Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 4.09895948724   b = 2.3465695253   c = 3.35

Area: T = 3.92990395488
Perimeter: p = 9.78552901254
Semiperimeter: s = 4.89326450627

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 55° = 0.96599310886 rad

Height: ha = 1.92114810618
Height: hb = 3.35
Height: hc = 2.3465695253

Median: ma = 2.04547974362
Median: mb = 3.54993762205
Median: mc = 2.88223447434

Inradius: r = 0.80330501903
Circumradius: R = 2.04547974362

Vertex coordinates: A[3.35; 0] B[0; 0] C[3.35; 2.3465695253]
Centroid: CG[2.23333333333; 0.78218984177]
Coordinates of the circumscribed circle: U[1.675; 1.17328476265]
Coordinates of the inscribed circle: I[2.54769498097; 0.80330501903]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 125° = 0.96599310886 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 35° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 35° = 55° ; ;

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

c = 3.35 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 3.35 * fraction{ sin(90° ) }{ sin (55° ) } = 4.09 ; ;

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 = 3.35 * fraction{ sin(35° ) }{ sin (55° ) } = 2.35 ; ;


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 = 4.09 ; ; b = 2.35 ; ; c = 3.35 ; ;

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

p = a+b+c = 4.09+2.35+3.35 = 9.79 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9.79 }{ 2 } = 4.89 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.89 * (4.89-4.09)(4.89-2.35)(4.89-3.35) } ; ; T = sqrt{ 15.44 } = 3.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.93 }{ 4.09 } = 1.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.93 }{ 2.35 } = 3.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.93 }{ 3.35 } = 2.35 ; ;

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{ 4.09**2-2.35**2-3.35**2 }{ 2 * 2.35 * 3.35 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.35**2-4.09**2-3.35**2 }{ 2 * 4.09 * 3.35 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.35**2-4.09**2-2.35**2 }{ 2 * 2.35 * 4.09 } ) = 55° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.93 }{ 4.89 } = 0.8 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.09 }{ 2 * sin 90° } = 2.04 ; ;




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