Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene Pythagorean triangle.

Sides: a = 0.5   b = 0.33000007146   c = 0.43999994641

Area: T = 0.06600000625
Perimeter: p = 1.22000001786
Semiperimeter: s = 0.66000000893

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 36.87° = 36°52'12″ = 0.64435028952 rad
Angle ∠ C = γ = 53.13° = 53°7'48″ = 0.92772934316 rad

Height: ha = 0.24400002501
Height: hb = 0.43999994641
Height: hc = 0.33000007146

Median: ma = 0.25
Median: mb = 0.42771998109
Median: mc = 0.36105555735

Inradius: r = 0.11000000893
Circumradius: R = 0.25

Vertex coordinates: A[0.43999994641; 0] B[0; 0] C[0.43999994641; 0.33000007146]
Centroid: CG[0.26766663094; 0.11000002382]
Coordinates of the circumscribed circle: U[0.2199999732; 0.15500003573]
Coordinates of the inscribed circle: I[0.32999993748; 0.11000000893]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 143.13° = 143°7'48″ = 0.64435028952 rad
∠ C' = γ' = 126.87° = 126°52'12″ = 0.92772934316 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 90° ; ; beta = 36° 52'12" ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 36° 52'12" = 53° 7'48" ; ;

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

a = 0.5 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 0.5 * fraction{ sin 36° 52'12" }{ sin 90° } = 0.3 ; ;

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 = 0.5 * fraction{ sin 53° 7'48" }{ sin 90° } = 0.4 ; ;


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 = 0.5 ; ; b = 0.3 ; ; c = 0.4 ; ;

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

p = a+b+c = 0.5+0.3+0.4 = 1.2 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1.2 }{ 2 } = 0.6 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.6 * (0.6-0.5)(0.6-0.3)(0.6-0.4) } ; ; T = sqrt{ 0 } = 0.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.06 }{ 0.5 } = 0.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.06 }{ 0.3 } = 0.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.06 }{ 0.4 } = 0.3 ; ;

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{ 0.3**2+0.4**2-0.5**2 }{ 2 * 0.3 * 0.4 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.5**2+0.4**2-0.3**2 }{ 2 * 0.5 * 0.4 } ) = 36° 52'12" ; ; gamma = 180° - alpha - beta = 180° - 90° - 36° 52'12" = 53° 7'48" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.06 }{ 0.6 } = 0.1 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 0.5 }{ 2 * sin 90° } = 0.25 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.3**2+2 * 0.4**2 - 0.5**2 } }{ 2 } = 0.25 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.4**2+2 * 0.5**2 - 0.3**2 } }{ 2 } = 0.427 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.3**2+2 * 0.5**2 - 0.4**2 } }{ 2 } = 0.361 ; ;
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