Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute isosceles triangle.

Sides: a = 3.7   b = 3.7   c = 2.53109490606

Area: T = 4.43998811883
Perimeter: p = 9.93109490606
Semiperimeter: s = 4.96554745303

Angle ∠ A = α = 70° = 1.22217304764 rad
Angle ∠ B = β = 70° = 1.22217304764 rad
Angle ∠ C = γ = 40° = 0.69881317008 rad

Height: ha = 2.37883141558
Height: hb = 2.37883141558
Height: hc = 3.47768626969

Median: ma = 2.57439758301
Median: mb = 2.57439758301
Median: mc = 3.47768626969

Inradius: r = 0.88660948055
Circumradius: R = 1.96987288791

Vertex coordinates: A[2.53109490606; 0] B[0; 0] C[1.26554745303; 3.47768626969]
Centroid: CG[1.26554745303; 1.15989542323]
Coordinates of the circumscribed circle: U[1.26554745303; 1.50881338178]
Coordinates of the inscribed circle: I[1.26554745303; 0.88660948055]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110° = 1.22217304764 rad
∠ B' = β' = 110° = 1.22217304764 rad
∠ C' = γ' = 140° = 0.69881317008 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 3.7 ; ; b = 3.7 ; ; beta = 70° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.7**2 = 3.7**2 + c**2 -2 * 3.7 * c * cos (70° ) ; ; ; ; c**2 -2.531c =0 ; ; p=1; q=-2.531; r=0 ; ; D = q**2 - 4pr = 2.531**2 - 4 * 1 * 0 = 6.4057031474 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 2.53 ± sqrt{ 6.41 } }{ 2 } ; ; c_{1,2} = 1.26547453 ± 1.2654745303 ; ; c_{1} = 2.5309490603 ; ; c_{2} = -3.04974712151E-10 ; ;
 ; ; text{ Factored form: } ; ; (c -2.5309490603) (c +3.04974712151E-10) = 0 ; ; ; ; c>0 ; ;


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 = 3.7 ; ; b = 3.7 ; ; c = 2.53 ; ;

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

p = a+b+c = 3.7+3.7+2.53 = 9.93 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9.93 }{ 2 } = 4.97 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.97 * (4.97-3.7)(4.97-3.7)(4.97-2.53) } ; ; T = sqrt{ 19.36 } = 4.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.4 }{ 3.7 } = 2.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.4 }{ 3.7 } = 2.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.4 }{ 2.53 } = 3.48 ; ;

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{ 3.7**2+2.53**2-3.7**2 }{ 2 * 3.7 * 2.53 } ) = 70° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.7**2+2.53**2-3.7**2 }{ 2 * 3.7 * 2.53 } ) = 70° ; ; gamma = 180° - alpha - beta = 180° - 70° - 70° = 40° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.4 }{ 4.97 } = 0.89 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.7 }{ 2 * sin 70° } = 1.97 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 2.53**2 - 3.7**2 } }{ 2 } = 2.574 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.53**2+2 * 3.7**2 - 3.7**2 } }{ 2 } = 2.574 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.7**2+2 * 3.7**2 - 2.53**2 } }{ 2 } = 3.477 ; ;
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