12 20 28 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 20   c = 28

Area: T = 103.9233048454
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 21.78767892983° = 21°47'12″ = 0.38802512067 rad
Angle ∠ B = β = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 17.32105080757
Height: hb = 10.39223048454
Height: hc = 7.42330748896

Median: ma = 23.58796522451
Median: mb = 19.07987840283
Median: mc = 8.71877978871

Inradius: r = 3.46441016151
Circumradius: R = 16.16658075373

Vertex coordinates: A[28; 0] B[0; 0] C[9.42985714286; 7.42330748896]
Centroid: CG[12.47661904762; 2.47443582965]
Coordinates of the circumscribed circle: U[14; -8.08329037687]
Coordinates of the inscribed circle: I[10; 3.46441016151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2133210702° = 158°12'48″ = 0.38802512067 rad
∠ B' = β' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ C' = γ' = 60° = 2.09443951024 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 = 20 ; ; c = 28 ; ;

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

p = a+b+c = 12+20+28 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-12)(30-20)(30-28) } ; ; T = sqrt{ 10800 } = 103.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.92 }{ 12 } = 17.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.92 }{ 20 } = 10.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.92 }{ 28 } = 7.42 ; ;

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**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 21° 47'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-12**2-28**2 }{ 2 * 12 * 28 } ) = 38° 12'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-12**2-20**2 }{ 2 * 20 * 12 } ) = 120° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.92 }{ 30 } = 3.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 21° 47'12" } = 16.17 ; ;




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