Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 960   b = 1108.513251684   c = 554.2566258422

Area: T = 266043.0044043
Perimeter: p = 2622.769877527
Semiperimeter: s = 1311.384438763

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 30° = 0.52435987756 rad

Height: ha = 554.2566258422
Height: hb = 480
Height: hc = 960

Median: ma = 733.2122111193
Median: mb = 554.2566258422
Median: mc = 999.2199679744

Inradius: r = 202.8721870789
Circumradius: R = 554.2566258422

Vertex coordinates: A[554.2566258422; 0] B[0; 0] C[-0; 960]
Centroid: CG[184.7522086141; 320]
Coordinates of the circumscribed circle: U[277.1288129211; 480]
Coordinates of the inscribed circle: I[202.8721870789; 202.8721870789]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 150° = 0.52435987756 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 60° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 60° - 90° = 30° ; ;

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

a = 960 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 960 * fraction{ sin(90° ) }{ sin (60° ) } = 1108.51 ; ;

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 = 960 * fraction{ sin(30° ) }{ sin (60° ) } = 554.26 ; ;


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 = 960 ; ; b = 1108.51 ; ; c = 554.26 ; ;

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

p = a+b+c = 960+1108.51+554.26 = 2622.77 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2622.77 }{ 2 } = 1311.38 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1311.38 * (1311.38-960)(1311.38-1108.51)(1311.38-554.26) } ; ; T = sqrt{ 70778880000 } = 266043 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 266043 }{ 960 } = 554.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 266043 }{ 1108.51 } = 480 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 266043 }{ 554.26 } = 960 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 960**2-1108.51**2-554.26**2 }{ 2 * 1108.51 * 554.26 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1108.51**2-960**2-554.26**2 }{ 2 * 960 * 554.26 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 554.26**2-960**2-1108.51**2 }{ 2 * 1108.51 * 960 } ) = 30° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 266043 }{ 1311.38 } = 202.87 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 960 }{ 2 * sin 60° } = 554.26 ; ;




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