Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 60   b = 60   c = 45.92220118838

Area: T = 1272.792220614
Perimeter: p = 165.9222011884
Semiperimeter: s = 82.96110059419

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 = 42.42664068712
Height: hb = 42.42664068712
Height: hc = 55.43327719507

Median: ma = 44.20987727462
Median: mb = 44.20987727462
Median: mc = 55.43327719507

Inradius: r = 15.34220536755
Circumradius: R = 32.47217660088

Vertex coordinates: A[45.92220118838; 0] B[0; 0] C[22.96110059419; 55.43327719507]
Centroid: CG[22.96110059419; 18.47875906502]
Coordinates of the circumscribed circle: U[22.96110059419; 22.96110059419]
Coordinates of the inscribed circle: I[22.96110059419; 15.34220536755]

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 = 60 ; ; b = 60 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 60**2+60**2 - 2 * 60 * 60 * cos(45° ) } ; ; c = 45.92 ; ;


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 = 60 ; ; b = 60 ; ; c = 45.92 ; ;

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

p = a+b+c = 60+60+45.92 = 165.92 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 165.92 }{ 2 } = 82.96 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 82.96 * (82.96-60)(82.96-60)(82.96-45.92) } ; ; T = sqrt{ 1620000 } = 1272.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1272.79 }{ 60 } = 42.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1272.79 }{ 60 } = 42.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1272.79 }{ 45.92 } = 55.43 ; ;

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{ 60**2-60**2-45.92**2 }{ 2 * 60 * 45.92 } ) = 67° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-60**2-45.92**2 }{ 2 * 60 * 45.92 } ) = 67° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45.92**2-60**2-60**2 }{ 2 * 60 * 60 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1272.79 }{ 82.96 } = 15.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 67° 30' } = 32.47 ; ;




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