Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, c and angle β.

Acute scalene triangle.

Sides: a = 125   b = 87.43435383235   c = 140

Area: T = 5387.03879091
Perimeter: p = 352.4343538323
Semiperimeter: s = 176.2176769162

Angle ∠ A = α = 61.66546816852° = 61°39'53″ = 1.07662517276 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 80.33553183148° = 80°20'7″ = 1.40221158102 rad

Height: ha = 86.19326065456
Height: hb = 123.2265892773
Height: hc = 76.95876844157

Median: ma = 98.57700857861
Median: mb = 125.3055004266
Median: mc = 82.06658992023

Inradius: r = 30.57105179747
Circumradius: R = 71.00878036613

Vertex coordinates: A[140; 0] B[0; 0] C[98.50113442008; 76.95876844157]
Centroid: CG[79.55004480669; 25.65325614719]
Coordinates of the circumscribed circle: U[70; 11.92109135892]
Coordinates of the inscribed circle: I[88.78332308383; 30.57105179747]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.3355318315° = 118°20'7″ = 1.07662517276 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 99.66546816852° = 99°39'53″ = 1.40221158102 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 125 ; ; c = 140 ; ; beta = 38° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 125**2+140**2 - 2 * 125 * 140 * cos 38° } ; ; b = 87.43 ; ;
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 = 125 ; ; b = 87.43 ; ; c = 140 ; ;

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

p = a+b+c = 125+87.43+140 = 352.43 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 352.43 }{ 2 } = 176.22 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 176.22 * (176.22-125)(176.22-87.43)(176.22-140) } ; ; T = sqrt{ 29020177.43 } = 5387.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5387.04 }{ 125 } = 86.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5387.04 }{ 87.43 } = 123.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5387.04 }{ 140 } = 76.96 ; ;

7. 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{ 87.43**2+140**2-125**2 }{ 2 * 87.43 * 140 } ) = 61° 39'53" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+140**2-87.43**2 }{ 2 * 125 * 140 } ) = 38° ; ;
gamma = 180° - alpha - beta = 180° - 61° 39'53" - 38° = 80° 20'7" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5387.04 }{ 176.22 } = 30.57 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 125 }{ 2 * sin 61° 39'53" } = 71.01 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 87.43**2+2 * 140**2 - 125**2 } }{ 2 } = 98.57 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 140**2+2 * 125**2 - 87.43**2 } }{ 2 } = 125.305 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 87.43**2+2 * 125**2 - 140**2 } }{ 2 } = 82.066 ; ;
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