Triangle calculator ASA

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


Equilateral triangle.

Sides: a = 60   b = 60   c = 60

Area: T = 1558.846572681
Perimeter: p = 180
Semiperimeter: s = 90

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 51.96215242271
Height: hb = 51.96215242271
Height: hc = 51.96215242271

Median: ma = 51.96215242271
Median: mb = 51.96215242271
Median: mc = 51.96215242271

Inradius: r = 17.32105080757
Circumradius: R = 34.64110161514

Vertex coordinates: A[60; 0] B[0; 0] C[30; 51.96215242271]
Centroid: CG[30; 17.32105080757]
Coordinates of the circumscribed circle: U[30; 17.32105080757]
Coordinates of the inscribed circle: I[30; 17.32105080757]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 60° ; ; beta = 60° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 60° - 60° = 60° ; ;

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

c = 60 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 60 * fraction{ sin(60° ) }{ sin (60° ) } = 60 ; ;

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 = 60 * fraction{ sin(60° ) }{ sin (60° ) } = 60 ; ;


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 = 60 ; ; b = 60 ; ; c = 60 ; ;

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

p = a+b+c = 60+60+60 = 180 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180 }{ 2 } = 90 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90 * (90-60)(90-60)(90-60) } ; ; T = sqrt{ 2430000 } = 1558.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1558.85 }{ 60 } = 51.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1558.85 }{ 60 } = 51.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1558.85 }{ 60 } = 51.96 ; ;

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{ 60**2-60**2-60**2 }{ 2 * 60 * 60 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-60**2-60**2 }{ 2 * 60 * 60 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 60**2-60**2-60**2 }{ 2 * 60 * 60 } ) = 60° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1558.85 }{ 90 } = 17.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 60° } = 34.64 ; ;




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