18 21 23 triangle

Acute scalene triangle.

Sides: a = 18   b = 21   c = 23

Area: T = 179.55550055
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 48.03303348284° = 48°1'49″ = 0.83882874836 rad
Angle ∠ B = β = 60.16596772223° = 60°9'35″ = 1.05499844445 rad
Angle ∠ C = γ = 71.81099879493° = 71°48'36″ = 1.25333207255 rad

Height: ha = 19.95105561666
Height: hb = 17.11004767143
Height: hc = 15.61334787391

Median: ma = 20.10997512422
Median: mb = 17.78334192438
Median: mc = 15.81992920196

Inradius: r = 5.79220969516
Circumradius: R = 12.10549256965

Vertex coordinates: A[23; 0] B[0; 0] C[8.95765217391; 15.61334787391]
Centroid: CG[10.6522173913; 5.2044492913]
Coordinates of the circumscribed circle: U[11.5; 3.77987863285]
Coordinates of the inscribed circle: I[10; 5.79220969516]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.9769665172° = 131°58'11″ = 0.83882874836 rad
∠ B' = β' = 119.8440322778° = 119°50'25″ = 1.05499844445 rad
∠ C' = γ' = 108.1990012051° = 108°11'24″ = 1.25333207255 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 = 18 ; ; b = 21 ; ; c = 23 ; ;

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

p = a+b+c = 18+21+23 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-18)(31-21)(31-23) } ; ; T = sqrt{ 32240 } = 179.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 179.56 }{ 18 } = 19.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 179.56 }{ 21 } = 17.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 179.56 }{ 23 } = 15.61 ; ;

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{ 18**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 48° 1'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-23**2 }{ 2 * 18 * 23 } ) = 60° 9'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 71° 48'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 179.56 }{ 31 } = 5.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 48° 1'49" } = 12.1 ; ;




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