15 23 24 triangle

Acute scalene triangle.

Sides: a = 15   b = 23   c = 24

Area: T = 166.6611333248
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 37.14658041715° = 37°8'45″ = 0.64883165861 rad
Angle ∠ B = β = 67.80438991432° = 67°48'14″ = 1.18334012857 rad
Angle ∠ C = γ = 75.05502966853° = 75°3'1″ = 1.31098747818 rad

Height: ha = 22.22215110997
Height: hb = 14.49222898477
Height: hc = 13.88884444373

Median: ma = 22.27766694099
Median: mb = 16.37883393542
Median: mc = 15.26443375225

Inradius: r = 5.37661720403
Circumradius: R = 12.42203974591

Vertex coordinates: A[24; 0] B[0; 0] C[5.66766666667; 13.88884444373]
Centroid: CG[9.88988888889; 4.62994814791]
Coordinates of the circumscribed circle: U[12; 3.20441025329]
Coordinates of the inscribed circle: I[8; 5.37661720403]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.8544195828° = 142°51'15″ = 0.64883165861 rad
∠ B' = β' = 112.1966100857° = 112°11'46″ = 1.18334012857 rad
∠ C' = γ' = 104.9549703315° = 104°56'59″ = 1.31098747818 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 = 23 ; ; c = 24 ; ;

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

p = a+b+c = 15+23+24 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-15)(31-23)(31-24) } ; ; T = sqrt{ 27776 } = 166.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 166.66 }{ 15 } = 22.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 166.66 }{ 23 } = 14.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 166.66 }{ 24 } = 13.89 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 23**2+24**2-15**2 }{ 2 * 23 * 24 } ) = 37° 8'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15**2+24**2-23**2 }{ 2 * 15 * 24 } ) = 67° 48'14" ; ; gamma = 180° - alpha - beta = 180° - 37° 8'45" - 67° 48'14" = 75° 3'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 166.66 }{ 31 } = 5.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15 }{ 2 * sin 37° 8'45" } = 12.42 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 24**2 - 15**2 } }{ 2 } = 22.277 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 15**2 - 23**2 } }{ 2 } = 16.378 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 15**2 - 24**2 } }{ 2 } = 15.264 ; ;
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