12 14 20 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 14   c = 20

Area: T = 82.65498638837
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 36.18222872212° = 36°10'56″ = 0.63215000429 rad
Angle ∠ B = β = 43.53111521674° = 43°31'52″ = 0.76597619325 rad
Angle ∠ C = γ = 100.2876560611° = 100°17'12″ = 1.75503306782 rad

Height: ha = 13.7754977314
Height: hb = 11.8077123412
Height: hc = 8.26549863884

Median: ma = 16.18664140562
Median: mb = 14.93331845231
Median: mc = 8.36766002653

Inradius: r = 3.59334723428
Circumradius: R = 10.1633356121

Vertex coordinates: A[20; 0] B[0; 0] C[8.7; 8.26549863884]
Centroid: CG[9.56766666667; 2.75549954628]
Coordinates of the circumscribed circle: U[10; -1.81548850216]
Coordinates of the inscribed circle: I[9; 3.59334723428]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.8187712779° = 143°49'4″ = 0.63215000429 rad
∠ B' = β' = 136.4698847833° = 136°28'8″ = 0.76597619325 rad
∠ C' = γ' = 79.71334393885° = 79°42'48″ = 1.75503306782 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 = 14 ; ; c = 20 ; ;

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

p = a+b+c = 12+14+20 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-12)(23-14)(23-20) } ; ; T = sqrt{ 6831 } = 82.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.65 }{ 12 } = 13.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.65 }{ 14 } = 11.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.65 }{ 20 } = 8.26 ; ;

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**2-14**2-20**2 }{ 2 * 14 * 20 } ) = 36° 10'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 43° 31'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-12**2-14**2 }{ 2 * 14 * 12 } ) = 100° 17'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.65 }{ 23 } = 3.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 36° 10'56" } = 10.16 ; ;




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