15 18 23 triangle

Acute scalene triangle.

Sides: a = 15   b = 18   c = 23

Area: T = 134.9077375632
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 40.67218958023° = 40°40'19″ = 0.71098584948 rad
Angle ∠ B = β = 51.45106555599° = 51°27'2″ = 0.89879833418 rad
Angle ∠ C = γ = 87.87774486379° = 87°52'39″ = 1.5343750817 rad

Height: ha = 17.98876500843
Height: hb = 14.99897084036
Height: hc = 11.73110761419

Median: ma = 19.24218814049
Median: mb = 17.20546505341
Median: mc = 11.92768604419

Inradius: r = 4.81881205583
Circumradius: R = 11.50878956412

Vertex coordinates: A[23; 0] B[0; 0] C[9.3487826087; 11.73110761419]
Centroid: CG[10.78326086957; 3.9110358714]
Coordinates of the circumscribed circle: U[11.5; 0.42662183571]
Coordinates of the inscribed circle: I[10; 4.81881205583]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.3288104198° = 139°19'41″ = 0.71098584948 rad
∠ B' = β' = 128.549934444° = 128°32'58″ = 0.89879833418 rad
∠ C' = γ' = 92.12325513621° = 92°7'21″ = 1.5343750817 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 = 18 ; ; c = 23 ; ;

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

p = a+b+c = 15+18+23 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-15)(28-18)(28-23) } ; ; T = sqrt{ 18200 } = 134.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.91 }{ 15 } = 17.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.91 }{ 18 } = 14.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.91 }{ 23 } = 11.73 ; ;

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**2-18**2-23**2 }{ 2 * 18 * 23 } ) = 40° 40'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 51° 27'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-15**2-18**2 }{ 2 * 18 * 15 } ) = 87° 52'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.91 }{ 28 } = 4.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 40° 40'19" } = 11.51 ; ;




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