Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 1.687   b = 1.253   c = 2.10114228513

Area: T = 1.05769055
Perimeter: p = 5.04114228513
Semiperimeter: s = 2.52107114257

Angle ∠ A = α = 53.39773025867° = 53°23'50″ = 0.93219587418 rad
Angle ∠ B = β = 36.60326974133° = 36°36'10″ = 0.6398837585 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.253
Height: hb = 1.687
Height: hc = 1.00658951242

Median: ma = 1.51104639188
Median: mb = 1.87995752971
Median: mc = 1.05107114257

Inradius: r = 0.41992885743
Circumradius: R = 1.05107114257

Vertex coordinates: A[2.10114228513; 0] B[0; 0] C[1.35443057259; 1.00658951242]
Centroid: CG[1.15219095257; 0.33552983747]
Coordinates of the circumscribed circle: U[1.05107114257; -0]
Coordinates of the inscribed circle: I[1.26877114257; 0.41992885743]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.6032697413° = 126°36'10″ = 0.93219587418 rad
∠ B' = β' = 143.3977302587° = 143°23'50″ = 0.6398837585 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 1.69 ; ; b = 1.25 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 1.69**2+1.25**2 - 2 * 1.69 * 1.25 * cos(90° ) } ; ; 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 = 1.69 ; ; b = 1.25 ; ; c = 2.1 ; ;

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

p = a+b+c = 1.69+1.25+2.1 = 5.04 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.04 }{ 2 } = 2.52 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.52 * (2.52-1.69)(2.52-1.25)(2.52-2.1) } ; ; T = sqrt{ 1.12 } = 1.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.06 }{ 1.69 } = 1.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.06 }{ 1.25 } = 1.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.06 }{ 2.1 } = 1.01 ; ;

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{ 1.69**2-1.25**2-2.1**2 }{ 2 * 1.25 * 2.1 } ) = 53° 23'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.25**2-1.69**2-2.1**2 }{ 2 * 1.69 * 2.1 } ) = 36° 36'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.1**2-1.69**2-1.25**2 }{ 2 * 1.25 * 1.69 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.06 }{ 2.52 } = 0.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.69 }{ 2 * sin 53° 23'50" } = 1.05 ; ;




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