21 21 30 triangle

Obtuse isosceles triangle.

Sides: a = 21   b = 21   c = 30

Area: T = 220.4544076851
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ B = β = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ C = γ = 91.16993828056° = 91°10'10″ = 1.5911205907 rad

Height: ha = 20.99656263667
Height: hb = 20.99656263667
Height: hc = 14.69769384567

Median: ma = 23.67696007571
Median: mb = 23.67696007571
Median: mc = 14.69769384567

Inradius: r = 6.1243724357
Circumradius: R = 15.00331246745

Vertex coordinates: A[30; 0] B[0; 0] C[15; 14.69769384567]
Centroid: CG[15; 4.89989794856]
Coordinates of the circumscribed circle: U[15; -0.30661862178]
Coordinates of the inscribed circle: I[15; 6.1243724357]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ B' = β' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ C' = γ' = 88.83106171944° = 88°49'50″ = 1.5911205907 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 = 21 ; ; c = 30 ; ;

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

p = a+b+c = 21+21+30 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-21)(36-21)(36-30) } ; ; T = sqrt{ 48600 } = 220.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 220.45 }{ 21 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 220.45 }{ 21 } = 21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 220.45 }{ 30 } = 14.7 ; ;

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{ 21**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 44° 24'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 44° 24'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 91° 10'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 220.45 }{ 36 } = 6.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 44° 24'55" } = 15 ; ;




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