Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 36   b = 72   c = 80.498844719

Area: T = 1296
Perimeter: p = 188.498844719
Semiperimeter: s = 94.2499223595

Angle ∠ A = α = 26.56550511771° = 26°33'54″ = 0.4643647609 rad
Angle ∠ B = β = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 72
Height: hb = 36
Height: hc = 32.1999378876

Median: ma = 74.21659012611
Median: mb = 50.91216882454
Median: mc = 40.2499223595

Inradius: r = 13.7510776405
Circumradius: R = 40.2499223595

Vertex coordinates: A[80.498844719; 0] B[0; 0] C[16.1099689438; 32.1999378876]
Centroid: CG[32.1999378876; 10.7333126292]
Coordinates of the circumscribed circle: U[40.2499223595; -0]
Coordinates of the inscribed circle: I[22.2499223595; 13.7510776405]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ B' = β' = 116.5655051177° = 116°33'54″ = 1.10771487178 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 = 36 ; ; b = 72 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 36**2+72**2 - 2 * 36 * 72 * cos(90° ) } ; ; c = 80.5 ; ;


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 = 36 ; ; b = 72 ; ; c = 80.5 ; ;

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

p = a+b+c = 36+72+80.5 = 188.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 188.5 }{ 2 } = 94.25 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.25 * (94.25-36)(94.25-72)(94.25-80.5) } ; ; T = sqrt{ 1679616 } = 1296 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1296 }{ 36 } = 72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1296 }{ 72 } = 36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1296 }{ 80.5 } = 32.2 ; ;

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{ 36**2-72**2-80.5**2 }{ 2 * 72 * 80.5 } ) = 26° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 72**2-36**2-80.5**2 }{ 2 * 36 * 80.5 } ) = 63° 26'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 80.5**2-36**2-72**2 }{ 2 * 72 * 36 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1296 }{ 94.25 } = 13.75 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 36 }{ 2 * sin 26° 33'54" } = 40.25 ; ;




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