13 20 24 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 20   c = 24

Area: T = 129.988822062
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 32.79438229603° = 32°47'38″ = 0.5722360185 rad
Angle ∠ B = β = 56.43548644036° = 56°26'5″ = 0.98549741968 rad
Angle ∠ C = γ = 90.77113126361° = 90°46'17″ = 1.58442582719 rad

Height: ha = 19.99881877877
Height: hb = 12.9998822062
Height: hc = 10.83223517183

Median: ma = 21.11327923307
Median: mb = 16.50875740192
Median: mc = 11.85332695911

Inradius: r = 4.56109901972
Circumradius: R = 12.00110874259

Vertex coordinates: A[24; 0] B[0; 0] C[7.18875; 10.83223517183]
Centroid: CG[10.39658333333; 3.61107839061]
Coordinates of the circumscribed circle: U[12; -0.16215531]
Coordinates of the inscribed circle: I[8.5; 4.56109901972]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.206617704° = 147°12'22″ = 0.5722360185 rad
∠ B' = β' = 123.5655135596° = 123°33'55″ = 0.98549741968 rad
∠ C' = γ' = 89.22986873639° = 89°13'43″ = 1.58442582719 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 = 13 ; ; b = 20 ; ; c = 24 ; ;

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

p = a+b+c = 13+20+24 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-13)(28.5-20)(28.5-24) } ; ; T = sqrt{ 16896.94 } = 129.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.99 }{ 13 } = 20 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.99 }{ 20 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.99 }{ 24 } = 10.83 ; ;

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{ 13**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 32° 47'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 56° 26'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-13**2-20**2 }{ 2 * 20 * 13 } ) = 90° 46'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.99 }{ 28.5 } = 4.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 32° 47'38" } = 12 ; ;




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