Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 21.25   b = 221.9996188337   c = 5.69439203392

Area: T = 60.49879036036
Perimeter: p = 48.94435391729
Semiperimeter: s = 24.47217695864

Angle ∠ A = α = 75° = 1.3098996939 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 15° = 0.26217993878 rad

Height: ha = 5.69439203392
Height: hb = 5.54999047084
Height: hc = 21.25

Median: ma = 12.05545159102
Median: mb = 110.9998094169
Median: mc = 21.44398619913

Inradius: r = 2.47221507527
Circumradius: R = 110.9998094169

Vertex coordinates: A[5.69439203392; 0] B[0; 0] C[-0; 21.25]
Centroid: CG[1.89879734464; 7.08333333333]
Coordinates of the circumscribed circle: U[2.84769601696; 10.625]
Coordinates of the inscribed circle: I[2.47221507527; 2.47221507527]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 165° = 0.26217993878 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 75° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 75° - 90° = 15° ; ;

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

a = 21.25 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 21.25 * fraction{ sin 90° }{ sin 75° } = 22 ; ;

3. By using the law of sines, we calculate last unknown side c

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 21.25 * fraction{ sin 15° }{ sin 75° } = 5.69 ; ;


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 = 21.25 ; ; b = 22 ; ; c = 5.69 ; ;

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

p = a+b+c = 21.25+22+5.69 = 48.94 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48.94 }{ 2 } = 24.47 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.47 * (24.47-21.25)(24.47-22)(24.47-5.69) } ; ; T = sqrt{ 3660 } = 60.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60.5 }{ 21.25 } = 5.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60.5 }{ 22 } = 5.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60.5 }{ 5.69 } = 21.25 ; ;

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{ 22**2+5.69**2-21.25**2 }{ 2 * 22 * 5.69 } ) = 75° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21.25**2+5.69**2-22**2 }{ 2 * 21.25 * 5.69 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 75° - 90° = 15° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60.5 }{ 24.47 } = 2.47 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 21.25 }{ 2 * sin 75° } = 11 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 5.69**2 - 21.25**2 } }{ 2 } = 12.055 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.69**2+2 * 21.25**2 - 22**2 } }{ 2 } = 11 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 21.25**2 - 5.69**2 } }{ 2 } = 21.44 ; ;
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