6 12 15 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 12   c = 15

Area: T = 34.19770393455
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 22.33216450092° = 22°19'54″ = 0.39897607328 rad
Angle ∠ B = β = 49.45883981265° = 49°27'30″ = 0.86332118901 rad
Angle ∠ C = γ = 108.2109956864° = 108°12'36″ = 1.88986200307 rad

Height: ha = 11.39990131152
Height: hb = 5.76995065576
Height: hc = 4.56596052461

Median: ma = 13.24876412995
Median: mb = 9.72111110476
Median: mc = 5.80994750193

Inradius: r = 2.07325478391
Circumradius: R = 7.89554203395

Vertex coordinates: A[15; 0] B[0; 0] C[3.9; 4.56596052461]
Centroid: CG[6.3; 1.52198684154]
Coordinates of the circumscribed circle: U[7.5; -2.46773188561]
Coordinates of the inscribed circle: I[4.5; 2.07325478391]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.6688354991° = 157°40'6″ = 0.39897607328 rad
∠ B' = β' = 130.5421601874° = 130°32'30″ = 0.86332118901 rad
∠ C' = γ' = 71.79900431357° = 71°47'24″ = 1.88986200307 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 = 12 ; ; c = 15 ; ;

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

p = a+b+c = 6+12+15 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-6)(16.5-12)(16.5-15) } ; ; T = sqrt{ 1169.44 } = 34.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.2 }{ 6 } = 11.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.2 }{ 12 } = 5.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.2 }{ 15 } = 4.56 ; ;

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**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 22° 19'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-6**2-15**2 }{ 2 * 6 * 15 } ) = 49° 27'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-6**2-12**2 }{ 2 * 12 * 6 } ) = 108° 12'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.2 }{ 16.5 } = 2.07 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 22° 19'54" } = 7.9 ; ;




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