12 15 23 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 15   c = 23

Area: T = 80.6232577483
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 27.86441830985° = 27°51'51″ = 0.48663217384 rad
Angle ∠ B = β = 35.74880169402° = 35°44'53″ = 0.62439205967 rad
Angle ∠ C = γ = 116.3887799961° = 116°23'16″ = 2.03113503185 rad

Height: ha = 13.43770962472
Height: hb = 10.75496769977
Height: hc = 7.01106589116

Median: ma = 18.46661853126
Median: mb = 16.74106690428
Median: mc = 7.22884161474

Inradius: r = 3.22549030993
Circumradius: R = 12.838759503

Vertex coordinates: A[23; 0] B[0; 0] C[9.73991304348; 7.01106589116]
Centroid: CG[10.91330434783; 2.33768863039]
Coordinates of the circumscribed circle: U[11.5; -5.70655977911]
Coordinates of the inscribed circle: I[10; 3.22549030993]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.1365816901° = 152°8'9″ = 0.48663217384 rad
∠ B' = β' = 144.252198306° = 144°15'7″ = 0.62439205967 rad
∠ C' = γ' = 63.61222000388° = 63°36'44″ = 2.03113503185 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 = 15 ; ; c = 23 ; ;

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

p = a+b+c = 12+15+23 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-12)(25-15)(25-23) } ; ; T = sqrt{ 6500 } = 80.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.62 }{ 12 } = 13.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.62 }{ 15 } = 10.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.62 }{ 23 } = 7.01 ; ;

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**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 27° 51'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 35° 44'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 116° 23'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.62 }{ 25 } = 3.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 27° 51'51" } = 12.84 ; ;




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