Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 100   b = 100   c = 76.5376686473

Area: T = 3535.534390593
Perimeter: p = 276.5376686473
Semiperimeter: s = 138.2688343236

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 = 70.71106781187
Height: hb = 70.71106781187
Height: hc = 92.38879532511

Median: ma = 73.68112879104
Median: mb = 73.68112879104
Median: mc = 92.38879532511

Inradius: r = 25.57700894592
Circumradius: R = 54.12196100146

Vertex coordinates: A[76.5376686473; 0] B[0; 0] C[38.26883432365; 92.38879532511]
Centroid: CG[38.26883432365; 30.7965984417]
Coordinates of the circumscribed circle: U[38.26883432365; 38.26883432365]
Coordinates of the inscribed circle: I[38.26883432365; 25.57700894592]

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 = 100 ; ; b = 100 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 100**2+100**2 - 2 * 100 * 100 * cos(45° ) } ; ; c = 76.54 ; ;


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 = 100 ; ; b = 100 ; ; c = 76.54 ; ;

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

p = a+b+c = 100+100+76.54 = 276.54 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 276.54 }{ 2 } = 138.27 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 138.27 * (138.27-100)(138.27-100)(138.27-76.54) } ; ; T = sqrt{ 12500000 } = 3535.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3535.53 }{ 100 } = 70.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3535.53 }{ 100 } = 70.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3535.53 }{ 76.54 } = 92.39 ; ;

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{ 100**2-100**2-76.54**2 }{ 2 * 100 * 76.54 } ) = 67° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-100**2-76.54**2 }{ 2 * 100 * 76.54 } ) = 67° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 76.54**2-100**2-100**2 }{ 2 * 100 * 100 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3535.53 }{ 138.27 } = 25.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 67° 30' } = 54.12 ; ;




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