Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 6.69989885908   b = 1.62106320185   c = 6.5

Area: T = 5.26770540601
Perimeter: p = 14.82196206093
Semiperimeter: s = 7.41098103046

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 14° = 0.24443460953 rad
Angle ∠ C = γ = 76° = 1.32664502315 rad

Height: ha = 1.57224923214
Height: hb = 6.5
Height: hc = 1.62106320185

Median: ma = 3.34994942954
Median: mb = 6.55503138883
Median: mc = 3.63216591442

Inradius: r = 0.71108217139
Circumradius: R = 3.34994942954

Vertex coordinates: A[6.5; 0] B[0; 0] C[6.5; 1.62106320185]
Centroid: CG[4.33333333333; 0.54402106728]
Coordinates of the circumscribed circle: U[3.25; 0.81103160092]
Coordinates of the inscribed circle: I[5.78991782861; 0.71108217139]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 166° = 0.24443460953 rad
∠ C' = γ' = 104° = 1.32664502315 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 14° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 14° = 76° ; ;

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

c = 6.5 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 6.5 * fraction{ sin(90° ) }{ sin (76° ) } = 6.7 ; ;

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 = 6.5 * fraction{ sin(14° ) }{ sin (76° ) } = 1.62 ; ;


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 = 6.7 ; ; b = 1.62 ; ; c = 6.5 ; ;

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

p = a+b+c = 6.7+1.62+6.5 = 14.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.82 }{ 2 } = 7.41 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.41 * (7.41-6.7)(7.41-1.62)(7.41-6.5) } ; ; T = sqrt{ 27.74 } = 5.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.27 }{ 6.7 } = 1.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.27 }{ 1.62 } = 6.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.27 }{ 6.5 } = 1.62 ; ;

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{ 6.7**2-1.62**2-6.5**2 }{ 2 * 1.62 * 6.5 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.62**2-6.7**2-6.5**2 }{ 2 * 6.7 * 6.5 } ) = 14° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.5**2-6.7**2-1.62**2 }{ 2 * 1.62 * 6.7 } ) = 76° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.27 }{ 7.41 } = 0.71 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.7 }{ 2 * sin 90° } = 3.35 ; ;




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