Triangle calculator ASA

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

Sides: a = 89.46657514841   b = 24.66601031753   c = 86

Area: T = 1060.384443654
Perimeter: p = 200.1265854659
Semiperimeter: s = 100.063292733

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 16° = 0.27992526803 rad
Angle ∠ C = γ = 74° = 1.29215436465 rad

Height: ha = 23.70548126003
Height: hb = 86
Height: hc = 24.66601031753

Median: ma = 44.7332875742
Median: mb = 86.879940016
Median: mc = 49.56993523118

Inradius: r = 10.59771758456
Circumradius: R = 44.7332875742

Vertex coordinates: A[86; 0] B[0; 0] C[86; 24.66601031753]
Centroid: CG[57.33333333333; 8.22200343918]
Coordinates of the circumscribed circle: U[43; 12.33300515876]
Coordinates of the inscribed circle: I[75.40328241544; 10.59771758456]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 164° = 0.27992526803 rad
∠ C' = γ' = 106° = 1.29215436465 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 16° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 16° = 74° ; ;

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

c = 86 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 86 * fraction{ sin(90° ) }{ sin (74° ) } = 89.47 ; ;

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 = 86 * fraction{ sin(16° ) }{ sin (74° ) } = 24.66 ; ;


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 = 89.47 ; ; b = 24.66 ; ; c = 86 ; ;

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

p = a+b+c = 89.47+24.66+86 = 200.13 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 200.13 }{ 2 } = 100.06 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.06 * (100.06-89.47)(100.06-24.66)(100.06-86) } ; ; T = sqrt{ 1124415.15 } = 1060.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1060.38 }{ 89.47 } = 23.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1060.38 }{ 24.66 } = 86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1060.38 }{ 86 } = 24.66 ; ;

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{ 89.47**2-24.66**2-86**2 }{ 2 * 24.66 * 86 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24.66**2-89.47**2-86**2 }{ 2 * 89.47 * 86 } ) = 16° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 86**2-89.47**2-24.66**2 }{ 2 * 24.66 * 89.47 } ) = 74° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1060.38 }{ 100.06 } = 10.6 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 89.47 }{ 2 * sin 90° } = 44.73 ; ;




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