Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 8.2   b = 9.4   c = 4.98664722496

Area: T = 20.42330884435
Perimeter: p = 22.58664722496
Semiperimeter: s = 11.29332361248

Angle ∠ A = α = 60.62546907843° = 60°37'29″ = 1.05881004622 rad
Angle ∠ B = β = 87.37553092157° = 87°22'31″ = 1.52549868308 rad
Angle ∠ C = γ = 32° = 0.55985053606 rad

Height: ha = 4.98112410838
Height: hb = 4.34553379667
Height: hc = 8.19113976138

Median: ma = 6.30989185086
Median: mb = 4.89551458352
Median: mc = 8.46107194508

Inradius: r = 1.80884354403
Circumradius: R = 4.7054935814

Vertex coordinates: A[4.98664722496; 0] B[0; 0] C[0.37655065013; 8.19113976138]
Centroid: CG[1.78773262503; 2.73304658713]
Coordinates of the circumscribed circle: U[2.49332361248; 3.99900118596]
Coordinates of the inscribed circle: I[1.89332361248; 1.80884354403]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119.3755309216° = 119°22'31″ = 1.05881004622 rad
∠ B' = β' = 92.62546907843° = 92°37'29″ = 1.52549868308 rad
∠ C' = γ' = 148° = 0.55985053606 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 = 8.2 ; ; b = 9.4 ; ; gamma = 32° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 8.2**2+9.4**2 - 2 * 8.2 * 9.4 * cos(32° ) } ; ; c = 4.99 ; ;


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 = 8.2 ; ; b = 9.4 ; ; c = 4.99 ; ;

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

p = a+b+c = 8.2+9.4+4.99 = 22.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.59 }{ 2 } = 11.29 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.29 * (11.29-8.2)(11.29-9.4)(11.29-4.99) } ; ; T = sqrt{ 417.1 } = 20.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.42 }{ 8.2 } = 4.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.42 }{ 9.4 } = 4.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.42 }{ 4.99 } = 8.19 ; ;

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{ 8.2**2-9.4**2-4.99**2 }{ 2 * 9.4 * 4.99 } ) = 60° 37'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.4**2-8.2**2-4.99**2 }{ 2 * 8.2 * 4.99 } ) = 87° 22'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.99**2-8.2**2-9.4**2 }{ 2 * 9.4 * 8.2 } ) = 32° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.42 }{ 11.29 } = 1.81 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.2 }{ 2 * sin 60° 37'29" } = 4.7 ; ;

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