9 15 21 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 15   c = 21

Area: T = 58.45767147554
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 21.78767892983° = 21°47'12″ = 0.38802512067 rad
Angle ∠ B = β = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 12.99903810568
Height: hb = 7.79442286341
Height: hc = 5.56773061672

Median: ma = 17.68547391838
Median: mb = 14.30990880213
Median: mc = 6.53883484153

Inradius: r = 2.59880762114
Circumradius: R = 12.1244355653

Vertex coordinates: A[21; 0] B[0; 0] C[7.07114285714; 5.56773061672]
Centroid: CG[9.35771428571; 1.85657687224]
Coordinates of the circumscribed circle: U[10.5; -6.06221778265]
Coordinates of the inscribed circle: I[7.5; 2.59880762114]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2133210702° = 158°12'48″ = 0.38802512067 rad
∠ B' = β' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ C' = γ' = 60° = 2.09443951024 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 = 9 ; ; b = 15 ; ; c = 21 ; ;

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

p = a+b+c = 9+15+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-9)(22.5-15)(22.5-21) } ; ; T = sqrt{ 3417.19 } = 58.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.46 }{ 9 } = 12.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.46 }{ 15 } = 7.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.46 }{ 21 } = 5.57 ; ;

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{ 9**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 21° 47'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-9**2-21**2 }{ 2 * 9 * 21 } ) = 38° 12'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-9**2-15**2 }{ 2 * 15 * 9 } ) = 120° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.46 }{ 22.5 } = 2.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 21° 47'12" } = 12.12 ; ;




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