3 20 21 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 20   c = 21

Area: T = 28.91436645896
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 7.91438581623° = 7°54'50″ = 0.13881228815 rad
Angle ∠ B = β = 66.62201318843° = 66°37'12″ = 1.16327406495 rad
Angle ∠ C = γ = 105.4666009953° = 105°27'58″ = 1.84107291226 rad

Height: ha = 19.27657763931
Height: hb = 2.8911366459
Height: hc = 2.75436823419

Median: ma = 20.45111613362
Median: mb = 11.18803398875
Median: mc = 9.70882439195

Inradius: r = 1.31442574813
Circumradius: R = 10.89545028059

Vertex coordinates: A[21; 0] B[0; 0] C[1.19904761905; 2.75436823419]
Centroid: CG[7.39768253968; 0.9187894114]
Coordinates of the circumscribed circle: U[10.5; -2.90552007482]
Coordinates of the inscribed circle: I[2; 1.31442574813]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172.0866141838° = 172°5'10″ = 0.13881228815 rad
∠ B' = β' = 113.3879868116° = 113°22'48″ = 1.16327406495 rad
∠ C' = γ' = 74.53439900466° = 74°32'2″ = 1.84107291226 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 = 3 ; ; b = 20 ; ; c = 21 ; ;

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

p = a+b+c = 3+20+21 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-3)(22-20)(22-21) } ; ; T = sqrt{ 836 } = 28.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.91 }{ 3 } = 19.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.91 }{ 20 } = 2.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.91 }{ 21 } = 2.75 ; ;

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{ 3**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 7° 54'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-3**2-21**2 }{ 2 * 3 * 21 } ) = 66° 37'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-3**2-20**2 }{ 2 * 20 * 3 } ) = 105° 27'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.91 }{ 22 } = 1.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 7° 54'50" } = 10.89 ; ;




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