Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute isosceles triangle.

Sides: a = 90   b = 90   c = 76.07112871133

Area: T = 3102.487999463
Perimeter: p = 256.0711287113
Semiperimeter: s = 128.0365643557

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 68.94439998807
Height: hb = 68.94439998807
Height: hc = 81.56877008333

Median: ma = 70.13114505877
Median: mb = 70.13114505877
Median: mc = 81.56877008333

Inradius: r = 24.23113773606
Circumradius: R = 49.65220063533

Vertex coordinates: A[76.07112871133; 0] B[0; 0] C[38.03656435567; 81.56877008333]
Centroid: CG[38.03656435567; 27.18992336111]
Coordinates of the circumscribed circle: U[38.03656435567; 31.916569448]
Coordinates of the inscribed circle: I[38.03656435567; 24.23113773606]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 90 ; ; b = 90 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 90**2 + c**2 -2 * 90 * c * cos (65° ) ; ; ; ; c**2 -76.071c =0 ; ; p=1; q=-76.0712871133; r=0 ; ; D = q**2 - 4pr = 76.071**2 - 4 * 1 * 0 = 5786.84072308 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 76.07 ± sqrt{ 5786.84 } }{ 2 } ; ; c_{1,2} = 38.0356435567 ± 38.0356435567 ; ; c_{1} = 76.0712871133 ; ;
c_{2} = 0 ; ; ; ; (c -76.0712871133) c = 0 ; ; ; ; c>0 ; ;


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 = 90 ; ; b = 90 ; ; c = 76.07 ; ;

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

p = a+b+c = 90+90+76.07 = 256.07 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 256.07 }{ 2 } = 128.04 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 128.04 * (128.04-90)(128.04-90)(128.04-76.07) } ; ; T = sqrt{ 9625382.12 } = 3102.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3102.48 }{ 90 } = 68.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3102.48 }{ 90 } = 68.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3102.48 }{ 76.07 } = 81.57 ; ;

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{ 90**2-90**2-76.07**2 }{ 2 * 90 * 76.07 } ) = 65° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-90**2-76.07**2 }{ 2 * 90 * 76.07 } ) = 65° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 76.07**2-90**2-90**2 }{ 2 * 90 * 90 } ) = 50° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3102.48 }{ 128.04 } = 24.23 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 65° } = 49.65 ; ;




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