12 27 27 triangle

Acute isosceles triangle.

Sides: a = 12   b = 27   c = 27

Area: T = 157.9499358973
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 25.67991768138° = 25°40'45″ = 0.44881861846 rad
Angle ∠ B = β = 77.16604115931° = 77°9'37″ = 1.34767032345 rad
Angle ∠ C = γ = 77.16604115931° = 77°9'37″ = 1.34767032345 rad

Height: ha = 26.32548931622
Height: hb = 11.76999525165
Height: hc = 11.76999525165

Median: ma = 26.32548931622
Median: mb = 15.94552187191
Median: mc = 15.94552187191

Inradius: r = 4.78663442113
Circumradius: R = 13.84662100398

Vertex coordinates: A[27; 0] B[0; 0] C[2.66766666667; 11.76999525165]
Centroid: CG[9.88988888889; 3.98999841722]
Coordinates of the circumscribed circle: U[13.5; 3.07769355644]
Coordinates of the inscribed circle: I[6; 4.78663442113]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.3210823186° = 154°19'15″ = 0.44881861846 rad
∠ B' = β' = 102.8439588407° = 102°50'23″ = 1.34767032345 rad
∠ C' = γ' = 102.8439588407° = 102°50'23″ = 1.34767032345 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 = 27 ; ; c = 27 ; ;

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

p = a+b+c = 12+27+27 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-12)(33-27)(33-27) } ; ; T = sqrt{ 24948 } = 157.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 157.95 }{ 12 } = 26.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 157.95 }{ 27 } = 11.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 157.95 }{ 27 } = 11.7 ; ;

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**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 25° 40'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 77° 9'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-27**2 }{ 2 * 27 * 12 } ) = 77° 9'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 157.95 }{ 33 } = 4.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 25° 40'45" } = 13.85 ; ;




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