12 13 24 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 13   c = 24

Area: T = 41.9643525829
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 15.60545888784° = 15°36'17″ = 0.27223514543 rad
Angle ∠ B = β = 16.94325916094° = 16°56'33″ = 0.29657040074 rad
Angle ∠ C = γ = 147.4532819512° = 147°27'10″ = 2.57435371918 rad

Height: ha = 6.99439209715
Height: hb = 6.45659270506
Height: hc = 3.49769604857

Median: ma = 18.34439363278
Median: mb = 17.826554347
Median: mc = 3.53655339059

Inradius: r = 1.71327969726
Circumradius: R = 22.30550847494

Vertex coordinates: A[24; 0] B[0; 0] C[11.47991666667; 3.49769604857]
Centroid: CG[11.82663888889; 1.16656534952]
Coordinates of the circumscribed circle: U[12; -18.80220425933]
Coordinates of the inscribed circle: I[11.5; 1.71327969726]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.3955411122° = 164°23'43″ = 0.27223514543 rad
∠ B' = β' = 163.0577408391° = 163°3'27″ = 0.29657040074 rad
∠ C' = γ' = 32.54771804879° = 32°32'50″ = 2.57435371918 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 = 13 ; ; c = 24 ; ;

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

p = a+b+c = 12+13+24 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-12)(24.5-13)(24.5-24) } ; ; T = sqrt{ 1760.94 } = 41.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.96 }{ 12 } = 6.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.96 }{ 13 } = 6.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.96 }{ 24 } = 3.5 ; ;

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**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 15° 36'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 16° 56'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 147° 27'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.96 }{ 24.5 } = 1.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 15° 36'17" } = 22.31 ; ;




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