11 12 21 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 12   c = 21

Area: T = 49.1933495505
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 22.98109198075° = 22°58'51″ = 0.40110927158 rad
Angle ∠ B = β = 25.20987652968° = 25°12'32″ = 0.44399759548 rad
Angle ∠ C = γ = 131.8110314896° = 131°48'37″ = 2.3010523983 rad

Height: ha = 8.944427191
Height: hb = 8.19989159175
Height: hc = 4.685509481

Median: ma = 16.19441347407
Median: mb = 15.65224758425
Median: mc = 4.7176990566

Inradius: r = 2.23660679775
Circumradius: R = 14.08772282582

Vertex coordinates: A[21; 0] B[0; 0] C[9.95223809524; 4.685509481]
Centroid: CG[10.31774603175; 1.562169827]
Coordinates of the circumscribed circle: U[10.5; -9.39114855055]
Coordinates of the inscribed circle: I[10; 2.23660679775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.0199080193° = 157°1'9″ = 0.40110927158 rad
∠ B' = β' = 154.7911234703° = 154°47'28″ = 0.44399759548 rad
∠ C' = γ' = 48.19896851042° = 48°11'23″ = 2.3010523983 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 = 12 ; ; c = 21 ; ;

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

p = a+b+c = 11+12+21 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-11)(22-12)(22-21) } ; ; T = sqrt{ 2420 } = 49.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.19 }{ 11 } = 8.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.19 }{ 12 } = 8.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.19 }{ 21 } = 4.69 ; ;

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{ 12**2+21**2-11**2 }{ 2 * 12 * 21 } ) = 22° 58'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11**2+21**2-12**2 }{ 2 * 11 * 21 } ) = 25° 12'32" ; ; gamma = 180° - alpha - beta = 180° - 22° 58'51" - 25° 12'32" = 131° 48'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.19 }{ 22 } = 2.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 11 }{ 2 * sin 22° 58'51" } = 14.09 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 21**2 - 11**2 } }{ 2 } = 16.194 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 11**2 - 12**2 } }{ 2 } = 15.652 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 11**2 - 21**2 } }{ 2 } = 4.717 ; ;
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