18 18 21 triangle

Acute isosceles triangle.

Sides: a = 18   b = 18   c = 21

Area: T = 153.5122010931
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 54.31546652873° = 54°18'53″ = 0.94879697414 rad
Angle ∠ B = β = 54.31546652873° = 54°18'53″ = 0.94879697414 rad
Angle ∠ C = γ = 71.37106694253° = 71°22'14″ = 1.24656531708 rad

Height: ha = 17.05768901034
Height: hb = 17.05768901034
Height: hc = 14.62201915172

Median: ma = 17.36437553542
Median: mb = 17.36437553542
Median: mc = 14.62201915172

Inradius: r = 5.38663863484
Circumradius: R = 11.08105662025

Vertex coordinates: A[21; 0] B[0; 0] C[10.5; 14.62201915172]
Centroid: CG[10.5; 4.87333971724]
Coordinates of the circumscribed circle: U[10.5; 3.54396253147]
Coordinates of the inscribed circle: I[10.5; 5.38663863484]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.6855334713° = 125°41'7″ = 0.94879697414 rad
∠ B' = β' = 125.6855334713° = 125°41'7″ = 0.94879697414 rad
∠ C' = γ' = 108.6299330575° = 108°37'46″ = 1.24656531708 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 = 18 ; ; b = 18 ; ; c = 21 ; ;

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

p = a+b+c = 18+18+21 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-18)(28.5-18)(28.5-21) } ; ; T = sqrt{ 23565.94 } = 153.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 153.51 }{ 18 } = 17.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 153.51 }{ 18 } = 17.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 153.51 }{ 21 } = 14.62 ; ;

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{ 18**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 54° 18'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 54° 18'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-18**2-18**2 }{ 2 * 18 * 18 } ) = 71° 22'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 153.51 }{ 28.5 } = 5.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 54° 18'53" } = 11.08 ; ;




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