Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute isosceles triangle.

Sides: a = 48   b = 70.17113056039   c = 70.17113056039

Area: T = 1582.547699361
Perimeter: p = 188.3432611208
Semiperimeter: s = 94.17113056039

Angle ∠ A = α = 40° = 0.69881317008 rad
Angle ∠ B = β = 70° = 1.22217304764 rad
Angle ∠ C = γ = 70° = 1.22217304764 rad

Height: ha = 65.93994580669
Height: hb = 45.10552457977
Height: hc = 45.10552457977

Median: ma = 65.93994580669
Median: mb = 48.81660120508
Median: mc = 48.81660120508

Inradius: r = 16.8054980917
Circumradius: R = 37.33773718446

Vertex coordinates: A[70.17113056039; 0] B[0; 0] C[16.41769668796; 45.10552457977]
Centroid: CG[28.86327574945; 15.03550819326]
Coordinates of the circumscribed circle: U[35.0865652802; 12.77701332697]
Coordinates of the inscribed circle: I[24; 16.8054980917]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140° = 0.69881317008 rad
∠ B' = β' = 110° = 1.22217304764 rad
∠ C' = γ' = 110° = 1.22217304764 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 40° ; ; beta = 70° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 40° - 70° = 70° ; ;

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

a = 48 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 48 * fraction{ sin(70° ) }{ sin (40° ) } = 70.17 ; ;

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 = 48 * fraction{ sin(70° ) }{ sin (40° ) } = 70.17 ; ;


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 = 48 ; ; b = 70.17 ; ; c = 70.17 ; ;

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

p = a+b+c = 48+70.17+70.17 = 188.34 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 188.34 }{ 2 } = 94.17 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.17 * (94.17-48)(94.17-70.17)(94.17-70.17) } ; ; T = sqrt{ 2504454.99 } = 1582.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1582.55 }{ 48 } = 65.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1582.55 }{ 70.17 } = 45.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1582.55 }{ 70.17 } = 45.11 ; ;

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{ 48**2-70.17**2-70.17**2 }{ 2 * 70.17 * 70.17 } ) = 40° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 70.17**2-48**2-70.17**2 }{ 2 * 48 * 70.17 } ) = 70° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 70.17**2-48**2-70.17**2 }{ 2 * 70.17 * 48 } ) = 70° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1582.55 }{ 94.17 } = 16.8 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 48 }{ 2 * sin 40° } = 37.34 ; ;




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