14 15 21 triangle

Obtuse scalene triangle.

Sides: a = 14   b = 15   c = 21

Area: T = 104.8810884817
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 41.7522205202° = 41°45'8″ = 0.72987134507 rad
Angle ∠ B = β = 45.51883921612° = 45°31'6″ = 0.79444458134 rad
Angle ∠ C = γ = 92.72994026368° = 92°43'46″ = 1.61884333894 rad

Height: ha = 14.98329835453
Height: hb = 13.98441179756
Height: hc = 9.98986556969

Median: ma = 16.85222995464
Median: mb = 16.19441347407
Median: mc = 10.01224921973

Inradius: r = 4.19552353927
Circumradius: R = 10.51219250464

Vertex coordinates: A[21; 0] B[0; 0] C[9.81095238095; 9.98986556969]
Centroid: CG[10.27698412698; 3.3329551899]
Coordinates of the circumscribed circle: U[10.5; -0.50105678594]
Coordinates of the inscribed circle: I[10; 4.19552353927]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2487794798° = 138°14'52″ = 0.72987134507 rad
∠ B' = β' = 134.4821607839° = 134°28'54″ = 0.79444458134 rad
∠ C' = γ' = 87.27105973632° = 87°16'14″ = 1.61884333894 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 = 14 ; ; b = 15 ; ; c = 21 ; ;

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

p = a+b+c = 14+15+21 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-14)(25-15)(25-21) } ; ; T = sqrt{ 11000 } = 104.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104.88 }{ 14 } = 14.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104.88 }{ 15 } = 13.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104.88 }{ 21 } = 9.99 ; ;

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{ 14**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 41° 45'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-14**2-21**2 }{ 2 * 14 * 21 } ) = 45° 31'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-14**2-15**2 }{ 2 * 15 * 14 } ) = 92° 43'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104.88 }{ 25 } = 4.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 41° 45'8" } = 10.51 ; ;




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