10 17 21 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 17   c = 21

Area: T = 84
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 28.07224869359° = 28°4'21″ = 0.49899573263 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 98.797741071° = 98°47'51″ = 1.72443401093 rad

Height: ha = 16.8
Height: hb = 9.88223529412
Height: hc = 8

Median: ma = 18.43990889146
Median: mb = 14.08801278403
Median: mc = 9.17987798753

Inradius: r = 3.5
Circumradius: R = 10.625

Vertex coordinates: A[21; 0] B[0; 0] C[6; 8]
Centroid: CG[9; 2.66766666667]
Coordinates of the circumscribed circle: U[10.5; -1.625]
Coordinates of the inscribed circle: I[7; 3.5]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.9287513064° = 151°55'39″ = 0.49899573263 rad
∠ B' = β' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ C' = γ' = 81.203258929° = 81°12'9″ = 1.72443401093 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 = 10 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 10+17+21 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-10)(24-17)(24-21) } ; ; T = sqrt{ 7056 } = 84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 84 }{ 10 } = 16.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 84 }{ 17 } = 9.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 84 }{ 21 } = 8 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 17**2+21**2-10**2 }{ 2 * 17 * 21 } ) = 28° 4'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+21**2-17**2 }{ 2 * 10 * 21 } ) = 53° 7'48" ; ; gamma = 180° - alpha - beta = 180° - 28° 4'21" - 53° 7'48" = 98° 47'51" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 84 }{ 24 } = 3.5 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 28° 4'21" } = 10.63 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 21**2 - 10**2 } }{ 2 } = 18.439 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 10**2 - 17**2 } }{ 2 } = 14.08 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 10**2 - 21**2 } }{ 2 } = 9.179 ; ;
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