15 22 27 triangle

Obtuse scalene triangle.

Sides: a = 15   b = 22   c = 27

Area: T = 164.9244225025
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 33.73114095226° = 33°43'53″ = 0.58987241575 rad
Angle ∠ B = β = 54.53220889015° = 54°31'56″ = 0.95217644993 rad
Angle ∠ C = γ = 91.73765015759° = 91°44'11″ = 1.60111039968 rad

Height: ha = 21.998989667
Height: hb = 14.99331113659
Height: hc = 12.21766092611

Median: ma = 23.45774082115
Median: mb = 18.86879622641
Median: mc = 13.12444047484

Inradius: r = 5.1543882032
Circumradius: R = 13.50662026192

Vertex coordinates: A[27; 0] B[0; 0] C[8.70437037037; 12.21766092611]
Centroid: CG[11.90112345679; 4.0722203087]
Coordinates of the circumscribed circle: U[13.5; -0.40992788672]
Coordinates of the inscribed circle: I[10; 5.1543882032]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.2698590477° = 146°16'7″ = 0.58987241575 rad
∠ B' = β' = 125.4687911099° = 125°28'4″ = 0.95217644993 rad
∠ C' = γ' = 88.26334984241° = 88°15'49″ = 1.60111039968 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 = 15 ; ; b = 22 ; ; c = 27 ; ;

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

p = a+b+c = 15+22+27 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-15)(32-22)(32-27) } ; ; T = sqrt{ 27200 } = 164.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 164.92 }{ 15 } = 21.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 164.92 }{ 22 } = 14.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 164.92 }{ 27 } = 12.22 ; ;

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{ 15**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 33° 43'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-15**2-27**2 }{ 2 * 15 * 27 } ) = 54° 31'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-15**2-22**2 }{ 2 * 22 * 15 } ) = 91° 44'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 164.92 }{ 32 } = 5.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 33° 43'53" } = 13.51 ; ;




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