Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 101   b = 65   c = 105.0132884628

Area: T = 3170.652152479
Perimeter: p = 271.0132884628
Semiperimeter: s = 135.5066442314

Angle ∠ A = α = 68.2821734543° = 68°16'54″ = 1.19217410867 rad
Angle ∠ B = β = 36.7188265457° = 36°43'6″ = 0.64108546278 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 62.78551787088
Height: hb = 97.55985084552
Height: hc = 60.38659523722

Median: ma = 71.24767751474
Median: mb = 97.76655510336
Median: mc = 66.75438277221

Inradius: r = 23.39985297721
Circumradius: R = 54.35986690455

Vertex coordinates: A[105.0132884628; 0] B[0; 0] C[80.96600936024; 60.38659523722]
Centroid: CG[61.99109927433; 20.12986507907]
Coordinates of the circumscribed circle: U[52.50664423138; 14.06990588154]
Coordinates of the inscribed circle: I[70.50664423138; 23.39985297721]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.7188265457° = 111°43'6″ = 1.19217410867 rad
∠ B' = β' = 143.2821734543° = 143°16'54″ = 0.64108546278 rad
∠ C' = γ' = 105° = 1.3098996939 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 = 101 ; ; b = 65 ; ; gamma = 75° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 101**2+65**2 - 2 * 101 * 65 * cos 75° } ; ; c = 105.01 ; ;


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 = 101 ; ; b = 65 ; ; c = 105.01 ; ;

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

p = a+b+c = 101+65+105.01 = 271.01 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 271.01 }{ 2 } = 135.51 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.51 * (135.51-101)(135.51-65)(135.51-105.01) } ; ; T = sqrt{ 10053031.09 } = 3170.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3170.65 }{ 101 } = 62.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3170.65 }{ 65 } = 97.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3170.65 }{ 105.01 } = 60.39 ; ;

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{ 65**2+105.01**2-101**2 }{ 2 * 65 * 105.01 } ) = 68° 16'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 101**2+105.01**2-65**2 }{ 2 * 101 * 105.01 } ) = 36° 43'6" ; ; gamma = 180° - alpha - beta = 180° - 68° 16'54" - 36° 43'6" = 75° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3170.65 }{ 135.51 } = 23.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 101 }{ 2 * sin 68° 16'54" } = 54.36 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 65**2+2 * 105.01**2 - 101**2 } }{ 2 } = 71.247 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 105.01**2+2 * 101**2 - 65**2 } }{ 2 } = 97.766 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 65**2+2 * 101**2 - 105.01**2 } }{ 2 } = 66.754 ; ;
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