Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 300   b = 77.64657135308   c = 289.7787747887

Area: T = 11250
Perimeter: p = 667.4233461417
Semiperimeter: s = 333.7121730709

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

Height: ha = 75
Height: hb = 289.7787747887
Height: hc = 77.64657135308

Median: ma = 150
Median: mb = 292.3676819899
Median: mc = 164.3832610462

Inradius: r = 33.71217307087
Circumradius: R = 150

Vertex coordinates: A[289.7787747887; 0] B[0; 0] C[289.7787747887; 77.64657135308]
Centroid: CG[193.1855165258; 25.88219045103]
Coordinates of the circumscribed circle: U[144.8898873943; 38.82328567654]
Coordinates of the inscribed circle: I[256.0666017178; 33.71217307087]

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

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

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

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

a = 300 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 300 * fraction{ sin 15° }{ sin 90° } = 77.65 ; ;

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 = 300 * fraction{ sin 75° }{ sin 90° } = 289.78 ; ;


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 = 300 ; ; b = 77.65 ; ; c = 289.78 ; ;

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

p = a+b+c = 300+77.65+289.78 = 667.42 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 667.42 }{ 2 } = 333.71 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 333.71 * (333.71-300)(333.71-77.65)(333.71-289.78) } ; ; T = sqrt{ 126562500 } = 11250 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11250 }{ 300 } = 75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11250 }{ 77.65 } = 289.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11250 }{ 289.78 } = 77.65 ; ;

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{ 77.65**2+289.78**2-300**2 }{ 2 * 77.65 * 289.78 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 300**2+289.78**2-77.65**2 }{ 2 * 300 * 289.78 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 90° - 15° = 75° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11250 }{ 333.71 } = 33.71 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 300 }{ 2 * sin 90° } = 150 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.65**2+2 * 289.78**2 - 300**2 } }{ 2 } = 150 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 289.78**2+2 * 300**2 - 77.65**2 } }{ 2 } = 292.367 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.65**2+2 * 300**2 - 289.78**2 } }{ 2 } = 164.383 ; ;
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