12 18 24 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 18   c = 24

Area: T = 104.5710550348
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 28.95550243719° = 28°57'18″ = 0.50553605103 rad
Angle ∠ B = β = 46.56774634422° = 46°34'3″ = 0.81327555614 rad
Angle ∠ C = γ = 104.4787512186° = 104°28'39″ = 1.82334765819 rad

Height: ha = 17.42884250579
Height: hb = 11.61989500386
Height: hc = 8.7144212529

Median: ma = 20.34769899494
Median: mb = 16.70332930885
Median: mc = 9.48768329805

Inradius: r = 3.87329833462
Circumradius: R = 12.39435467079

Vertex coordinates: A[24; 0] B[0; 0] C[8.25; 8.7144212529]
Centroid: CG[10.75; 2.90547375097]
Coordinates of the circumscribed circle: U[12; -3.0988386677]
Coordinates of the inscribed circle: I[9; 3.87329833462]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.0454975628° = 151°2'42″ = 0.50553605103 rad
∠ B' = β' = 133.4332536558° = 133°25'57″ = 0.81327555614 rad
∠ C' = γ' = 75.52224878141° = 75°31'21″ = 1.82334765819 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 = 18 ; ; c = 24 ; ;

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

p = a+b+c = 12+18+24 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-12)(27-18)(27-24) } ; ; T = sqrt{ 10935 } = 104.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104.57 }{ 12 } = 17.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104.57 }{ 18 } = 11.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104.57 }{ 24 } = 8.71 ; ;

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{ 18**2+24**2-12**2 }{ 2 * 18 * 24 } ) = 28° 57'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+24**2-18**2 }{ 2 * 12 * 24 } ) = 46° 34'3" ; ; gamma = 180° - alpha - beta = 180° - 28° 57'18" - 46° 34'3" = 104° 28'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104.57 }{ 27 } = 3.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 28° 57'18" } = 12.39 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 24**2 - 12**2 } }{ 2 } = 20.347 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 12**2 - 18**2 } }{ 2 } = 16.703 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 12**2 - 24**2 } }{ 2 } = 9.487 ; ;
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