12 21 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 21   c = 27

Area: T = 120.7487670785
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 25.20987652968° = 25°12'32″ = 0.44399759548 rad
Angle ∠ B = β = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ C = γ = 106.6021549599° = 106°36'6″ = 1.86105480282 rad

Height: ha = 20.12546117975
Height: hb = 11.549977817
Height: hc = 8.944427191

Median: ma = 23.43107490277
Median: mb = 18.06223918682
Median: mc = 10.5

Inradius: r = 4.02549223595
Circumradius: R = 14.08772282582

Vertex coordinates: A[27; 0] B[0; 0] C[8; 8.944427191]
Centroid: CG[11.66766666667; 2.981142397]
Coordinates of the circumscribed circle: U[13.5; -4.02549223595]
Coordinates of the inscribed circle: I[9; 4.02549223595]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.7911234703° = 154°47'28″ = 0.44399759548 rad
∠ B' = β' = 131.8110314896° = 131°48'37″ = 0.84110686706 rad
∠ C' = γ' = 73.3988450401° = 73°23'54″ = 1.86105480282 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 = 12 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 12+21+27 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-12)(30-21)(30-27) } ; ; T = sqrt{ 14580 } = 120.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 120.75 }{ 12 } = 20.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 120.75 }{ 21 } = 11.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 120.75 }{ 27 } = 8.94 ; ;

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{ 21**2+27**2-12**2 }{ 2 * 21 * 27 } ) = 25° 12'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+27**2-21**2 }{ 2 * 12 * 27 } ) = 48° 11'23" ; ; gamma = 180° - alpha - beta = 180° - 25° 12'32" - 48° 11'23" = 106° 36'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 120.75 }{ 30 } = 4.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 25° 12'32" } = 14.09 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 27**2 - 12**2 } }{ 2 } = 23.431 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 12**2 - 21**2 } }{ 2 } = 18.062 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 12**2 - 27**2 } }{ 2 } = 10.5 ; ;
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