Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 67.5   b = 45.5   c = 81.40333168857

Area: T = 1535.625
Perimeter: p = 194.4033316886
Semiperimeter: s = 97.20216584429

Angle ∠ A = α = 56.01771116188° = 56°1'2″ = 0.97876830352 rad
Angle ∠ B = β = 33.98328883812° = 33°58'58″ = 0.59331132916 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 45.5
Height: hb = 67.5
Height: hc = 37.72988066076

Median: ma = 56.65107943457
Median: mb = 71.23106991402
Median: mc = 40.70216584429

Inradius: r = 15.79883415571
Circumradius: R = 40.70216584429

Vertex coordinates: A[81.40333168857; 0] B[0; 0] C[55.97113065058; 37.72988066076]
Centroid: CG[45.79215411305; 12.57662688692]
Coordinates of the circumscribed circle: U[40.70216584429; 0]
Coordinates of the inscribed circle: I[51.70216584429; 15.79883415571]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.9832888381° = 123°58'58″ = 0.97876830352 rad
∠ B' = β' = 146.0177111619° = 146°1'2″ = 0.59331132916 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 = 67.5 ; ; b = 45.5 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 67.5**2+45.5**2 - 2 * 67.5 * 45.5 * cos 90° } ; ; c = 81.4 ; ;
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 = 67.5 ; ; b = 45.5 ; ; c = 81.4 ; ;

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

p = a+b+c = 67.5+45.5+81.4 = 194.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 194.4 }{ 2 } = 97.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 97.2 * (97.2-67.5)(97.2-45.5)(97.2-81.4) } ; ; T = sqrt{ 2358144.14 } = 1535.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1535.63 }{ 67.5 } = 45.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1535.63 }{ 45.5 } = 67.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1535.63 }{ 81.4 } = 37.73 ; ;

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{ 45.5**2+81.4**2-67.5**2 }{ 2 * 45.5 * 81.4 } ) = 56° 1'2" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 67.5**2+81.4**2-45.5**2 }{ 2 * 67.5 * 81.4 } ) = 33° 58'58" ; ;
 gamma = 180° - alpha - beta = 180° - 56° 1'2" - 33° 58'58" = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1535.63 }{ 97.2 } = 15.8 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 67.5 }{ 2 * sin 56° 1'2" } = 40.7 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.5**2+2 * 81.4**2 - 67.5**2 } }{ 2 } = 56.651 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 81.4**2+2 * 67.5**2 - 45.5**2 } }{ 2 } = 71.231 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.5**2+2 * 67.5**2 - 81.4**2 } }{ 2 } = 40.702 ; ;
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