16 20 23 triangle

Acute scalene triangle.

Sides: a = 16   b = 20   c = 23

Area: T = 156.8188166996
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 42.98658810327° = 42°59'9″ = 0.75502451559 rad
Angle ∠ B = β = 58.46597221093° = 58°27'35″ = 1.02203146306 rad
Angle ∠ C = γ = 78.55443968581° = 78°33'16″ = 1.37110328671 rad

Height: ha = 19.60222708745
Height: hb = 15.68218166996
Height: hc = 13.63663623475

Median: ma = 20.01224960962
Median: mb = 17.10326313765
Median: mc = 13.99110685796

Inradius: r = 5.31658700677
Circumradius: R = 11.73333344423

Vertex coordinates: A[23; 0] B[0; 0] C[8.37695652174; 13.63663623475]
Centroid: CG[10.45765217391; 4.54554541158]
Coordinates of the circumscribed circle: U[11.5; 2.32883335534]
Coordinates of the inscribed circle: I[9.5; 5.31658700677]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.0144118967° = 137°51″ = 0.75502451559 rad
∠ B' = β' = 121.5440277891° = 121°32'25″ = 1.02203146306 rad
∠ C' = γ' = 101.4465603142° = 101°26'44″ = 1.37110328671 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 = 16 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 16+20+23 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-16)(29.5-20)(29.5-23) } ; ; T = sqrt{ 24591.94 } = 156.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 156.82 }{ 16 } = 19.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 156.82 }{ 20 } = 15.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 156.82 }{ 23 } = 13.64 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 20**2+23**2-16**2 }{ 2 * 20 * 23 } ) = 42° 59'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16**2+23**2-20**2 }{ 2 * 16 * 23 } ) = 58° 27'35" ; ; gamma = 180° - alpha - beta = 180° - 42° 59'9" - 58° 27'35" = 78° 33'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 156.82 }{ 29.5 } = 5.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16 }{ 2 * sin 42° 59'9" } = 11.73 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 23**2 - 16**2 } }{ 2 } = 20.012 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 16**2 - 20**2 } }{ 2 } = 17.103 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 16**2 - 23**2 } }{ 2 } = 13.991 ; ;
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