Triangle calculator ASA

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

Sides: a = 28.28442712475   b = 28.28442712475   c = 40

Area: T = 400
Perimeter: p = 96.56985424949
Semiperimeter: s = 48.28442712475

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

Height: ha = 28.28442712475
Height: hb = 28.28442712475
Height: hc = 20

Median: ma = 31.62327766017
Median: mb = 31.62327766017
Median: mc = 20

Inradius: r = 8.28442712475
Circumradius: R = 20

Vertex coordinates: A[40; 0] B[0; 0] C[20; 20]
Centroid: CG[20; 6.66766666667]
Coordinates of the circumscribed circle: U[20; -0]
Coordinates of the inscribed circle: I[20; 8.28442712475]

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 = 40 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 40 * fraction{ sin(45° ) }{ sin (90° ) } = 28.28 ; ;

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 = 40 * fraction{ sin(45° ) }{ sin (90° ) } = 28.28 ; ;


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 = 28.28 ; ; b = 28.28 ; ; c = 40 ; ;

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

p = a+b+c = 28.28+28.28+40 = 96.57 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 96.57 }{ 2 } = 48.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 48.28 * (48.28-28.28)(48.28-28.28)(48.28-40) } ; ; T = sqrt{ 160000 } = 400 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 400 }{ 28.28 } = 28.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 400 }{ 28.28 } = 28.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 400 }{ 40 } = 20 ; ;

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{ 28.28**2-28.28**2-40**2 }{ 2 * 28.28 * 40 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.28**2-28.28**2-40**2 }{ 2 * 28.28 * 40 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 40**2-28.28**2-28.28**2 }{ 2 * 28.28 * 28.28 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 400 }{ 48.28 } = 8.28 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 28.28 }{ 2 * sin 45° } = 20 ; ;




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