21 27 27 triangle

Acute isosceles triangle.

Sides: a = 21   b = 27   c = 27

Area: T = 261.184420224
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 45.77107609523° = 45°46'15″ = 0.79988504798 rad
Angle ∠ B = β = 67.11546195238° = 67°6'53″ = 1.17113710869 rad
Angle ∠ C = γ = 67.11546195238° = 67°6'53″ = 1.17113710869 rad

Height: ha = 24.87546859277
Height: hb = 19.34769779437
Height: hc = 19.34769779437

Median: ma = 24.87546859277
Median: mb = 20.06986322404
Median: mc = 20.06986322404

Inradius: r = 6.96549120597
Circumradius: R = 14.65334513465

Vertex coordinates: A[27; 0] B[0; 0] C[8.16766666667; 19.34769779437]
Centroid: CG[11.72222222222; 6.44989926479]
Coordinates of the circumscribed circle: U[13.5; 5.69985644125]
Coordinates of the inscribed circle: I[10.5; 6.96549120597]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.2299239048° = 134°13'45″ = 0.79988504798 rad
∠ B' = β' = 112.8855380476° = 112°53'7″ = 1.17113710869 rad
∠ C' = γ' = 112.8855380476° = 112°53'7″ = 1.17113710869 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 = 21 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 21+27+27 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-21)(37.5-27)(37.5-27) } ; ; T = sqrt{ 68217.19 } = 261.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 261.18 }{ 21 } = 24.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 261.18 }{ 27 } = 19.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 261.18 }{ 27 } = 19.35 ; ;

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{ 21**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 45° 46'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 67° 6'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-21**2-27**2 }{ 2 * 27 * 21 } ) = 67° 6'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 261.18 }{ 37.5 } = 6.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 45° 46'15" } = 14.65 ; ;




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