10 22 27 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 22   c = 27

Area: T = 103.8555368181
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 20.46878277899° = 20°28'4″ = 0.35772309857 rad
Angle ∠ B = β = 50.29109834469° = 50°17'28″ = 0.87877432452 rad
Angle ∠ C = γ = 109.2411188763° = 109°14'28″ = 1.90766184227 rad

Height: ha = 20.77110736362
Height: hb = 9.44113971074
Height: hc = 7.69329902356

Median: ma = 24.11443111036
Median: mb = 17.1321841699
Median: mc = 10.47661634199

Inradius: r = 3.52105209553
Circumradius: R = 14.29987312645

Vertex coordinates: A[27; 0] B[0; 0] C[6.38988888889; 7.69329902356]
Centroid: CG[11.13296296296; 2.56443300785]
Coordinates of the circumscribed circle: U[13.5; -4.7122081894]
Coordinates of the inscribed circle: I[7.5; 3.52105209553]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.532217221° = 159°31'56″ = 0.35772309857 rad
∠ B' = β' = 129.7099016553° = 129°42'32″ = 0.87877432452 rad
∠ C' = γ' = 70.75988112369° = 70°45'32″ = 1.90766184227 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 = 10 ; ; b = 22 ; ; c = 27 ; ;

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

p = a+b+c = 10+22+27 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-10)(29.5-22)(29.5-27) } ; ; T = sqrt{ 10785.94 } = 103.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.86 }{ 10 } = 20.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.86 }{ 22 } = 9.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.86 }{ 27 } = 7.69 ; ;

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{ 10**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 20° 28'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-10**2-27**2 }{ 2 * 10 * 27 } ) = 50° 17'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-10**2-22**2 }{ 2 * 22 * 10 } ) = 109° 14'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.86 }{ 29.5 } = 3.52 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 20° 28'4" } = 14.3 ; ;




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