12 18 19 triangle

Acute scalene triangle.

Sides: a = 12   b = 18   c = 19

Area: T = 104.6354781502
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 37.72769622912° = 37°43'37″ = 0.65884597088 rad
Angle ∠ B = β = 66.61436132504° = 66°36'49″ = 1.16326268779 rad
Angle ∠ C = γ = 75.65994244584° = 75°39'34″ = 1.3210506067 rad

Height: ha = 17.43991302504
Height: hb = 11.62660868336
Height: hc = 11.01441875265

Median: ma = 17.50771414
Median: mb = 13.09658008537
Median: mc = 11.99895788083

Inradius: r = 4.27108074083
Circumradius: R = 9.80655348831

Vertex coordinates: A[19; 0] B[0; 0] C[4.76331578947; 11.01441875265]
Centroid: CG[7.92110526316; 3.67113958422]
Coordinates of the circumscribed circle: U[9.5; 2.42986857233]
Coordinates of the inscribed circle: I[6.5; 4.27108074083]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.2733037709° = 142°16'23″ = 0.65884597088 rad
∠ B' = β' = 113.386638675° = 113°23'11″ = 1.16326268779 rad
∠ C' = γ' = 104.3410575542° = 104°20'26″ = 1.3210506067 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 = 18 ; ; c = 19 ; ;

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

p = a+b+c = 12+18+19 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-12)(24.5-18)(24.5-19) } ; ; T = sqrt{ 10948.44 } = 104.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104.63 }{ 12 } = 17.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104.63 }{ 18 } = 11.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104.63 }{ 19 } = 11.01 ; ;

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{ 12**2-18**2-19**2 }{ 2 * 18 * 19 } ) = 37° 43'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-12**2-19**2 }{ 2 * 12 * 19 } ) = 66° 36'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-12**2-18**2 }{ 2 * 18 * 12 } ) = 75° 39'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104.63 }{ 24.5 } = 4.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 37° 43'37" } = 9.81 ; ;




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