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: h_a = 19.60222708745
Height: h_b = 15.68218166996
Height: h_c = 13.63663623475

Median: m_a = 20.01224960962
Median: m_b = 17.10326313765
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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