Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Right isosceles triangle.

Sides: a = 1.47107821049   b = 1.47107821049   c = 2.08

Area: T = 1.08216
Perimeter: p = 5.02215642097
Semiperimeter: s = 2.51107821049

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

Height: ha = 1.47107821049
Height: hb = 1.47107821049
Height: hc = 1.04

Median: ma = 1.64443843833
Median: mb = 1.64443843833
Median: mc = 1.04

Inradius: r = 0.43107821049
Circumradius: R = 1.04

Vertex coordinates: A[2.08; 0] B[0; 0] C[1.04; 1.04]
Centroid: CG[1.04; 0.34766666667]
Coordinates of the circumscribed circle: U[1.04; -0]
Coordinates of the inscribed circle: I[1.04; 0.43107821049]

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. Calculate the third unknown inner angle

alpha = 45° ; ; beta = 45° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

2. By using the law of sines, we calculate unknown side a

c = 2.08 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 2.08 * fraction{ sin 45° }{ sin 90° } = 1.47 ; ;

3. By using the law of sines, we calculate last unknown side b

fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 2.08 * fraction{ sin 45° }{ sin 90° } = 1.47 ; ;
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 = 1.47 ; ; b = 1.47 ; ; c = 2.08 ; ;

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

p = a+b+c = 1.47+1.47+2.08 = 5.02 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.02 }{ 2 } = 2.51 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.51 * (2.51-1.47)(2.51-1.47)(2.51-2.08) } ; ; T = sqrt{ 1.17 } = 1.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.08 }{ 1.47 } = 1.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.08 }{ 1.47 } = 1.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.08 }{ 2.08 } = 1.04 ; ;

8. 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 1.47**2+2.08**2-1.47**2 }{ 2 * 1.47 * 2.08 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.47**2+2.08**2-1.47**2 }{ 2 * 1.47 * 2.08 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.08 }{ 2.51 } = 0.43 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.47 }{ 2 * sin 45° } = 1.04 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.47**2+2 * 2.08**2 - 1.47**2 } }{ 2 } = 1.644 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.08**2+2 * 1.47**2 - 1.47**2 } }{ 2 } = 1.644 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.47**2+2 * 1.47**2 - 2.08**2 } }{ 2 } = 1.04 ; ;
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