Triangle calculator SSS - result

Please enter the triangle sides:


Acute scalene triangle.

Sides: a = 9   b = 11   c = 12

Area: T = 47.32986382648
Perimeter: p = 32
Semiperimeter: s = 16

Angle ∠ A = α = 45.81656148467° = 45°48'56″ = 0.87996333279 rad
Angle ∠ B = β = 61.21877953194° = 61°13'4″ = 1.06884520891 rad
Angle ∠ C = γ = 72.96765898339° = 72°58' = 1.27435072366 rad

Height: ha = 10.517747517
Height: hb = 8.60552069572
Height: hc = 7.88881063775

Median: ma = 10.59548100502
Median: mb = 9.06991785736
Median: mc = 8.06222577483

Inradius: r = 2.95880398915
Circumradius: R = 6.27552703414

Vertex coordinates: A[12; 0] B[0; 0] C[4.33333333333; 7.88881063775]
Centroid: CG[5.44444444444; 2.62993687925]
Coordinates of the circumscribed circle: U[6; 1.8388210504]
Coordinates of the inscribed circle: I[5; 2.95880398915]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.1844385153° = 134°11'4″ = 0.87996333279 rad
∠ B' = β' = 118.7822204681° = 118°46'56″ = 1.06884520891 rad
∠ C' = γ' = 107.0333410166° = 107°2' = 1.27435072366 rad

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How did we calculate this triangle?

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

p = a+b+c = 9+11+12 = 32 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32 }{ 2 } = 16 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16 * (16-9)(16-11)(16-12) } ; ; T = sqrt{ 2240 } = 47.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 47.33 }{ 9 } = 10.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 47.33 }{ 11 } = 8.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 47.33 }{ 12 } = 7.89 ; ;

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{ 11**2+12**2-9**2 }{ 2 * 11 * 12 } ) = 45° 48'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+12**2-11**2 }{ 2 * 9 * 12 } ) = 61° 13'4" ; ; gamma = 180° - alpha - beta = 180° - 45° 48'56" - 61° 13'4" = 72° 58' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 47.33 }{ 16 } = 2.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9 }{ 2 * sin 45° 48'56" } = 6.28 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11**2+2 * 12**2 - 9**2 } }{ 2 } = 10.595 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 9**2 - 11**2 } }{ 2 } = 9.069 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11**2+2 * 9**2 - 12**2 } }{ 2 } = 8.062 ; ;
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