8 21 28 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 21   c = 28

Area: T = 46.80774513299
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 9.16109877988° = 9°9'40″ = 0.16598893998 rad
Angle ∠ B = β = 24.7043569498° = 24°42'13″ = 0.43111586247 rad
Angle ∠ C = γ = 146.1355442703° = 146°8'8″ = 2.5510544629 rad

Height: ha = 11.70218628325
Height: hb = 4.45878525076
Height: hc = 3.34333893807

Median: ma = 24.42333494836
Median: mb = 17.71329895839
Median: mc = 7.51766481892

Inradius: r = 1.64223667133
Circumradius: R = 25.12442049415

Vertex coordinates: A[28; 0] B[0; 0] C[7.26878571429; 3.34333893807]
Centroid: CG[11.7565952381; 1.11444631269]
Coordinates of the circumscribed circle: U[14; -20.86220630318]
Coordinates of the inscribed circle: I[7.5; 1.64223667133]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.8399012201° = 170°50'20″ = 0.16598893998 rad
∠ B' = β' = 155.2966430502° = 155°17'47″ = 0.43111586247 rad
∠ C' = γ' = 33.86545572968° = 33°51'52″ = 2.5510544629 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 = 8 ; ; b = 21 ; ; c = 28 ; ;

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

p = a+b+c = 8+21+28 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-8)(28.5-21)(28.5-28) } ; ; T = sqrt{ 2190.94 } = 46.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46.81 }{ 8 } = 11.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46.81 }{ 21 } = 4.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46.81 }{ 28 } = 3.34 ; ;

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{ 8**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 9° 9'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-8**2-28**2 }{ 2 * 8 * 28 } ) = 24° 42'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-8**2-21**2 }{ 2 * 21 * 8 } ) = 146° 8'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46.81 }{ 28.5 } = 1.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 9° 9'40" } = 25.12 ; ;




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