18 21 21 triangle

Acute isosceles triangle.

Sides: a = 18   b = 21   c = 21

Area: T = 170.7632993649
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 50.75438670503° = 50°45'14″ = 0.88658220881 rad
Angle ∠ B = β = 64.62330664748° = 64°37'23″ = 1.12878852827 rad
Angle ∠ C = γ = 64.62330664748° = 64°37'23″ = 1.12878852827 rad

Height: ha = 18.9743665961
Height: hb = 16.26331422523
Height: hc = 16.26331422523

Median: ma = 18.9743665961
Median: mb = 16.5
Median: mc = 16.5

Inradius: r = 5.69220997883
Circumradius: R = 11.62113704011

Vertex coordinates: A[21; 0] B[0; 0] C[7.71442857143; 16.26331422523]
Centroid: CG[9.57114285714; 5.42110474174]
Coordinates of the circumscribed circle: U[10.5; 4.98105873148]
Coordinates of the inscribed circle: I[9; 5.69220997883]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.246613295° = 129°14'46″ = 0.88658220881 rad
∠ B' = β' = 115.3776933525° = 115°22'37″ = 1.12878852827 rad
∠ C' = γ' = 115.3776933525° = 115°22'37″ = 1.12878852827 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 = 21 ; ; c = 21 ; ;

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

p = a+b+c = 18+21+21 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-18)(30-21)(30-21) } ; ; T = sqrt{ 29160 } = 170.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 170.76 }{ 18 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 170.76 }{ 21 } = 16.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 170.76 }{ 21 } = 16.26 ; ;

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**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 50° 45'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 64° 37'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 64° 37'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 170.76 }{ 30 } = 5.69 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 50° 45'14" } = 11.62 ; ;




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