12 26 27 triangle

Acute scalene triangle.

Sides: a = 12   b = 26   c = 27

Area: T = 154.3322230918
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 26.08442927774° = 26°5'3″ = 0.4555256792 rad
Angle ∠ B = β = 72.30112458697° = 72°18'4″ = 1.26218947937 rad
Angle ∠ C = γ = 81.61444613529° = 81°36'52″ = 1.42444410679 rad

Height: ha = 25.72220384863
Height: hb = 11.87217100706
Height: hc = 11.4322017105

Median: ma = 25.81766612869
Median: mb = 16.35554272338
Median: mc = 15.09113882728

Inradius: r = 4.74986840282
Circumradius: R = 13.64658858106

Vertex coordinates: A[27; 0] B[0; 0] C[3.64881481481; 11.4322017105]
Centroid: CG[10.21660493827; 3.81106723683]
Coordinates of the circumscribed circle: U[13.5; 1.9990025014]
Coordinates of the inscribed circle: I[6.5; 4.74986840282]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.9165707223° = 153°54'57″ = 0.4555256792 rad
∠ B' = β' = 107.699875413° = 107°41'55″ = 1.26218947937 rad
∠ C' = γ' = 98.38655386471° = 98°23'8″ = 1.42444410679 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 = 26 ; ; c = 27 ; ;

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

p = a+b+c = 12+26+27 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-12)(32.5-26)(32.5-27) } ; ; T = sqrt{ 23818.44 } = 154.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 154.33 }{ 12 } = 25.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 154.33 }{ 26 } = 11.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 154.33 }{ 27 } = 11.43 ; ;

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**2-26**2-27**2 }{ 2 * 26 * 27 } ) = 26° 5'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 72° 18'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-26**2 }{ 2 * 26 * 12 } ) = 81° 36'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 154.33 }{ 32.5 } = 4.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 26° 5'3" } = 13.65 ; ;




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