15 22 23 triangle

Acute scalene triangle.

Sides: a = 15   b = 22   c = 23

Area: T = 158.7455078664
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 38.86223042378° = 38°51'44″ = 0.67882751639 rad
Angle ∠ B = β = 66.96443158941° = 66°57'52″ = 1.16987477937 rad
Angle ∠ C = γ = 74.17333798681° = 74°10'24″ = 1.2954569696 rad

Height: ha = 21.16660104885
Height: hb = 14.43113707876
Height: hc = 13.80439198838

Median: ma = 21.21990951739
Median: mb = 16
Median: mc = 14.90880515159

Inradius: r = 5.29215026221
Circumradius: R = 11.95331264589

Vertex coordinates: A[23; 0] B[0; 0] C[5.87695652174; 13.80439198838]
Centroid: CG[9.62331884058; 4.60113066279]
Coordinates of the circumscribed circle: U[11.5; 3.26599435797]
Coordinates of the inscribed circle: I[8; 5.29215026221]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.1387695762° = 141°8'16″ = 0.67882751639 rad
∠ B' = β' = 113.0365684106° = 113°2'8″ = 1.16987477937 rad
∠ C' = γ' = 105.8276620132° = 105°49'36″ = 1.2954569696 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 = 23 ; ;

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

p = a+b+c = 15+22+23 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-15)(30-22)(30-23) } ; ; T = sqrt{ 25200 } = 158.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 158.75 }{ 15 } = 21.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 158.75 }{ 22 } = 14.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 158.75 }{ 23 } = 13.8 ; ;

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**2-23**2 }{ 2 * 22 * 23 } ) = 38° 51'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 66° 57'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-15**2-22**2 }{ 2 * 22 * 15 } ) = 74° 10'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 158.75 }{ 30 } = 5.29 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 38° 51'44" } = 11.95 ; ;




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