Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 8.124404   b = 4.699042   c = 9.38108350214

Area: T = 19.05325798484
Perimeter: p = 22.19552950214
Semiperimeter: s = 11.09876475107

Angle ∠ A = α = 609.9999824438° = 60° = 1.04771972448 rad
Angle ∠ B = β = 300.0000175562° = 30° = 0.5243599082 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 4.699042
Height: hb = 8.124404
Height: hc = 4.06220221558

Median: ma = 6.20548405505
Median: mb = 8.45657693834
Median: mc = 4.69904175107

Inradius: r = 1.71768124893
Circumradius: R = 4.69904175107

Vertex coordinates: A[9.38108350214; 0] B[0; 0] C[7.03656237767; 4.06220221558]
Centroid: CG[5.47221529327; 1.35440073853]
Coordinates of the circumscribed circle: U[4.69904175107; 0]
Coordinates of the inscribed circle: I[6.40772275107; 1.71768124893]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 1200.000017556° = 120° = 1.04771972448 rad
∠ B' = β' = 1509.999982444° = 150° = 0.5243599082 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 = 8.12 ; ; b = 4.69 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 8.12**2+4.69**2 - 2 * 8.12 * 4.69 * cos(90° ) } ; ; c = 9.38 ; ;


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 = 8.12 ; ; b = 4.69 ; ; c = 9.38 ; ;

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

p = a+b+c = 8.12+4.69+9.38 = 22.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.2 }{ 2 } = 11.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.1 * (11.1-8.12)(11.1-4.69)(11.1-9.38) } ; ; T = sqrt{ 363 } = 19.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19.05 }{ 8.12 } = 4.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19.05 }{ 4.69 } = 8.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19.05 }{ 9.38 } = 4.06 ; ;

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{ 8.12**2-4.69**2-9.38**2 }{ 2 * 4.69 * 9.38 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.69**2-8.12**2-9.38**2 }{ 2 * 8.12 * 9.38 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.38**2-8.12**2-4.69**2 }{ 2 * 4.69 * 8.12 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19.05 }{ 11.1 } = 1.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.12 }{ 2 * sin 60° } = 4.69 ; ;




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