11 18 21 triangle

Acute scalene triangle.

Sides: a = 11   b = 18   c = 21

Area: T = 98.99549493661
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 31.58663380965° = 31°35'11″ = 0.55112855984 rad
Angle ∠ B = β = 58.99224169931° = 58°59'33″ = 1.03296119102 rad
Angle ∠ C = γ = 89.42112449103° = 89°25'16″ = 1.56106951449 rad

Height: ha = 17.99990817029
Height: hb = 10.99994388185
Height: hc = 9.42880904158

Median: ma = 18.76883243791
Median: mb = 14.14221356237
Median: mc = 10.59548100502

Inradius: r = 3.96597979746
Circumradius: R = 10.50105357006

Vertex coordinates: A[21; 0] B[0; 0] C[5.66766666667; 9.42880904158]
Centroid: CG[8.88988888889; 3.14326968053]
Coordinates of the circumscribed circle: U[10.5; 0.10660660172]
Coordinates of the inscribed circle: I[7; 3.96597979746]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.4143661903° = 148°24'49″ = 0.55112855984 rad
∠ B' = β' = 121.0087583007° = 121°27″ = 1.03296119102 rad
∠ C' = γ' = 90.57987550897° = 90°34'44″ = 1.56106951449 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 = 11 ; ; b = 18 ; ; c = 21 ; ;

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

p = a+b+c = 11+18+21 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-11)(25-18)(25-21) } ; ; T = sqrt{ 9800 } = 98.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.99 }{ 11 } = 18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.99 }{ 18 } = 11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.99 }{ 21 } = 9.43 ; ;

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{ 11**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 31° 35'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-11**2-21**2 }{ 2 * 11 * 21 } ) = 58° 59'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-11**2-18**2 }{ 2 * 18 * 11 } ) = 89° 25'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.99 }{ 25 } = 3.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 31° 35'11" } = 10.5 ; ;




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