Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 1.1   b = 1.3   c = 2.10113255568

Area: T = 0.60663543888
Perimeter: p = 4.50113255568
Semiperimeter: s = 2.25106627784

Angle ∠ A = α = 26.35552492346° = 26°21'19″ = 0.46599858743 rad
Angle ∠ B = β = 31.64547507654° = 31°38'41″ = 0.55223050918 rad
Angle ∠ C = γ = 122° = 2.12993016874 rad

Height: ha = 1.1022462525
Height: hb = 0.93328529058
Height: hc = 0.57771160845

Median: ma = 1.65883981874
Median: mb = 1.546605451
Median: mc = 0.58883092096

Inradius: r = 0.26994114794
Circumradius: R = 1.23989188575

Vertex coordinates: A[2.10113255568; 0] B[0; 0] C[0.93664491578; 0.57771160845]
Centroid: CG[1.01325915715; 0.19223720282]
Coordinates of the circumscribed circle: U[1.05106627784; -0.65765269694]
Coordinates of the inscribed circle: I[0.95106627784; 0.26994114794]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.6454750765° = 153°38'41″ = 0.46599858743 rad
∠ B' = β' = 148.3555249235° = 148°21'19″ = 0.55223050918 rad
∠ C' = γ' = 58° = 2.12993016874 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 = 1.3 ; ; gamma = 122° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 1.1**2+1.3**2 - 2 * 1.1 * 1.3 * cos(122° ) } ; ; c = 2.1 ; ;


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 = 1.3 ; ; c = 2.1 ; ;

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

p = a+b+c = 1.1+1.3+2.1 = 4.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.5 }{ 2 } = 2.25 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.25 * (2.25-1.1)(2.25-1.3)(2.25-2.1) } ; ; T = sqrt{ 0.37 } = 0.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.61 }{ 1.1 } = 1.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.61 }{ 1.3 } = 0.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.61 }{ 2.1 } = 0.58 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1.1**2-1.3**2-2.1**2 }{ 2 * 1.3 * 2.1 } ) = 26° 21'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.3**2-1.1**2-2.1**2 }{ 2 * 1.1 * 2.1 } ) = 31° 38'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.1**2-1.1**2-1.3**2 }{ 2 * 1.3 * 1.1 } ) = 122° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.61 }{ 2.25 } = 0.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.1 }{ 2 * sin 26° 21'19" } = 1.24 ; ;




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