12 14 21 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 14   c = 21

Area: T = 80.11551515008
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 33.02547340605° = 33°1'29″ = 0.5766390344 rad
Angle ∠ B = β = 39.48219079378° = 39°28'55″ = 0.68990892885 rad
Angle ∠ C = γ = 107.4933358002° = 107°29'36″ = 1.87661130212 rad

Height: ha = 13.35325252501
Height: hb = 11.4455021643
Height: hc = 7.63300144286

Median: ma = 16.8087736314
Median: mb = 15.60444865343
Median: mc = 7.73298124169

Inradius: r = 3.4099155383
Circumradius: R = 11.00991534932

Vertex coordinates: A[21; 0] B[0; 0] C[9.26219047619; 7.63300144286]
Centroid: CG[10.08773015873; 2.54333381429]
Coordinates of the circumscribed circle: U[10.5; -3.30992991155]
Coordinates of the inscribed circle: I[9.5; 3.4099155383]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.9755265939° = 146°58'31″ = 0.5766390344 rad
∠ B' = β' = 140.5188092062° = 140°31'5″ = 0.68990892885 rad
∠ C' = γ' = 72.50766419983° = 72°30'24″ = 1.87661130212 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 = 14 ; ; c = 21 ; ;

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

p = a+b+c = 12+14+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-12)(23.5-14)(23.5-21) } ; ; T = sqrt{ 6418.44 } = 80.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.12 }{ 12 } = 13.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.12 }{ 14 } = 11.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.12 }{ 21 } = 7.63 ; ;

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{ 14**2+21**2-12**2 }{ 2 * 14 * 21 } ) = 33° 1'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+21**2-14**2 }{ 2 * 12 * 21 } ) = 39° 28'55" ; ; gamma = 180° - alpha - beta = 180° - 33° 1'29" - 39° 28'55" = 107° 29'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.12 }{ 23.5 } = 3.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 33° 1'29" } = 11.01 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 21**2 - 12**2 } }{ 2 } = 16.808 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 12**2 - 14**2 } }{ 2 } = 15.604 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 12**2 - 21**2 } }{ 2 } = 7.73 ; ;
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