8 21 23 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 21   c = 23

Area: T = 83.78554402626
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 20.33001215246° = 20°18' = 0.35443039592 rad
Angle ∠ B = β = 65.60438347173° = 65°36'14″ = 1.14550029178 rad
Angle ∠ C = γ = 94.09660437582° = 94°5'46″ = 1.64222857767 rad

Height: ha = 20.94663600657
Height: hb = 7.98795657393
Height: hc = 7.28656904576

Median: ma = 21.65664078277
Median: mb = 13.6477344064
Median: mc = 10.96658560997

Inradius: r = 3.22325169332
Circumradius: R = 11.5299449472

Vertex coordinates: A[23; 0] B[0; 0] C[3.30443478261; 7.28656904576]
Centroid: CG[8.7688115942; 2.42985634859]
Coordinates of the circumscribed circle: U[11.5; -0.82435321051]
Coordinates of the inscribed circle: I[5; 3.22325169332]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.7699878475° = 159°42' = 0.35443039592 rad
∠ B' = β' = 114.3966165283° = 114°23'46″ = 1.14550029178 rad
∠ C' = γ' = 85.90439562418° = 85°54'14″ = 1.64222857767 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 = 8 ; ; b = 21 ; ; c = 23 ; ;

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

p = a+b+c = 8+21+23 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-8)(26-21)(26-23) } ; ; T = sqrt{ 7020 } = 83.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 83.79 }{ 8 } = 20.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 83.79 }{ 21 } = 7.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 83.79 }{ 23 } = 7.29 ; ;

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{ 8**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 20° 18' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-8**2-23**2 }{ 2 * 8 * 23 } ) = 65° 36'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-8**2-21**2 }{ 2 * 21 * 8 } ) = 94° 5'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 83.79 }{ 26 } = 3.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 20° 18' } = 11.53 ; ;




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