Triangle calculator ASA

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


Right isosceles triangle.

Sides: a = 2.47548737342   b = 2.47548737342   c = 3.5

Area: T = 3.06325
Perimeter: p = 8.45497474683
Semiperimeter: s = 4.22548737342

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

Height: ha = 2.47548737342
Height: hb = 2.47548737342
Height: hc = 1.75

Median: ma = 2.76769929526
Median: mb = 2.76769929526
Median: mc = 1.75

Inradius: r = 0.72548737342
Circumradius: R = 1.75

Vertex coordinates: A[3.5; 0] B[0; 0] C[1.75; 1.75]
Centroid: CG[1.75; 0.58333333333]
Coordinates of the circumscribed circle: U[1.75; -0]
Coordinates of the inscribed circle: I[1.75; 0.72548737342]

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 = 3.5 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 3.5 * fraction{ sin(45° ) }{ sin (90° ) } = 2.47 ; ;

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 = 3.5 * fraction{ sin(45° ) }{ sin (90° ) } = 2.47 ; ;


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 = 2.47 ; ; b = 2.47 ; ; c = 3.5 ; ;

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

p = a+b+c = 2.47+2.47+3.5 = 8.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.45 }{ 2 } = 4.22 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.22 * (4.22-2.47)(4.22-2.47)(4.22-3.5) } ; ; T = sqrt{ 9.38 } = 3.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.06 }{ 2.47 } = 2.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.06 }{ 2.47 } = 2.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.06 }{ 3.5 } = 1.75 ; ;

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{ 2.47**2-2.47**2-3.5**2 }{ 2 * 2.47 * 3.5 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.47**2-2.47**2-3.5**2 }{ 2 * 2.47 * 3.5 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.5**2-2.47**2-2.47**2 }{ 2 * 2.47 * 2.47 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.06 }{ 4.22 } = 0.72 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.47 }{ 2 * sin 45° } = 1.75 ; ;




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