Triangle calculator ASA

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


Acute isosceles triangle.

Sides: a = 5.42220864676   b = 5.42220864676   c = 6.97105

Area: T = 14.47661922317
Perimeter: p = 17.81546729351
Semiperimeter: s = 8.90773364676

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 5.34397127908
Height: hb = 5.34397127908
Height: hc = 4.15435592086

Median: ma = 5.62552724859
Median: mb = 5.62552724859
Median: mc = 4.15435592086

Inradius: r = 1.62551987656
Circumradius: R = 3.53990155991

Vertex coordinates: A[6.97105; 0] B[0; 0] C[3.485525; 4.15435592086]
Centroid: CG[3.485525; 1.38545197362]
Coordinates of the circumscribed circle: U[3.485525; 0.61545436095]
Coordinates of the inscribed circle: I[3.485525; 1.62551987656]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 100° = 1.39662634016 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 50° ; ; beta = 50° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 50° - 50° = 80° ; ;

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

c = 6.97 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 6.97 * fraction{ sin 50° }{ sin 80° } = 5.42 ; ;

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 = 6.97 * fraction{ sin 50° }{ sin 80° } = 5.42 ; ;


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 = 5.42 ; ; b = 5.42 ; ; c = 6.97 ; ;

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

p = a+b+c = 5.42+5.42+6.97 = 17.81 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.81 }{ 2 } = 8.91 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.91 * (8.91-5.42)(8.91-5.42)(8.91-6.97) } ; ; T = sqrt{ 209.56 } = 14.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.48 }{ 5.42 } = 5.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.48 }{ 5.42 } = 5.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.48 }{ 6.97 } = 4.15 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.42**2+6.97**2-5.42**2 }{ 2 * 5.42 * 6.97 } ) = 50° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.42**2+6.97**2-5.42**2 }{ 2 * 5.42 * 6.97 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 50° - 50° = 80° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.48 }{ 8.91 } = 1.63 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.42 }{ 2 * sin 50° } = 3.54 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.42**2+2 * 6.97**2 - 5.42**2 } }{ 2 } = 5.625 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.97**2+2 * 5.42**2 - 5.42**2 } }{ 2 } = 5.625 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.42**2+2 * 5.42**2 - 6.97**2 } }{ 2 } = 4.154 ; ;
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