9 15 19 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 15   c = 19

Area: T = 66.08546994394
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 27.63295013938° = 27°37'46″ = 0.482222577 rad
Angle ∠ B = β = 50.61768729893° = 50°37'1″ = 0.88334310907 rad
Angle ∠ C = γ = 101.7543625617° = 101°45'13″ = 1.77659357929 rad

Height: ha = 14.68554887643
Height: hb = 8.81112932586
Height: hc = 6.95662841515

Median: ma = 16.51551445649
Median: mb = 12.8355497653
Median: mc = 7.92114897589

Inradius: r = 3.07437069507
Circumradius: R = 9.70334564043

Vertex coordinates: A[19; 0] B[0; 0] C[5.71105263158; 6.95662841515]
Centroid: CG[8.23768421053; 2.31987613838]
Coordinates of the circumscribed circle: U[9.5; -1.97766300083]
Coordinates of the inscribed circle: I[6.5; 3.07437069507]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.3770498606° = 152°22'14″ = 0.482222577 rad
∠ B' = β' = 129.3833127011° = 129°22'59″ = 0.88334310907 rad
∠ C' = γ' = 78.24663743831° = 78°14'47″ = 1.77659357929 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 = 9 ; ; b = 15 ; ; c = 19 ; ;

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

p = a+b+c = 9+15+19 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-9)(21.5-15)(21.5-19) } ; ; T = sqrt{ 4367.19 } = 66.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66.08 }{ 9 } = 14.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66.08 }{ 15 } = 8.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66.08 }{ 19 } = 6.96 ; ;

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{ 15**2+19**2-9**2 }{ 2 * 15 * 19 } ) = 27° 37'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+19**2-15**2 }{ 2 * 9 * 19 } ) = 50° 37'1" ; ; gamma = 180° - alpha - beta = 180° - 27° 37'46" - 50° 37'1" = 101° 45'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66.08 }{ 21.5 } = 3.07 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9 }{ 2 * sin 27° 37'46" } = 9.7 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 19**2 - 9**2 } }{ 2 } = 16.515 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 9**2 - 15**2 } }{ 2 } = 12.835 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 9**2 - 19**2 } }{ 2 } = 7.921 ; ;
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