Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 135   b = 86.77663273077   c = 103.4165999821

Area: T = 4487.033032466
Perimeter: p = 325.1922327129
Semiperimeter: s = 162.5966163564

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 66.47545233283
Height: hb = 103.4165999821
Height: hc = 86.77663273077

Median: ma = 67.5
Median: mb = 112.1499015886
Median: mc = 101.0144099193

Inradius: r = 27.59661635644
Circumradius: R = 67.5

Vertex coordinates: A[103.4165999821; 0] B[0; 0] C[103.4165999821; 86.77663273077]
Centroid: CG[68.94439998807; 28.92554424359]
Coordinates of the circumscribed circle: U[51.70879999105; 43.38881636538]
Coordinates of the inscribed circle: I[75.82198362567; 27.59661635644]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 130° = 0.8732664626 rad

Calculate another triangle




How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 40° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 40° = 50° ; ;

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

a = 135 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 135 * fraction{ sin(40° ) }{ sin (90° ) } = 86.78 ; ;

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 = 135 * fraction{ sin(50° ) }{ sin (90° ) } = 103.42 ; ;


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 = 135 ; ; b = 86.78 ; ; c = 103.42 ; ;

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

p = a+b+c = 135+86.78+103.42 = 325.19 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 325.19 }{ 2 } = 162.6 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 162.6 * (162.6-135)(162.6-86.78)(162.6-103.42) } ; ; T = sqrt{ 20133441.13 } = 4487.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4487.03 }{ 135 } = 66.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4487.03 }{ 86.78 } = 103.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4487.03 }{ 103.42 } = 86.78 ; ;

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{ 135**2-86.78**2-103.42**2 }{ 2 * 86.78 * 103.42 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 86.78**2-135**2-103.42**2 }{ 2 * 135 * 103.42 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 103.42**2-135**2-86.78**2 }{ 2 * 86.78 * 135 } ) = 50° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4487.03 }{ 162.6 } = 27.6 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 135 }{ 2 * sin 90° } = 67.5 ; ;




Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.