Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right isosceles triangle.

Sides: a = 3.25   b = 4.59661940777   c = 3.25

Area: T = 5.281125
Perimeter: p = 11.09661940777
Semiperimeter: s = 5.54880970389

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 3.25
Height: hb = 2.29880970389
Height: hc = 3.25

Median: ma = 3.63436104634
Median: mb = 2.29880970389
Median: mc = 3.63436104634

Inradius: r = 0.95219029611
Circumradius: R = 2.29880970389

Vertex coordinates: A[3.25; 0] B[0; 0] C[-0; 3.25]
Centroid: CG[1.08333333333; 1.08333333333]
Coordinates of the circumscribed circle: U[1.625; 1.625]
Coordinates of the inscribed circle: I[0.95219029611; 0.95219029611]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 45° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 45° - 90° = 45° ; ;

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

a = 3.25 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 3.25 * fraction{ sin(90° ) }{ sin (45° ) } = 4.6 ; ;

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 = 3.25 * fraction{ sin(45° ) }{ sin (45° ) } = 3.25 ; ;


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 = 3.25 ; ; b = 4.6 ; ; c = 3.25 ; ;

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

p = a+b+c = 3.25+4.6+3.25 = 11.1 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.1 }{ 2 } = 5.55 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.55 * (5.55-3.25)(5.55-4.6)(5.55-3.25) } ; ; T = sqrt{ 27.89 } = 5.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.28 }{ 3.25 } = 3.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.28 }{ 4.6 } = 2.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.28 }{ 3.25 } = 3.25 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 3.25**2-4.6**2-3.25**2 }{ 2 * 4.6 * 3.25 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.6**2-3.25**2-3.25**2 }{ 2 * 3.25 * 3.25 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.25**2-3.25**2-4.6**2 }{ 2 * 4.6 * 3.25 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.28 }{ 5.55 } = 0.95 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.25 }{ 2 * sin 45° } = 2.3 ; ;




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