 } = 13.97 ; ;

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**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 41° 41'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-30**2 }{ 2 * 20 * 30 } ) = 44° 17'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 94° 1'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 209.48 }{ 35.5 } = 5.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 41° 41'2" } = 15.04 ; ;




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