Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 110   b = 110   c = 84.19903551203

Area: T = 4277.996602618
Perimeter: p = 304.199035512
Semiperimeter: s = 152.095517756

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 77.78217459305
Height: hb = 77.78217459305
Height: hc = 101.6276748576

Median: ma = 81.04994167014
Median: mb = 81.04994167014
Median: mc = 101.6276748576

Inradius: r = 28.12770984051
Circumradius: R = 59.53215710161

Vertex coordinates: A[84.19903551203; 0] B[0; 0] C[42.09551775602; 101.6276748576]
Centroid: CG[42.09551775602; 33.87655828587]
Coordinates of the circumscribed circle: U[42.09551775602; 42.09551775602]
Coordinates of the inscribed circle: I[42.09551775602; 28.12770984051]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 135° = 0.78553981634 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 = 110 ; ; b = 110 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 110**2+110**2 - 2 * 110 * 110 * cos 45° } ; ; c = 84.19 ; ;
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 = 110 ; ; b = 110 ; ; c = 84.19 ; ;

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

p = a+b+c = 110+110+84.19 = 304.19 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 304.19 }{ 2 } = 152.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 152.1 * (152.1-110)(152.1-110)(152.1-84.19) } ; ; T = sqrt{ 18301250 } = 4278 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4278 }{ 110 } = 77.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4278 }{ 110 } = 77.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4278 }{ 84.19 } = 101.63 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 110**2+84.19**2-110**2 }{ 2 * 110 * 84.19 } ) = 67° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 110**2+84.19**2-110**2 }{ 2 * 110 * 84.19 } ) = 67° 30' ; ;
gamma = 180° - alpha - beta = 180° - 67° 30' - 67° 30' = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4278 }{ 152.1 } = 28.13 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 110 }{ 2 * sin 67° 30' } = 59.53 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 84.19**2 - 110**2 } }{ 2 } = 81.049 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 84.19**2+2 * 110**2 - 110**2 } }{ 2 } = 81.049 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 110**2 - 84.19**2 } }{ 2 } = 101.627 ; ;
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