Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 74.121   b = 75.19547   c = 105.5854873676

Area: T = 2786.753317935
Perimeter: p = 254.9010573676
Semiperimeter: s = 127.4550286838

Angle ∠ A = α = 44.58880043547° = 44°35'17″ = 0.77882074829 rad
Angle ∠ B = β = 45.41219956453° = 45°24'43″ = 0.79325888439 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 75.19547
Height: hb = 74.121
Height: hc = 52.78769775724

Median: ma = 83.83215189433
Median: mb = 83.11112710047
Median: mc = 52.79224368378

Inradius: r = 21.86554131622
Circumradius: R = 52.79224368378

Vertex coordinates: A[105.5854873676; 0] B[0; 0] C[52.03332359148; 52.78769775724]
Centroid: CG[52.53993698635; 17.59656591908]
Coordinates of the circumscribed circle: U[52.79224368378; 0]
Coordinates of the inscribed circle: I[52.25655868378; 21.86554131622]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.4121995645° = 135°24'43″ = 0.77882074829 rad
∠ B' = β' = 134.5888004355° = 134°35'17″ = 0.79325888439 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 74.12 ; ; b = 75.19 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 74.12**2+75.19**2 - 2 * 74.12 * 75.19 * cos(90° ) } ; ; c = 105.58 ; ;


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 = 74.12 ; ; b = 75.19 ; ; c = 105.58 ; ;

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

p = a+b+c = 74.12+75.19+105.58 = 254.9 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 254.9 }{ 2 } = 127.45 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 127.45 * (127.45-74.12)(127.45-75.19)(127.45-105.58) } ; ; T = sqrt{ 7765993.28 } = 2786.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2786.75 }{ 74.12 } = 75.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2786.75 }{ 75.19 } = 74.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2786.75 }{ 105.58 } = 52.79 ; ;

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{ 74.12**2-75.19**2-105.58**2 }{ 2 * 75.19 * 105.58 } ) = 44° 35'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 75.19**2-74.12**2-105.58**2 }{ 2 * 74.12 * 105.58 } ) = 45° 24'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 105.58**2-74.12**2-75.19**2 }{ 2 * 75.19 * 74.12 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2786.75 }{ 127.45 } = 21.87 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 74.12 }{ 2 * sin 44° 35'17" } = 52.79 ; ;




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