12 20 23 triangle

Acute scalene triangle.

Sides: a = 12   b = 20   c = 23

Area: T = 119.9411391938
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 31.43218690268° = 31°25'55″ = 0.54985896046 rad
Angle ∠ B = β = 60.35989156325° = 60°21'32″ = 1.05334618107 rad
Angle ∠ C = γ = 88.20992153407° = 88°12'33″ = 1.54395412383 rad

Height: ha = 19.99902319896
Height: hb = 11.99441391938
Height: hc = 10.43296862555

Median: ma = 20.77002415445
Median: mb = 15.37985564992
Median: mc = 11.82215904175

Inradius: r = 4.36215051614
Circumradius: R = 11.50656193504

Vertex coordinates: A[23; 0] B[0; 0] C[5.93547826087; 10.43296862555]
Centroid: CG[9.64549275362; 3.47765620852]
Coordinates of the circumscribed circle: U[11.5; 0.36595506047]
Coordinates of the inscribed circle: I[7.5; 4.36215051614]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.5688130973° = 148°34'5″ = 0.54985896046 rad
∠ B' = β' = 119.6411084367° = 119°38'28″ = 1.05334618107 rad
∠ C' = γ' = 91.79107846593° = 91°47'27″ = 1.54395412383 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 = 12 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 12+20+23 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-12)(27.5-20)(27.5-23) } ; ; T = sqrt{ 14385.94 } = 119.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 119.94 }{ 12 } = 19.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 119.94 }{ 20 } = 11.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 119.94 }{ 23 } = 10.43 ; ;

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{ 12**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 31° 25'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 60° 21'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-20**2 }{ 2 * 20 * 12 } ) = 88° 12'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 119.94 }{ 27.5 } = 4.36 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 31° 25'55" } = 11.51 ; ;




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