4 21 23 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 21   c = 23

Area: T = 37.9477331922
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 9.04404551853° = 9°2'26″ = 0.15877857089 rad
Angle ∠ B = β = 55.58326112896° = 55°34'57″ = 0.97700995739 rad
Angle ∠ C = γ = 115.3776933525° = 115°22'37″ = 2.01437073709 rad

Height: ha = 18.9743665961
Height: hb = 3.61440316116
Height: hc = 3.32997679932

Median: ma = 21.93217121995
Median: mb = 12.73877392029
Median: mc = 9.81107084352

Inradius: r = 1.58111388301
Circumradius: R = 12.72881675822

Vertex coordinates: A[23; 0] B[0; 0] C[2.26108695652; 3.32997679932]
Centroid: CG[8.42202898551; 1.10999226644]
Coordinates of the circumscribed circle: U[11.5; -5.45549289638]
Coordinates of the inscribed circle: I[3; 1.58111388301]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.9659544815° = 170°57'34″ = 0.15877857089 rad
∠ B' = β' = 124.417738871° = 124°25'3″ = 0.97700995739 rad
∠ C' = γ' = 64.62330664748° = 64°37'23″ = 2.01437073709 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 = 21 ; ; c = 23 ; ;

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

p = a+b+c = 4+21+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 * (24-4)(24-21)(24-23) } ; ; T = sqrt{ 1440 } = 37.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 37.95 }{ 4 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 37.95 }{ 21 } = 3.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 37.95 }{ 23 } = 3.3 ; ;

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{ 21**2+23**2-4**2 }{ 2 * 21 * 23 } ) = 9° 2'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4**2+23**2-21**2 }{ 2 * 4 * 23 } ) = 55° 34'57" ; ; gamma = 180° - alpha - beta = 180° - 9° 2'26" - 55° 34'57" = 115° 22'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 37.95 }{ 24 } = 1.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4 }{ 2 * sin 9° 2'26" } = 12.73 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 23**2 - 4**2 } }{ 2 } = 21.932 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 4**2 - 21**2 } }{ 2 } = 12.738 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 4**2 - 23**2 } }{ 2 } = 9.811 ; ;
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