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 = 288.0076944361   b = 258   c = 128

Area: T = 16512
Perimeter: p = 674.0076944361
Semiperimeter: s = 337.003347218

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 63.61328854543° = 63°36'46″ = 1.11102542979 rad
Angle ∠ C = γ = 26.38771145457° = 26°23'14″ = 0.46105420289 rad

Height: ha = 114.6643901849
Height: hb = 128
Height: hc = 258

Median: ma = 144.003347218
Median: mb = 181.7287818454
Median: mc = 265.8199487623

Inradius: r = 48.99765278196
Circumradius: R = 144.003347218

Vertex coordinates: A[128; 0] B[0; 0] C[128; 258]
Centroid: CG[85.33333333333; 86]
Coordinates of the circumscribed circle: U[64; 129]
Coordinates of the inscribed circle: I[79.00334721804; 48.99765278196]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 116.3877114546° = 116°23'14″ = 1.11102542979 rad
∠ C' = γ' = 153.6132885454° = 153°36'46″ = 0.46105420289 rad

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How did we calculate this triangle?

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

b = 258 ; ; c = 128 ; ; 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{ 258**2+128**2 - 2 * 258 * 128 * cos 90° } ; ; a = 288.01 ; ;
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 = 288.01 ; ; b = 258 ; ; c = 128 ; ;

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

p = a+b+c = 288.01+258+128 = 674.01 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 674.01 }{ 2 } = 337 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 337 * (337-288.01)(337-258)(337-128) } ; ; T = sqrt{ 272646144 } = 16512 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16512 }{ 288.01 } = 114.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16512 }{ 258 } = 128 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16512 }{ 128 } = 258 ; ;

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{ 258**2+128**2-288.01**2 }{ 2 * 258 * 128 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 288.01**2+128**2-258**2 }{ 2 * 288.01 * 128 } ) = 63° 36'46" ; ; gamma = 180° - alpha - beta = 180° - 90° - 63° 36'46" = 26° 23'14" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16512 }{ 337 } = 49 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 288.01 }{ 2 * sin 90° } = 144 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 258**2+2 * 128**2 - 288.01**2 } }{ 2 } = 144.003 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 128**2+2 * 288.01**2 - 258**2 } }{ 2 } = 181.728 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 258**2+2 * 288.01**2 - 128**2 } }{ 2 } = 265.819 ; ;
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