Triangle calculator ASA

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


Obtuse isosceles triangle.

Sides: a = 105.0439736939   b = 105.0439736939   c = 200.9

Area: T = 3084.884448638
Perimeter: p = 410.9799473878
Semiperimeter: s = 205.4989736939

Angle ∠ A = α = 17° = 0.29767059728 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 146° = 2.54881807079 rad

Height: ha = 58.73774754788
Height: hb = 58.73774754788
Height: hc = 30.71106469525

Median: ma = 151.4555411208
Median: mb = 151.4555411208
Median: mc = 30.71106469525

Inradius: r = 15.01223530855
Circumradius: R = 179.634389624

Vertex coordinates: A[200.9; 0] B[0; 0] C[100.45; 30.71106469525]
Centroid: CG[100.45; 10.23768823175]
Coordinates of the circumscribed circle: U[100.45; -148.9233249287]
Coordinates of the inscribed circle: I[100.45; 15.01223530855]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163° = 0.29767059728 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 34° = 2.54881807079 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 17° ; ; beta = 17° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 17° - 17° = 146° ; ;

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

c = 200.9 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 200.9 * fraction{ sin(17° ) }{ sin (146° ) } = 105.04 ; ;

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 = 200.9 * fraction{ sin(17° ) }{ sin (146° ) } = 105.04 ; ;


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 = 105.04 ; ; b = 105.04 ; ; c = 200.9 ; ;

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

p = a+b+c = 105.04+105.04+200.9 = 410.98 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 410.98 }{ 2 } = 205.49 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 205.49 * (205.49-105.04)(205.49-105.04)(205.49-200.9) } ; ; T = sqrt{ 9516512.29 } = 3084.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3084.88 }{ 105.04 } = 58.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3084.88 }{ 105.04 } = 58.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3084.88 }{ 200.9 } = 30.71 ; ;

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{ 105.04**2-105.04**2-200.9**2 }{ 2 * 105.04 * 200.9 } ) = 17° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 105.04**2-105.04**2-200.9**2 }{ 2 * 105.04 * 200.9 } ) = 17° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200.9**2-105.04**2-105.04**2 }{ 2 * 105.04 * 105.04 } ) = 146° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3084.88 }{ 205.49 } = 15.01 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105.04 }{ 2 * sin 17° } = 179.63 ; ;




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