Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Right isosceles triangle.

Sides: a = 31.82198051534   b = 31.82198051534   c = 45

Area: T = 506.25
Perimeter: p = 108.6439610307
Semiperimeter: s = 54.32198051534

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

Height: ha = 31.82198051534
Height: hb = 31.82198051534
Height: hc = 22.5

Median: ma = 35.57656236769
Median: mb = 35.57656236769
Median: mc = 22.5

Inradius: r = 9.32198051534
Circumradius: R = 22.5

Vertex coordinates: A[45; 0] B[0; 0] C[22.5; 22.5]
Centroid: CG[22.5; 7.5]
Coordinates of the circumscribed circle: U[22.5; -0]
Coordinates of the inscribed circle: I[22.5; 9.32198051534]

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

Calculate another triangle




How did we calculate this triangle?

1. Calculate the third unknown inner angle

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

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

c = 45 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 45 * fraction{ sin(45° ) }{ sin (90° ) } = 31.82 ; ;

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

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 45 * fraction{ sin(45° ) }{ sin (90° ) } = 31.82 ; ;


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 = 31.82 ; ; b = 31.82 ; ; c = 45 ; ;

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

p = a+b+c = 31.82+31.82+45 = 108.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 108.64 }{ 2 } = 54.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.32 * (54.32-31.82)(54.32-31.82)(54.32-45) } ; ; T = sqrt{ 256289.06 } = 506.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 506.25 }{ 31.82 } = 31.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 506.25 }{ 31.82 } = 31.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 506.25 }{ 45 } = 22.5 ; ;

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{ 31.82**2-31.82**2-45**2 }{ 2 * 31.82 * 45 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31.82**2-31.82**2-45**2 }{ 2 * 31.82 * 45 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45**2-31.82**2-31.82**2 }{ 2 * 31.82 * 31.82 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 506.25 }{ 54.32 } = 9.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 31.82 }{ 2 * sin 45° } = 22.5 ; ;




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

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