Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 45   b = 45   c = 34.44215089129

Area: T = 715.9465615951
Perimeter: p = 124.4421508913
Semiperimeter: s = 62.22107544564

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 = 31.82198051534
Height: hb = 31.82198051534
Height: hc = 41.5754578963

Median: ma = 33.15765795597
Median: mb = 33.15765795597
Median: mc = 41.5754578963

Inradius: r = 11.50765402566
Circumradius: R = 24.35438245066

Vertex coordinates: A[34.44215089129; 0] B[0; 0] C[17.22107544564; 41.5754578963]
Centroid: CG[17.22107544564; 13.85881929877]
Coordinates of the circumscribed circle: U[17.22107544564; 17.22107544564]
Coordinates of the inscribed circle: I[17.22107544564; 11.50765402566]

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 = 45 ; ; b = 45 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 45**2+45**2 - 2 * 45 * 45 * cos 45° } ; ; c = 34.44 ; ;


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 = 45 ; ; b = 45 ; ; c = 34.44 ; ;

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

p = a+b+c = 45+45+34.44 = 124.44 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 124.44 }{ 2 } = 62.22 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.22 * (62.22-45)(62.22-45)(62.22-34.44) } ; ; T = sqrt{ 512578.13 } = 715.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 715.95 }{ 45 } = 31.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 715.95 }{ 45 } = 31.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 715.95 }{ 34.44 } = 41.57 ; ;

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{ 45**2+34.44**2-45**2 }{ 2 * 45 * 34.44 } ) = 67° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+34.44**2-45**2 }{ 2 * 45 * 34.44 } ) = 67° 30' ; ; gamma = 180° - alpha - beta = 180° - 67° 30' - 67° 30' = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 715.95 }{ 62.22 } = 11.51 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 67° 30' } = 24.35 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 34.44**2 - 45**2 } }{ 2 } = 33.157 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.44**2+2 * 45**2 - 45**2 } }{ 2 } = 33.157 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 45**2 - 34.44**2 } }{ 2 } = 41.575 ; ;
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