15 16 23 triangle

Obtuse scalene triangle.

Sides: a = 15   b = 16   c = 23

Area: T = 119.3988492453
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 40.45990830808° = 40°27'33″ = 0.70661442121 rad
Angle ∠ B = β = 43.80217464419° = 43°48'6″ = 0.76444846935 rad
Angle ∠ C = γ = 95.73991704773° = 95°44'21″ = 1.6710963748 rad

Height: ha = 15.92197989937
Height: hb = 14.92548115566
Height: hc = 10.38224776046

Median: ma = 18.33771208209
Median: mb = 17.6921806013
Median: mc = 10.40443260233

Inradius: r = 4.42221663871
Circumradius: R = 11.55879348755

Vertex coordinates: A[23; 0] B[0; 0] C[10.82660869565; 10.38224776046]
Centroid: CG[11.27553623188; 3.46108258682]
Coordinates of the circumscribed circle: U[11.5; -1.15657934875]
Coordinates of the inscribed circle: I[11; 4.42221663871]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5410916919° = 139°32'27″ = 0.70661442121 rad
∠ B' = β' = 136.1988253558° = 136°11'54″ = 0.76444846935 rad
∠ C' = γ' = 84.26108295227° = 84°15'39″ = 1.6710963748 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 = 16 ; ; c = 23 ; ;

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

p = a+b+c = 15+16+23 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-15)(27-16)(27-23) } ; ; T = sqrt{ 14256 } = 119.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 119.4 }{ 15 } = 15.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 119.4 }{ 16 } = 14.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 119.4 }{ 23 } = 10.38 ; ;

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**2-16**2-23**2 }{ 2 * 16 * 23 } ) = 40° 27'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 43° 48'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-15**2-16**2 }{ 2 * 16 * 15 } ) = 95° 44'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 119.4 }{ 27 } = 4.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 40° 27'33" } = 11.56 ; ;




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