12 28 30 triangle

Acute scalene triangle.

Sides: a = 12   b = 28   c = 30

Area: T = 167.8544103316
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 23.55664643091° = 23°33'23″ = 0.41111378623 rad
Angle ∠ B = β = 68.83215511542° = 68°49'54″ = 1.20113371969 rad
Angle ∠ C = γ = 87.61219845367° = 87°36'43″ = 1.52991175944 rad

Height: ha = 27.9765683886
Height: hb = 11.99895788083
Height: hc = 11.19902735544

Median: ma = 28.39901391332
Median: mb = 18.05554700853
Median: mc = 15.46596248337

Inradius: r = 4.79658315233
Circumradius: R = 15.01330378121

Vertex coordinates: A[30; 0] B[0; 0] C[4.33333333333; 11.19902735544]
Centroid: CG[11.44444444444; 3.73300911848]
Coordinates of the circumscribed circle: U[15; 0.62655432422]
Coordinates of the inscribed circle: I[7; 4.79658315233]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.4443535691° = 156°26'37″ = 0.41111378623 rad
∠ B' = β' = 111.1688448846° = 111°10'6″ = 1.20113371969 rad
∠ C' = γ' = 92.38880154633° = 92°23'17″ = 1.52991175944 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 = 28 ; ; c = 30 ; ;

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

p = a+b+c = 12+28+30 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-12)(35-28)(35-30) } ; ; T = sqrt{ 28175 } = 167.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 167.85 }{ 12 } = 27.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 167.85 }{ 28 } = 11.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 167.85 }{ 30 } = 11.19 ; ;

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**2-28**2-30**2 }{ 2 * 28 * 30 } ) = 23° 33'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 68° 49'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-28**2 }{ 2 * 28 * 12 } ) = 87° 36'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 167.85 }{ 35 } = 4.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 23° 33'23" } = 15.01 ; ;




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