Triangle calculator

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

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 311.2621947562   b = 140   c = 278

Area: T = 19460
Perimeter: p = 729.2621947562
Semiperimeter: s = 364.6310973781

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 26.73296937685° = 26°43'47″ = 0.46765211643 rad
Angle ∠ C = γ = 63.27703062315° = 63°16'13″ = 1.10442751625 rad

Height: ha = 125.0399376978
Height: hb = 278
Height: hc = 140

Median: ma = 155.6310973781
Median: mb = 286.6787519174
Median: mc = 197.2844059163

Inradius: r = 53.36990262191
Circumradius: R = 155.6310973781

Vertex coordinates: A[278; 0] B[0; 0] C[278; 140]
Centroid: CG[185.3333333333; 46.66766666667]
Coordinates of the circumscribed circle: U[139; 70]
Coordinates of the inscribed circle: I[224.6310973781; 53.36990262191]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 153.2770306232° = 153°16'13″ = 0.46765211643 rad
∠ C' = γ' = 116.7329693768° = 116°43'47″ = 1.10442751625 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 140 ; ; c = 278 ; ; alpha = 90° ; ;

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

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 140**2+278**2 - 2 * 140 * 278 * cos 90° } ; ; a = 311.26 ; ;
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 = 311.26 ; ; b = 140 ; ; c = 278 ; ;

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

p = a+b+c = 311.26+140+278 = 729.26 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 729.26 }{ 2 } = 364.63 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 364.63 * (364.63-311.26)(364.63-140)(364.63-278) } ; ; T = sqrt{ 378691600 } = 19460 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19460 }{ 311.26 } = 125.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19460 }{ 140 } = 278 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19460 }{ 278 } = 140 ; ;

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{ 140**2+278**2-311.26**2 }{ 2 * 140 * 278 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 311.26**2+278**2-140**2 }{ 2 * 311.26 * 278 } ) = 26° 43'47" ; ; gamma = 180° - alpha - beta = 180° - 90° - 26° 43'47" = 63° 16'13" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19460 }{ 364.63 } = 53.37 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 311.26 }{ 2 * sin 90° } = 155.63 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 140**2+2 * 278**2 - 311.26**2 } }{ 2 } = 155.631 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 278**2+2 * 311.26**2 - 140**2 } }{ 2 } = 286.678 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 140**2+2 * 311.26**2 - 278**2 } }{ 2 } = 197.284 ; ;
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