Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 150   b = 150   c = 212.1322034356

Area: T = 11250
Perimeter: p = 512.1322034356
Semiperimeter: s = 256.0666017178

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

Height: ha = 150
Height: hb = 150
Height: hc = 106.0666017178

Median: ma = 167.7055098313
Median: mb = 167.7055098313
Median: mc = 106.0666017178

Inradius: r = 43.9343982822
Circumradius: R = 106.0666017178

Vertex coordinates: A[212.1322034356; 0] B[0; 0] C[106.0666017178; 106.0666017178]
Centroid: CG[106.0666017178; 35.35553390593]
Coordinates of the circumscribed circle: U[106.0666017178; 0]
Coordinates of the inscribed circle: I[106.0666017178; 43.9343982822]

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 = 150 ; ; b = 150 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 150**2+150**2 - 2 * 150 * 150 * cos 90° } ; ; c = 212.13 ; ;
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 = 150 ; ; b = 150 ; ; c = 212.13 ; ;

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

p = a+b+c = 150+150+212.13 = 512.13 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 512.13 }{ 2 } = 256.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 256.07 * (256.07-150)(256.07-150)(256.07-212.13) } ; ; T = sqrt{ 126562500 } = 11250 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11250 }{ 150 } = 150 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11250 }{ 150 } = 150 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11250 }{ 212.13 } = 106.07 ; ;

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{ 150**2+212.13**2-150**2 }{ 2 * 150 * 212.13 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 150**2+212.13**2-150**2 }{ 2 * 150 * 212.13 } ) = 45° ; ;
gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11250 }{ 256.07 } = 43.93 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 150 }{ 2 * sin 45° } = 106.07 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 150**2+2 * 212.13**2 - 150**2 } }{ 2 } = 167.705 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 212.13**2+2 * 150**2 - 150**2 } }{ 2 } = 167.705 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 150**2+2 * 150**2 - 212.13**2 } }{ 2 } = 106.066 ; ;
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