12 21 28 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 21   c = 28

Area: T = 115.762241834
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 23.18880202295° = 23°11'17″ = 0.40547073 rad
Angle ∠ B = β = 43.55659049333° = 43°33'21″ = 0.76601939498 rad
Angle ∠ C = γ = 113.2566074837° = 113°15'22″ = 1.97766914038 rad

Height: ha = 19.29437363901
Height: hb = 11.02549922229
Height: hc = 8.26987441672

Median: ma = 24.01104144071
Median: mb = 18.80882428738
Median: mc = 9.82334413522

Inradius: r = 3.79554891259
Circumradius: R = 15.23881059872

Vertex coordinates: A[28; 0] B[0; 0] C[8.69664285714; 8.26987441672]
Centroid: CG[12.23221428571; 2.75662480557]
Coordinates of the circumscribed circle: U[14; -6.01766331179]
Coordinates of the inscribed circle: I[9.5; 3.79554891259]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.8121979771° = 156°48'43″ = 0.40547073 rad
∠ B' = β' = 136.4444095067° = 136°26'39″ = 0.76601939498 rad
∠ C' = γ' = 66.74439251627° = 66°44'38″ = 1.97766914038 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 = 21 ; ; c = 28 ; ;

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

p = a+b+c = 12+21+28 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-12)(30.5-21)(30.5-28) } ; ; T = sqrt{ 13400.94 } = 115.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 115.76 }{ 12 } = 19.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 115.76 }{ 21 } = 11.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 115.76 }{ 28 } = 8.27 ; ;

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**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 23° 11'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-12**2-28**2 }{ 2 * 12 * 28 } ) = 43° 33'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-12**2-21**2 }{ 2 * 21 * 12 } ) = 113° 15'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 115.76 }{ 30.5 } = 3.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 23° 11'17" } = 15.24 ; ;




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