Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 6.5   b = 6.5   c = 9.19223881554

Area: T = 21.125
Perimeter: p = 22.19223881554
Semiperimeter: s = 11.09661940777

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 6.5
Height: hb = 6.5
Height: hc = 4.59661940777

Median: ma = 7.26772209269
Median: mb = 7.26772209269
Median: mc = 4.59661940777

Inradius: r = 1.90438059223
Circumradius: R = 4.59661940777

Vertex coordinates: A[9.19223881554; 0] B[0; 0] C[4.59661940777; 4.59661940777]
Centroid: CG[4.59661940777; 1.53220646926]
Coordinates of the circumscribed circle: U[4.59661940777; -0]
Coordinates of the inscribed circle: I[4.59661940777; 1.90438059223]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 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 = 6.5 ; ; b = 6.5 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 6.5**2+6.5**2 - 2 * 6.5 * 6.5 * cos(90° ) } ; ; c = 9.19 ; ;


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 = 6.5 ; ; b = 6.5 ; ; c = 9.19 ; ;

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

p = a+b+c = 6.5+6.5+9.19 = 22.19 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.19 }{ 2 } = 11.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.1 * (11.1-6.5)(11.1-6.5)(11.1-9.19) } ; ; T = sqrt{ 446.27 } = 21.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.13 }{ 6.5 } = 6.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.13 }{ 6.5 } = 6.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.13 }{ 9.19 } = 4.6 ; ;

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{ 6.5**2-6.5**2-9.19**2 }{ 2 * 6.5 * 9.19 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.5**2-6.5**2-9.19**2 }{ 2 * 6.5 * 9.19 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.19**2-6.5**2-6.5**2 }{ 2 * 6.5 * 6.5 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.13 }{ 11.1 } = 1.9 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.5 }{ 2 * sin 45° } = 4.6 ; ;




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