Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 24.37333577895   b = 88.42554520482   c = 85

Area: T = 1035.868770605
Perimeter: p = 197.7998809838
Semiperimeter: s = 98.89994049189

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

Height: ha = 85
Height: hb = 23.42991752444
Height: hc = 24.37333577895

Median: ma = 85.86991745767
Median: mb = 44.21327260241
Median: mc = 48.99329644943

Inradius: r = 10.47439528706
Circumradius: R = 44.21327260241

Vertex coordinates: A[85; 0] B[0; 0] C[-0; 24.37333577895]
Centroid: CG[28.33333333333; 8.12444525965]
Coordinates of the circumscribed circle: U[42.5; 12.18766788947]
Coordinates of the inscribed circle: I[10.47439528706; 10.47439528706]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164° = 0.27992526803 rad
∠ B' = β' = 90° = 1.57107963268 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 = 16° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 16° - 90° = 74° ; ;

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

c = 85 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 85 * fraction{ sin(16° ) }{ sin (74° ) } = 24.37 ; ;

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 = 85 * fraction{ sin(90° ) }{ sin (74° ) } = 88.43 ; ;


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 = 24.37 ; ; b = 88.43 ; ; c = 85 ; ;

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

p = a+b+c = 24.37+88.43+85 = 197.8 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 197.8 }{ 2 } = 98.9 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 98.9 * (98.9-24.37)(98.9-88.43)(98.9-85) } ; ; T = sqrt{ 1073021.9 } = 1035.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1035.87 }{ 24.37 } = 85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1035.87 }{ 88.43 } = 23.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1035.87 }{ 85 } = 24.37 ; ;

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{ 24.37**2-88.43**2-85**2 }{ 2 * 88.43 * 85 } ) = 16° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 88.43**2-24.37**2-85**2 }{ 2 * 24.37 * 85 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 85**2-24.37**2-88.43**2 }{ 2 * 88.43 * 24.37 } ) = 74° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1035.87 }{ 98.9 } = 10.47 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 24.37 }{ 2 * sin 16° } = 44.21 ; ;




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