4 20 23 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 20   c = 23

Area: T = 28.31985010197
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 7.07224272592° = 7°4'21″ = 0.12334371418 rad
Angle ∠ B = β = 37.99769544166° = 37°59'49″ = 0.66331719603 rad
Angle ∠ C = γ = 134.9310618324° = 134°55'50″ = 2.35549835515 rad

Height: ha = 14.15992505098
Height: hb = 2.8321850102
Height: hc = 2.46224783495

Median: ma = 21.45992637339
Median: mb = 13.13439255366
Median: mc = 8.70334475928

Inradius: r = 1.20550425966
Circumradius: R = 16.24437976389

Vertex coordinates: A[23; 0] B[0; 0] C[3.1522173913; 2.46224783495]
Centroid: CG[8.71773913043; 0.82108261165]
Coordinates of the circumscribed circle: U[11.5; -11.47221820825]
Coordinates of the inscribed circle: I[3.5; 1.20550425966]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172.9287572741° = 172°55'39″ = 0.12334371418 rad
∠ B' = β' = 142.0033045583° = 142°11″ = 0.66331719603 rad
∠ C' = γ' = 45.06993816758° = 45°4'10″ = 2.35549835515 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 = 4 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 4+20+23 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-4)(23.5-20)(23.5-23) } ; ; T = sqrt{ 801.94 } = 28.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.32 }{ 4 } = 14.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.32 }{ 20 } = 2.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.32 }{ 23 } = 2.46 ; ;

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**2-4**2 }{ 2 * 20 * 23 } ) = 7° 4'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4**2+23**2-20**2 }{ 2 * 4 * 23 } ) = 37° 59'49" ; ; gamma = 180° - alpha - beta = 180° - 7° 4'21" - 37° 59'49" = 134° 55'50" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.32 }{ 23.5 } = 1.21 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4 }{ 2 * sin 7° 4'21" } = 16.24 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 23**2 - 4**2 } }{ 2 } = 21.459 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 4**2 - 20**2 } }{ 2 } = 13.134 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 4**2 - 23**2 } }{ 2 } = 8.703 ; ;
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