11 16 21 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 16   c = 21

Area: T = 86.53332306111
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 31.00327191339° = 31°10″ = 0.5411099526 rad
Angle ∠ B = β = 48.52215991697° = 48°31'18″ = 0.84768616638 rad
Angle ∠ C = γ = 100.4765681696° = 100°28'32″ = 1.75436314638 rad

Height: ha = 15.73333146566
Height: hb = 10.81766538264
Height: hc = 8.24112600582

Median: ma = 17.84395627749
Median: mb = 14.73109198627
Median: mc = 8.84659030065

Inradius: r = 3.60655512755
Circumradius: R = 10.67879787773

Vertex coordinates: A[21; 0] B[0; 0] C[7.28657142857; 8.24112600582]
Centroid: CG[9.42985714286; 2.74770866861]
Coordinates of the circumscribed circle: U[10.5; -1.94114506868]
Coordinates of the inscribed circle: I[8; 3.60655512755]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.9977280866° = 148°59'50″ = 0.5411099526 rad
∠ B' = β' = 131.478840083° = 131°28'42″ = 0.84768616638 rad
∠ C' = γ' = 79.52443183036° = 79°31'28″ = 1.75436314638 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 = 16 ; ; c = 21 ; ;

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

p = a+b+c = 11+16+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 * (24-11)(24-16)(24-21) } ; ; T = sqrt{ 7488 } = 86.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.53 }{ 11 } = 15.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.53 }{ 16 } = 10.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.53 }{ 21 } = 8.24 ; ;

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{ 16**2+21**2-11**2 }{ 2 * 16 * 21 } ) = 31° 10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11**2+21**2-16**2 }{ 2 * 11 * 21 } ) = 48° 31'18" ; ; gamma = 180° - alpha - beta = 180° - 31° 10" - 48° 31'18" = 100° 28'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.53 }{ 24 } = 3.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 11 }{ 2 * sin 31° 10" } = 10.68 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 21**2 - 11**2 } }{ 2 } = 17.84 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 11**2 - 16**2 } }{ 2 } = 14.731 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 11**2 - 21**2 } }{ 2 } = 8.846 ; ;
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