4 12 15 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 12   c = 15

Area: T = 17.66217524612
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 11.31772690307° = 11°19'2″ = 0.19875236069 rad
Angle ∠ B = β = 36.06765882598° = 36°4' = 0.62994807151 rad
Angle ∠ C = γ = 132.6166142709° = 132°36'58″ = 2.31545883316 rad

Height: ha = 8.83108762306
Height: hb = 2.94436254102
Height: hc = 2.35549003282

Median: ma = 13.43550288425
Median: mb = 9.19223881554
Median: mc = 4.87333971724

Inradius: r = 1.13994679007
Circumradius: R = 10.19215141431

Vertex coordinates: A[15; 0] B[0; 0] C[3.23333333333; 2.35549003282]
Centroid: CG[6.07877777778; 0.78549667761]
Coordinates of the circumscribed circle: U[7.5; -6.90105043677]
Coordinates of the inscribed circle: I[3.5; 1.13994679007]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.6832730969° = 168°40'58″ = 0.19875236069 rad
∠ B' = β' = 143.933341174° = 143°56' = 0.62994807151 rad
∠ C' = γ' = 47.38438572905° = 47°23'2″ = 2.31545883316 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 = 12 ; ; c = 15 ; ;

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

p = a+b+c = 4+12+15 = 31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-4)(15.5-12)(15.5-15) } ; ; T = sqrt{ 311.94 } = 17.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.66 }{ 4 } = 8.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.66 }{ 12 } = 2.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.66 }{ 15 } = 2.35 ; ;

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{ 4**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 11° 19'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-4**2-15**2 }{ 2 * 4 * 15 } ) = 36° 4' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-4**2-12**2 }{ 2 * 12 * 4 } ) = 132° 36'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.66 }{ 15.5 } = 1.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 11° 19'2" } = 10.19 ; ;




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