Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Right isosceles triangle.

Sides: a = 6.01104076401   b = 6.01104076401   c = 8.5

Area: T = 18.06325
Perimeter: p = 20.52108152802
Semiperimeter: s = 10.26604076401

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

Height: ha = 6.01104076401
Height: hb = 6.01104076401
Height: hc = 4.25

Median: ma = 6.72198400279
Median: mb = 6.72198400279
Median: mc = 4.25

Inradius: r = 1.76604076401
Circumradius: R = 4.25

Vertex coordinates: A[8.5; 0] B[0; 0] C[4.25; 4.25]
Centroid: CG[4.25; 1.41766666667]
Coordinates of the circumscribed circle: U[4.25; -0]
Coordinates of the inscribed circle: I[4.25; 1.76604076401]

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 = 8.5 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 8.5 * fraction{ sin(45° ) }{ sin (90° ) } = 6.01 ; ;

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 = 8.5 * fraction{ sin(45° ) }{ sin (90° ) } = 6.01 ; ;


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 = 6.01 ; ; b = 6.01 ; ; c = 8.5 ; ;

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

p = a+b+c = 6.01+6.01+8.5 = 20.52 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.52 }{ 2 } = 10.26 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.26 * (10.26-6.01)(10.26-6.01)(10.26-8.5) } ; ; T = sqrt{ 326.25 } = 18.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.06 }{ 6.01 } = 6.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.06 }{ 6.01 } = 6.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.06 }{ 8.5 } = 4.25 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 6.01**2-6.01**2-8.5**2 }{ 2 * 6.01 * 8.5 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.01**2-6.01**2-8.5**2 }{ 2 * 6.01 * 8.5 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.5**2-6.01**2-6.01**2 }{ 2 * 6.01 * 6.01 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.06 }{ 10.26 } = 1.76 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.01 }{ 2 * sin 45° } = 4.25 ; ;




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