8 17 21 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 17   c = 21

Area: T = 64.34328317686
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 21.12986972543° = 21°7'43″ = 0.36987653337 rad
Angle ∠ B = β = 49.99547991151° = 49°59'41″ = 0.87325738534 rad
Angle ∠ C = γ = 108.8776503631° = 108°52'35″ = 1.99002534664 rad

Height: ha = 16.08657079421
Height: hb = 7.5769744914
Height: hc = 6.12878887399

Median: ma = 18.68215416923
Median: mb = 13.42657215821
Median: mc = 8.1399410298

Inradius: r = 2.79875144247
Circumradius: R = 11.09768072181

Vertex coordinates: A[21; 0] B[0; 0] C[5.14328571429; 6.12878887399]
Centroid: CG[8.71442857143; 2.043262958]
Coordinates of the circumscribed circle: U[10.5; -3.59901435117]
Coordinates of the inscribed circle: I[6; 2.79875144247]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.8711302746° = 158°52'17″ = 0.36987653337 rad
∠ B' = β' = 130.0055200885° = 130°19″ = 0.87325738534 rad
∠ C' = γ' = 71.12334963694° = 71°7'25″ = 1.99002534664 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 = 8 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 8+17+21 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-8)(23-17)(23-21) } ; ; T = sqrt{ 4140 } = 64.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 64.34 }{ 8 } = 16.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 64.34 }{ 17 } = 7.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 64.34 }{ 21 } = 6.13 ; ;

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**2-8**2 }{ 2 * 17 * 21 } ) = 21° 7'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8**2+21**2-17**2 }{ 2 * 8 * 21 } ) = 49° 59'41" ; ; gamma = 180° - alpha - beta = 180° - 21° 7'43" - 49° 59'41" = 108° 52'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 64.34 }{ 23 } = 2.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8 }{ 2 * sin 21° 7'43" } = 11.1 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 21**2 - 8**2 } }{ 2 } = 18.682 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 8**2 - 17**2 } }{ 2 } = 13.426 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 8**2 - 21**2 } }{ 2 } = 8.139 ; ;
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