15 27 27 triangle

Acute isosceles triangle.

Sides: a = 15   b = 27   c = 27

Area: T = 194.531068524
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 32.25552404263° = 32°15'19″ = 0.56329601465 rad
Angle ∠ B = β = 73.87223797868° = 73°52'21″ = 1.28993162536 rad
Angle ∠ C = γ = 73.87223797868° = 73°52'21″ = 1.28993162536 rad

Height: ha = 25.93774246987
Height: hb = 14.41096803882
Height: hc = 14.41096803882

Median: ma = 25.93774246987
Median: mb = 17.16882847134
Median: mc = 17.16882847134

Inradius: r = 5.63985705867
Circumradius: R = 14.05330528468

Vertex coordinates: A[27; 0] B[0; 0] C[4.16766666667; 14.41096803882]
Centroid: CG[10.38988888889; 4.80332267961]
Coordinates of the circumscribed circle: U[13.5; 3.90436257908]
Coordinates of the inscribed circle: I[7.5; 5.63985705867]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.7454759574° = 147°44'41″ = 0.56329601465 rad
∠ B' = β' = 106.1287620213° = 106°7'39″ = 1.28993162536 rad
∠ C' = γ' = 106.1287620213° = 106°7'39″ = 1.28993162536 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 = 15 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 15+27+27 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-15)(34.5-27)(34.5-27) } ; ; T = sqrt{ 37842.19 } = 194.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 194.53 }{ 15 } = 25.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 194.53 }{ 27 } = 14.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 194.53 }{ 27 } = 14.41 ; ;

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{ 15**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 32° 15'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-15**2-27**2 }{ 2 * 15 * 27 } ) = 73° 52'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-15**2-27**2 }{ 2 * 27 * 15 } ) = 73° 52'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 194.53 }{ 34.5 } = 5.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 32° 15'19" } = 14.05 ; ;




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