12 19 29 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 19   c = 29

Area: T = 77.0711395472
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 16.24553430883° = 16°14'43″ = 0.2843534725 rad
Angle ∠ B = β = 26.29215552747° = 26°17'30″ = 0.4598874205 rad
Angle ∠ C = γ = 137.4633101637° = 137°27'47″ = 2.39991837236 rad

Height: ha = 12.84552325787
Height: hb = 8.11327784707
Height: hc = 5.31552686532

Median: ma = 23.7769728648
Median: mb = 20.05661711201
Median: mc = 6.5

Inradius: r = 2.56990465157
Circumradius: R = 21.44876459117

Vertex coordinates: A[29; 0] B[0; 0] C[10.75986206897; 5.31552686532]
Centroid: CG[13.25328735632; 1.77217562177]
Coordinates of the circumscribed circle: U[14.5; -15.80435285665]
Coordinates of the inscribed circle: I[11; 2.56990465157]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.7554656912° = 163°45'17″ = 0.2843534725 rad
∠ B' = β' = 153.7088444725° = 153°42'30″ = 0.4598874205 rad
∠ C' = γ' = 42.5376898363° = 42°32'13″ = 2.39991837236 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 = 19 ; ; c = 29 ; ;

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

p = a+b+c = 12+19+29 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-12)(30-19)(30-29) } ; ; T = sqrt{ 5940 } = 77.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 77.07 }{ 12 } = 12.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 77.07 }{ 19 } = 8.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 77.07 }{ 29 } = 5.32 ; ;

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{ 19**2+29**2-12**2 }{ 2 * 19 * 29 } ) = 16° 14'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+29**2-19**2 }{ 2 * 12 * 29 } ) = 26° 17'30" ; ; gamma = 180° - alpha - beta = 180° - 16° 14'43" - 26° 17'30" = 137° 27'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 77.07 }{ 30 } = 2.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 16° 14'43" } = 21.45 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 29**2 - 12**2 } }{ 2 } = 23.77 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29**2+2 * 12**2 - 19**2 } }{ 2 } = 20.056 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 12**2 - 29**2 } }{ 2 } = 6.5 ; ;
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