12 17 24 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 17   c = 24

Area: T = 95.5329772846
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 27.92329129398° = 27°55'22″ = 0.48773467675 rad
Angle ∠ B = β = 41.56597864393° = 41°33'35″ = 0.72553551098 rad
Angle ∠ C = γ = 110.5177300621° = 110°31'2″ = 1.92988907763 rad

Height: ha = 15.92216288077
Height: hb = 11.23987968054
Height: hc = 7.96108144038

Median: ma = 19.91223077517
Median: mb = 16.96331954537
Median: mc = 8.5154693183

Inradius: r = 3.60548970885
Circumradius: R = 12.81327594522

Vertex coordinates: A[24; 0] B[0; 0] C[8.97991666667; 7.96108144038]
Centroid: CG[10.99330555556; 2.65436048013]
Coordinates of the circumscribed circle: U[12; -4.49107465727]
Coordinates of the inscribed circle: I[9.5; 3.60548970885]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.077708706° = 152°4'38″ = 0.48773467675 rad
∠ B' = β' = 138.4440213561° = 138°26'25″ = 0.72553551098 rad
∠ C' = γ' = 69.48326993791° = 69°28'58″ = 1.92988907763 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 = 17 ; ; c = 24 ; ;

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

p = a+b+c = 12+17+24 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-12)(26.5-17)(26.5-24) } ; ; T = sqrt{ 9125.94 } = 95.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 95.53 }{ 12 } = 15.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 95.53 }{ 17 } = 11.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 95.53 }{ 24 } = 7.96 ; ;

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**2-17**2-24**2 }{ 2 * 17 * 24 } ) = 27° 55'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 41° 33'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-17**2 }{ 2 * 17 * 12 } ) = 110° 31'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 95.53 }{ 26.5 } = 3.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 27° 55'22" } = 12.81 ; ;




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