20 21 23 triangle

Acute scalene triangle.

Sides: a = 20   b = 21   c = 23

Area: T = 194.9776921711
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 53.83985840336° = 53°50'19″ = 0.9439660556 rad
Angle ∠ B = β = 57.96551639558° = 57°57'55″ = 1.01216829625 rad
Angle ∠ C = γ = 68.19662520106° = 68°11'47″ = 1.19902491351 rad

Height: ha = 19.49876921711
Height: hb = 18.56992306392
Height: hc = 16.95545149314

Median: ma = 19.62114168703
Median: mb = 18.82215302247
Median: mc = 16.97879268463

Inradius: r = 6.09330288035
Circumradius: R = 12.38660812798

Vertex coordinates: A[23; 0] B[0; 0] C[10.60986956522; 16.95545149314]
Centroid: CG[11.20328985507; 5.65215049771]
Coordinates of the circumscribed circle: U[11.5; 4.60105444754]
Coordinates of the inscribed circle: I[11; 6.09330288035]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.1611415966° = 126°9'41″ = 0.9439660556 rad
∠ B' = β' = 122.0354836044° = 122°2'5″ = 1.01216829625 rad
∠ C' = γ' = 111.8043747989° = 111°48'13″ = 1.19902491351 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 = 23 ; ;

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

p = a+b+c = 20+21+23 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-20)(32-21)(32-23) } ; ; T = sqrt{ 38016 } = 194.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 194.98 }{ 20 } = 19.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 194.98 }{ 21 } = 18.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 194.98 }{ 23 } = 16.95 ; ;

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**2-23**2 }{ 2 * 21 * 23 } ) = 53° 50'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 57° 57'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 68° 11'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 194.98 }{ 32 } = 6.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 53° 50'19" } = 12.39 ; ;




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