Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Right isosceles triangle.

Sides: a = 67.88222509939   b = 48   c = 48

Area: T = 1152
Perimeter: p = 163.8822250994
Semiperimeter: s = 81.9411125497

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

Height: ha = 33.9411125497
Height: hb = 48
Height: hc = 48

Median: ma = 33.9411125497
Median: mb = 53.666563146
Median: mc = 53.666563146

Inradius: r = 14.0598874503
Circumradius: R = 33.9411125497

Vertex coordinates: A[48; 0] B[0; 0] C[48; 48]
Centroid: CG[32; 16]
Coordinates of the circumscribed circle: U[24; 24]
Coordinates of the inscribed circle: I[33.9411125497; 14.0598874503]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 135° = 0.78553981634 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 = 90° ; ; beta = 45° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 45° = 45° ; ;

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

c = 48 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 48 * fraction{ sin(90° ) }{ sin (45° ) } = 67.88 ; ;

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 = 48 * fraction{ sin(45° ) }{ sin (45° ) } = 48 ; ;


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 = 67.88 ; ; b = 48 ; ; c = 48 ; ;

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

p = a+b+c = 67.88+48+48 = 163.88 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 163.88 }{ 2 } = 81.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.94 * (81.94-67.88)(81.94-48)(81.94-48) } ; ; T = sqrt{ 1327104 } = 1152 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1152 }{ 67.88 } = 33.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1152 }{ 48 } = 48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1152 }{ 48 } = 48 ; ;

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{ 67.88**2-48**2-48**2 }{ 2 * 48 * 48 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 48**2-67.88**2-48**2 }{ 2 * 67.88 * 48 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48**2-67.88**2-48**2 }{ 2 * 48 * 67.88 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1152 }{ 81.94 } = 14.06 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 67.88 }{ 2 * sin 90° } = 33.94 ; ;




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