5 20 23 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 20   c = 23

Area: T = 42.70883130081
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 10.70112654404° = 10°42'5″ = 0.18767723161 rad
Angle ∠ B = β = 47.9666483062° = 47°57'59″ = 0.837717306 rad
Angle ∠ C = γ = 121.3322251498° = 121°19'56″ = 2.11876472775 rad

Height: ha = 17.08333252033
Height: hb = 4.27108313008
Height: hc = 3.71437663485

Median: ma = 21.40767746286
Median: mb = 13.30441346957
Median: mc = 8.95882364336

Inradius: r = 1.7879513042
Circumradius: R = 13.46334210415

Vertex coordinates: A[23; 0] B[0; 0] C[3.3487826087; 3.71437663485]
Centroid: CG[8.78326086957; 1.23879221162]
Coordinates of the circumscribed circle: U[11.5; -7.00109789416]
Coordinates of the inscribed circle: I[4; 1.7879513042]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.299873456° = 169°17'55″ = 0.18767723161 rad
∠ B' = β' = 132.0343516938° = 132°2'1″ = 0.837717306 rad
∠ C' = γ' = 58.66877485024° = 58°40'4″ = 2.11876472775 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 = 5 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 5+20+23 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-5)(24-20)(24-23) } ; ; T = sqrt{ 1824 } = 42.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.71 }{ 5 } = 17.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.71 }{ 20 } = 4.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.71 }{ 23 } = 3.71 ; ;

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**2-5**2 }{ 2 * 20 * 23 } ) = 10° 42'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+23**2-20**2 }{ 2 * 5 * 23 } ) = 47° 57'59" ; ; gamma = 180° - alpha - beta = 180° - 10° 42'5" - 47° 57'59" = 121° 19'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.71 }{ 24 } = 1.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 10° 42'5" } = 13.46 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 23**2 - 5**2 } }{ 2 } = 21.407 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 5**2 - 20**2 } }{ 2 } = 13.304 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 5**2 - 23**2 } }{ 2 } = 8.958 ; ;
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