Triangle calculator

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

You have entered side a, b and angle γ.

Acute scalene triangle.

Sides: a = 3400   b = 2400   c = 3148.154422816

Area: T = 3635306.619869
Perimeter: p = 8948.154422816
Semiperimeter: s = 4474.077711408

Angle ∠ A = α = 74.21441457956° = 74°12'51″ = 1.29552811957 rad
Angle ∠ B = β = 42.78658542044° = 42°47'9″ = 0.74767540291 rad
Angle ∠ C = γ = 63° = 1.10995574288 rad

Height: ha = 2138.416565805
Height: hb = 3029.422218224
Height: hc = 2309.48444504

Median: ma = 2223.83439691
Median: mb = 3048.842199691
Median: mc = 2486.419936104

Inradius: r = 812.5276589506
Circumradius: R = 1766.628804519

Vertex coordinates: A[3148.154422816; 0] B[0; 0] C[2495.25218056; 2309.48444504]
Centroid: CG[1881.135534459; 769.8288150133]
Coordinates of the circumscribed circle: U[1574.077711408; 802.0322349089]
Coordinates of the inscribed circle: I[2074.077711408; 812.5276589506]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.7865854204° = 105°47'9″ = 1.29552811957 rad
∠ B' = β' = 137.2144145796° = 137°12'51″ = 0.74767540291 rad
∠ C' = γ' = 117° = 1.10995574288 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 3400 ; ; b = 2400 ; ; gamma = 63° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 3400**2+2400**2 - 2 * 3400 * 2400 * cos 63° } ; ; c = 3148.15 ; ;


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 = 3400 ; ; b = 2400 ; ; c = 3148.15 ; ;

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

p = a+b+c = 3400+2400+3148.15 = 8948.15 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8948.15 }{ 2 } = 4474.08 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4474.08 * (4474.08-3400)(4474.08-2400)(4474.08-3148.15) } ; ; T = sqrt{ 1.322 * 10**{ 13 } } = 3635306.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3635306.62 }{ 3400 } = 2138.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3635306.62 }{ 2400 } = 3029.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3635306.62 }{ 3148.15 } = 2309.48 ; ;

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{ 2400**2+3148.15**2-3400**2 }{ 2 * 2400 * 3148.15 } ) = 74° 12'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3400**2+3148.15**2-2400**2 }{ 2 * 3400 * 3148.15 } ) = 42° 47'9" ; ; gamma = 180° - alpha - beta = 180° - 74° 12'51" - 42° 47'9" = 63° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3635306.62 }{ 4474.08 } = 812.53 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3400 }{ 2 * sin 74° 12'51" } = 1766.63 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2400**2+2 * 3148.15**2 - 3400**2 } }{ 2 } = 2223.834 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3148.15**2+2 * 3400**2 - 2400**2 } }{ 2 } = 3048.842 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2400**2+2 * 3400**2 - 3148.15**2 } }{ 2 } = 2486.419 ; ;
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