12 20 21 triangle

Acute scalene triangle.

Sides: a = 12   b = 20   c = 21

Area: T = 117.2054682074
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 33.92657123692° = 33°55'33″ = 0.59221153819 rad
Angle ∠ B = β = 68.46553711739° = 68°27'55″ = 1.19549461506 rad
Angle ∠ C = γ = 77.60989164569° = 77°36'32″ = 1.35545311211 rad

Height: ha = 19.5344113679
Height: hb = 11.72204682074
Height: hc = 11.16223506737

Median: ma = 19.60986715511
Median: mb = 13.87444369255
Median: mc = 12.7188097342

Inradius: r = 4.42328181915
Circumradius: R = 10.7550423769

Vertex coordinates: A[21; 0] B[0; 0] C[4.40547619048; 11.16223506737]
Centroid: CG[8.46882539683; 3.72107835579]
Coordinates of the circumscribed circle: U[10.5; 2.30768617671]
Coordinates of the inscribed circle: I[6.5; 4.42328181915]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.0744287631° = 146°4'27″ = 0.59221153819 rad
∠ B' = β' = 111.5354628826° = 111°32'5″ = 1.19549461506 rad
∠ C' = γ' = 102.3911083543° = 102°23'28″ = 1.35545311211 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 12+20+21 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-12)(26.5-20)(26.5-21) } ; ; T = sqrt{ 13736.94 } = 117.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 117.2 }{ 12 } = 19.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 117.2 }{ 20 } = 11.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 117.2 }{ 21 } = 11.16 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 33° 55'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 68° 27'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-20**2 }{ 2 * 20 * 12 } ) = 77° 36'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 117.2 }{ 26.5 } = 4.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 33° 55'33" } = 10.75 ; ;




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