Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 71.79774009133   b = 46.15504797148   c = 55

Area: T = 1269.138819216
Perimeter: p = 172.9487880628
Semiperimeter: s = 86.4743940314

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 35.35333185328
Height: hb = 55
Height: hc = 46.15504797148

Median: ma = 35.89987004566
Median: mb = 59.64545026342
Median: mc = 53.72325909455

Inradius: r = 14.67765394007
Circumradius: R = 35.89987004566

Vertex coordinates: A[55; 0] B[0; 0] C[55; 46.15504797148]
Centroid: CG[36.66766666667; 15.38334932383]
Coordinates of the circumscribed circle: U[27.5; 23.07552398574]
Coordinates of the inscribed circle: I[40.32334605993; 14.67765394007]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 40° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 40° = 50° ; ;

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

c = 55 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 55 * fraction{ sin 90° }{ sin 50° } = 71.8 ; ;

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 = 55 * fraction{ sin 40° }{ sin 50° } = 46.15 ; ;


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 = 71.8 ; ; b = 46.15 ; ; c = 55 ; ;

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

p = a+b+c = 71.8+46.15+55 = 172.95 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 172.95 }{ 2 } = 86.47 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 86.47 * (86.47-71.8)(86.47-46.15)(86.47-55) } ; ; T = sqrt{ 1610711.75 } = 1269.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1269.14 }{ 71.8 } = 35.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1269.14 }{ 46.15 } = 55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1269.14 }{ 55 } = 46.15 ; ;

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{ 46.15**2+55**2-71.8**2 }{ 2 * 46.15 * 55 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 71.8**2+55**2-46.15**2 }{ 2 * 71.8 * 55 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 90° - 40° = 50° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1269.14 }{ 86.47 } = 14.68 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 71.8 }{ 2 * sin 90° } = 35.9 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 46.15**2+2 * 55**2 - 71.8**2 } }{ 2 } = 35.899 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 71.8**2 - 46.15**2 } }{ 2 } = 59.645 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 46.15**2+2 * 71.8**2 - 55**2 } }{ 2 } = 53.723 ; ;
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