Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute scalene triangle.

Sides: a = 10.1   b = 9.8   c = 13.76999844563

Area: T = 49.4222175675
Perimeter: p = 33.65999844563
Semiperimeter: s = 16.87999922281

Angle ∠ A = α = 47.41101317895° = 47°24'36″ = 0.82774628985 rad
Angle ∠ B = β = 45.59898682105° = 45°35'24″ = 0.79656933058 rad
Angle ∠ C = γ = 87° = 1.51884364492 rad

Height: ha = 9.78765694406
Height: hb = 10.0866158301
Height: hc = 7.21549243428

Median: ma = 10.78771352569
Median: mb = 10.9932715181
Median: mc = 7.21882135238

Inradius: r = 2.9421797532
Circumradius: R = 6.85993927876

Vertex coordinates: A[13.76999844563; 0] B[0; 0] C[7.0687875687; 7.21549243428]
Centroid: CG[6.92326200478; 2.40549747809]
Coordinates of the circumscribed circle: U[6.85499922281; 0.35989928808]
Coordinates of the inscribed circle: I[76.9999922281; 2.9421797532]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.598986821° = 132°35'24″ = 0.82774628985 rad
∠ B' = β' = 134.411013179° = 134°24'36″ = 0.79656933058 rad
∠ C' = γ' = 93° = 1.51884364492 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 10.1 ; ; b = 9.8 ; ; gamma = 87° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 10.1**2+9.8**2 - 2 * 10.1 * 9.8 * cos(87° ) } ; ; c = 13.7 ; ;


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 = 10.1 ; ; b = 9.8 ; ; c = 13.7 ; ;

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

p = a+b+c = 10.1+9.8+13.7 = 33.6 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.6 }{ 2 } = 16.8 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.8 * (16.8-10.1)(16.8-9.8)(16.8-13.7) } ; ; T = sqrt{ 2442.55 } = 49.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.42 }{ 10.1 } = 9.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.42 }{ 9.8 } = 10.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.42 }{ 13.7 } = 7.21 ; ;

6. 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{ 10.1**2-9.8**2-13.7**2 }{ 2 * 9.8 * 13.7 } ) = 47° 24'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.8**2-10.1**2-13.7**2 }{ 2 * 10.1 * 13.7 } ) = 45° 35'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.7**2-10.1**2-9.8**2 }{ 2 * 9.8 * 10.1 } ) = 87° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.42 }{ 16.8 } = 2.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.1 }{ 2 * sin 47° 24'36" } = 6.86 ; ;




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