17 21 23 triangle

Acute scalene triangle.

Sides: a = 17   b = 21   c = 23

Area: T = 171.2811019088
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 45.17329648542° = 45°10'23″ = 0.78884169696 rad
Angle ∠ B = β = 61.17875376083° = 61°10'39″ = 1.06877494595 rad
Angle ∠ C = γ = 73.64994975375° = 73°38'58″ = 1.28554262245 rad

Height: ha = 20.15107081281
Height: hb = 16.31224780084
Height: hc = 14.89440016599

Median: ma = 20.31662496539
Median: mb = 17.28443860174
Median: mc = 15.25661463024

Inradius: r = 5.61657711177
Circumradius: R = 11.98546904866

Vertex coordinates: A[23; 0] B[0; 0] C[8.19656521739; 14.89440016599]
Centroid: CG[10.39985507246; 4.965466722]
Coordinates of the circumscribed circle: U[11.5; 3.37438414395]
Coordinates of the inscribed circle: I[9.5; 5.61657711177]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.8277035146° = 134°49'37″ = 0.78884169696 rad
∠ B' = β' = 118.8222462392° = 118°49'21″ = 1.06877494595 rad
∠ C' = γ' = 106.3510502463° = 106°21'2″ = 1.28554262245 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 = 17 ; ; b = 21 ; ; c = 23 ; ;

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

p = a+b+c = 17+21+23 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-17)(30.5-21)(30.5-23) } ; ; T = sqrt{ 29337.19 } = 171.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 171.28 }{ 17 } = 20.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 171.28 }{ 21 } = 16.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 171.28 }{ 23 } = 14.89 ; ;

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{ 17**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 45° 10'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-17**2-23**2 }{ 2 * 17 * 23 } ) = 61° 10'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-17**2-21**2 }{ 2 * 21 * 17 } ) = 73° 38'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 171.28 }{ 30.5 } = 5.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 45° 10'23" } = 11.98 ; ;




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