12 22 28 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 22   c = 28

Area: T = 126.1077097342
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 24.17695751524° = 24°10'10″ = 0.42218386652 rad
Angle ∠ B = β = 48.64656289465° = 48°38'44″ = 0.84990263918 rad
Angle ∠ C = γ = 107.1854795901° = 107°11'5″ = 1.87107275966 rad

Height: ha = 21.0187849557
Height: hb = 11.46442815765
Height: hc = 9.00876498101

Median: ma = 24.45440385213
Median: mb = 18.52202591775
Median: mc = 10.86327804912

Inradius: r = 4.0687970882
Circumradius: R = 14.65442108966

Vertex coordinates: A[28; 0] B[0; 0] C[7.92985714286; 9.00876498101]
Centroid: CG[11.97661904762; 3.00325499367]
Coordinates of the circumscribed circle: U[14; -4.33296532194]
Coordinates of the inscribed circle: I[9; 4.0687970882]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.8330424848° = 155°49'50″ = 0.42218386652 rad
∠ B' = β' = 131.3544371054° = 131°21'16″ = 0.84990263918 rad
∠ C' = γ' = 72.81552040989° = 72°48'55″ = 1.87107275966 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 = 22 ; ; c = 28 ; ;

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

p = a+b+c = 12+22+28 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-12)(31-22)(31-28) } ; ; T = sqrt{ 15903 } = 126.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 126.11 }{ 12 } = 21.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 126.11 }{ 22 } = 11.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 126.11 }{ 28 } = 9.01 ; ;

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**2-22**2-28**2 }{ 2 * 22 * 28 } ) = 24° 10'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-12**2-28**2 }{ 2 * 12 * 28 } ) = 48° 38'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-12**2-22**2 }{ 2 * 22 * 12 } ) = 107° 11'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 126.11 }{ 31 } = 4.07 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 24° 10'10" } = 14.65 ; ;




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