12 14 23 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 14   c = 23

Area: T = 69.45109719154
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 25.55546937848° = 25°33'17″ = 0.44660135459 rad
Angle ∠ B = β = 30.21664398874° = 30°12'59″ = 0.52773763643 rad
Angle ∠ C = γ = 124.2298866328° = 124°13'44″ = 2.16882027434 rad

Height: ha = 11.57551619859
Height: hb = 9.92215674165
Height: hc = 6.03992149492

Median: ma = 18.06993109996
Median: mb = 16.95658249578
Median: mc = 6.14441028637

Inradius: r = 2.83547335476
Circumradius: R = 13.90990926067

Vertex coordinates: A[23; 0] B[0; 0] C[10.37695652174; 6.03992149492]
Centroid: CG[11.12331884058; 2.01330716497]
Coordinates of the circumscribed circle: U[11.5; -7.82438645913]
Coordinates of the inscribed circle: I[10.5; 2.83547335476]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.4455306215° = 154°26'43″ = 0.44660135459 rad
∠ B' = β' = 149.7843560113° = 149°47'1″ = 0.52773763643 rad
∠ C' = γ' = 55.77111336722° = 55°46'16″ = 2.16882027434 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 = 14 ; ; c = 23 ; ;

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

p = a+b+c = 12+14+23 = 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)(24.5-14)(24.5-23) } ; ; T = sqrt{ 4823.44 } = 69.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.45 }{ 12 } = 11.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.45 }{ 14 } = 9.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.45 }{ 23 } = 6.04 ; ;

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**2-14**2-23**2 }{ 2 * 14 * 23 } ) = 25° 33'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 30° 12'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-14**2 }{ 2 * 14 * 12 } ) = 124° 13'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.45 }{ 24.5 } = 2.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 25° 33'17" } = 13.91 ; ;




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