12 12 23 triangle

Obtuse isosceles triangle.

Sides: a = 12   b = 12   c = 23

Area: T = 39.42200139523
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 16.59878421359° = 16°35'52″ = 0.2989686994 rad
Angle ∠ B = β = 16.59878421359° = 16°35'52″ = 0.2989686994 rad
Angle ∠ C = γ = 146.8044315728° = 146°48'16″ = 2.56222186656 rad

Height: ha = 6.57700023254
Height: hb = 6.57700023254
Height: hc = 3.42878273002

Median: ma = 17.33549358234
Median: mb = 17.33549358234
Median: mc = 3.42878273002

Inradius: r = 1.67774474022
Circumradius: R = 21.00545587757

Vertex coordinates: A[23; 0] B[0; 0] C[11.5; 3.42878273002]
Centroid: CG[11.5; 1.14326091001]
Coordinates of the circumscribed circle: U[11.5; -17.57767314755]
Coordinates of the inscribed circle: I[11.5; 1.67774474022]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.4022157864° = 163°24'8″ = 0.2989686994 rad
∠ B' = β' = 163.4022157864° = 163°24'8″ = 0.2989686994 rad
∠ C' = γ' = 33.19656842717° = 33°11'44″ = 2.56222186656 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 = 12 ; ; c = 23 ; ;

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

p = a+b+c = 12+12+23 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-12)(23.5-12)(23.5-23) } ; ; T = sqrt{ 1553.94 } = 39.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 39.42 }{ 12 } = 6.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 39.42 }{ 12 } = 6.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 39.42 }{ 23 } = 3.43 ; ;

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**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 16° 35'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 16° 35'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 146° 48'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 39.42 }{ 23.5 } = 1.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 16° 35'52" } = 21 ; ;




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