6 9 12 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 9   c = 12

Area: T = 26.14326375869
Perimeter: p = 27
Semiperimeter: s = 13.5

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 = 8.7144212529
Height: hb = 5.80994750193
Height: hc = 4.35771062645

Median: ma = 10.17334949747
Median: mb = 8.35216465442
Median: mc = 4.74334164903

Inradius: r = 1.93664916731
Circumradius: R = 6.19767733539

Vertex coordinates: A[12; 0] B[0; 0] C[4.125; 4.35771062645]
Centroid: CG[5.375; 1.45223687548]
Coordinates of the circumscribed circle: U[6; -1.54991933385]
Coordinates of the inscribed circle: I[4.5; 1.93664916731]

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 = 6 ; ; b = 9 ; ; c = 12 ; ;

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

p = a+b+c = 6+9+12 = 27 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27 }{ 2 } = 13.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.5 * (13.5-6)(13.5-9)(13.5-12) } ; ; T = sqrt{ 683.44 } = 26.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.14 }{ 6 } = 8.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.14 }{ 9 } = 5.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.14 }{ 12 } = 4.36 ; ;

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{ 6**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 28° 57'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-6**2-12**2 }{ 2 * 6 * 12 } ) = 46° 34'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-6**2-9**2 }{ 2 * 9 * 6 } ) = 104° 28'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.14 }{ 13.5 } = 1.94 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 28° 57'18" } = 6.2 ; ;




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