Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 1.2   b = 1.2   c = 1.69770562748

Area: T = 0.72
Perimeter: p = 4.09770562748
Semiperimeter: s = 2.04985281374

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.2
Height: hb = 1.2
Height: hc = 0.84985281374

Median: ma = 1.34216407865
Median: mb = 1.34216407865
Median: mc = 0.84985281374

Inradius: r = 0.35114718626
Circumradius: R = 0.84985281374

Vertex coordinates: A[1.69770562748; 0] B[0; 0] C[0.84985281374; 0.84985281374]
Centroid: CG[0.84985281374; 0.28328427125]
Coordinates of the circumscribed circle: U[0.84985281374; 0]
Coordinates of the inscribed circle: I[0.84985281374; 0.35114718626]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 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 = 1.2 ; ; b = 1.2 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 1.2**2+1.2**2 - 2 * 1.2 * 1.2 * cos(90° ) } ; ; c = 1.7 ; ;


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 = 1.2 ; ; b = 1.2 ; ; c = 1.7 ; ;

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

p = a+b+c = 1.2+1.2+1.7 = 4.1 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.1 }{ 2 } = 2.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.05 * (2.05-1.2)(2.05-1.2)(2.05-1.7) } ; ; T = sqrt{ 0.52 } = 0.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.72 }{ 1.2 } = 1.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.72 }{ 1.2 } = 1.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.72 }{ 1.7 } = 0.85 ; ;

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{ 1.2**2-1.2**2-1.7**2 }{ 2 * 1.2 * 1.7 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.2**2-1.2**2-1.7**2 }{ 2 * 1.2 * 1.7 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.7**2-1.2**2-1.2**2 }{ 2 * 1.2 * 1.2 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.72 }{ 2.05 } = 0.35 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.2 }{ 2 * sin 45° } = 0.85 ; ;




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