Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute scalene triangle.

Sides: a = 1.1   b = 1.6   c = 1.74330292161

Area: T = 0.86107698886
Perimeter: p = 4.44330292161
Semiperimeter: s = 2.22215146081

Angle ∠ A = α = 38.11988331036° = 38°7'8″ = 0.66552991447 rad
Angle ∠ B = β = 63.88111668964° = 63°52'52″ = 1.11549366924 rad
Angle ∠ C = γ = 78° = 1.36113568166 rad

Height: ha = 1.56550361612
Height: hb = 1.07659623608
Height: hc = 0.98876712113

Median: ma = 1.5880055513
Median: mb = 1.21882263436
Median: mc = 1.0610878074

Inradius: r = 0.38774698305
Circumradius: R = 0.89109847628

Vertex coordinates: A[1.74330292161; 0] B[0; 0] C[0.48442577602; 0.98876712113]
Centroid: CG[0.74224289921; 0.32992237371]
Coordinates of the circumscribed circle: U[0.87215146081; 0.18552461485]
Coordinates of the inscribed circle: I[0.62215146081; 0.38774698305]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.8811166896° = 141°52'52″ = 0.66552991447 rad
∠ B' = β' = 116.1198833104° = 116°7'8″ = 1.11549366924 rad
∠ C' = γ' = 102° = 1.36113568166 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 1.1 ; ; b = 1.6 ; ; gamma = 78° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 1.1**2+1.6**2 - 2 * 1.1 * 1.6 * cos(78° ) } ; ; c = 1.74 ; ;


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 = 1.1 ; ; b = 1.6 ; ; c = 1.74 ; ;

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

p = a+b+c = 1.1+1.6+1.74 = 4.44 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.44 }{ 2 } = 2.22 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.22 * (2.22-1.1)(2.22-1.6)(2.22-1.74) } ; ; T = sqrt{ 0.74 } = 0.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.86 }{ 1.1 } = 1.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.86 }{ 1.6 } = 1.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.86 }{ 1.74 } = 0.99 ; ;

6. 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{ 1.6**2+1.74**2-1.1**2 }{ 2 * 1.6 * 1.74 } ) = 38° 7'8" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.1**2+1.74**2-1.6**2 }{ 2 * 1.1 * 1.74 } ) = 63° 52'52" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 1.1**2+1.6**2-1.74**2 }{ 2 * 1.1 * 1.6 } ) = 78° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.86 }{ 2.22 } = 0.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.1 }{ 2 * sin 38° 7'8" } = 0.89 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.6**2+2 * 1.74**2 - 1.1**2 } }{ 2 } = 1.58 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.74**2+2 * 1.1**2 - 1.6**2 } }{ 2 } = 1.218 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.6**2+2 * 1.1**2 - 1.74**2 } }{ 2 } = 1.061 ; ;
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