12 23 24 triangle

Acute scalene triangle.

Sides: a = 12   b = 23   c = 24

Area: T = 135.8532631554
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 29.48765793699° = 29°29'12″ = 0.51546378952 rad
Angle ∠ B = β = 70.63442509213° = 70°38'3″ = 1.23328002433 rad
Angle ∠ C = γ = 79.87991697087° = 79°52'45″ = 1.39441545152 rad

Height: ha = 22.64221052589
Height: hb = 11.8133272309
Height: hc = 11.32110526295

Median: ma = 22.72766363547
Median: mb = 15.09113882728
Median: mc = 13.87444369255

Inradius: r = 4.6055173951
Circumradius: R = 12.19896792213

Vertex coordinates: A[24; 0] B[0; 0] C[3.97991666667; 11.32110526295]
Centroid: CG[9.32663888889; 3.77436842098]
Coordinates of the circumscribed circle: U[12; 2.14220269646]
Coordinates of the inscribed circle: I[6.5; 4.6055173951]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.513342063° = 150°30'48″ = 0.51546378952 rad
∠ B' = β' = 109.3665749079° = 109°21'57″ = 1.23328002433 rad
∠ C' = γ' = 100.1210830291° = 100°7'15″ = 1.39441545152 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 = 12 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 12+23+24 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-12)(29.5-23)(29.5-24) } ; ; T = sqrt{ 18455.94 } = 135.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 135.85 }{ 12 } = 22.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 135.85 }{ 23 } = 11.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 135.85 }{ 24 } = 11.32 ; ;

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{ 12**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 29° 29'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 70° 38'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-23**2 }{ 2 * 23 * 12 } ) = 79° 52'45" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 135.85 }{ 29.5 } = 4.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 29° 29'12" } = 12.19 ; ;




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