Triangle calculator

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

You have entered side a, b and c.

Acute scalene triangle.

Sides: a = 1000   b = 820   c = 1090

Area: T = 391715.2599308
Perimeter: p = 2910
Semiperimeter: s = 1455

Angle ∠ A = α = 61.22549735797° = 61°13'30″ = 1.06985773734 rad
Angle ∠ B = β = 45.95108465095° = 45°57'3″ = 0.80219935657 rad
Angle ∠ C = γ = 72.82441799109° = 72°49'27″ = 1.27110217145 rad

Height: ha = 783.4310518617
Height: hb = 955.4033071484
Height: hc = 718.7443595061

Median: ma = 824.7732695959
Median: mb = 962.2632957824
Median: mc = 734.285536687

Inradius: r = 269.222010949
Circumradius: R = 570.4439865923

Vertex coordinates: A[1090; 0] B[0; 0] C[695.2755229358; 718.7443595061]
Centroid: CG[595.0921743119; 239.5811198354]
Coordinates of the circumscribed circle: U[545; 168.4543675041]
Coordinates of the inscribed circle: I[635; 269.222010949]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.775502642° = 118°46'30″ = 1.06985773734 rad
∠ B' = β' = 134.0499153491° = 134°2'57″ = 0.80219935657 rad
∠ C' = γ' = 107.1765820089° = 107°10'33″ = 1.27110217145 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and c.

a = 1000 ; ; b = 820 ; ; c = 1090 ; ;

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

p = a+b+c = 1000+820+1090 = 2910 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2910 }{ 2 } = 1455 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1455 * (1455-1000)(1455-820)(1455-1090) } ; ; T = sqrt{ 153440844375 } = 391715.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 391715.26 }{ 1000 } = 783.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 391715.26 }{ 820 } = 955.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 391715.26 }{ 1090 } = 718.74 ; ;

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{ 820**2+1090**2-1000**2 }{ 2 * 820 * 1090 } ) = 61° 13'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1000**2+1090**2-820**2 }{ 2 * 1000 * 1090 } ) = 45° 57'3" ; ; gamma = 180° - alpha - beta = 180° - 61° 13'30" - 45° 57'3" = 72° 49'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 391715.26 }{ 1455 } = 269.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1000 }{ 2 * sin 61° 13'30" } = 570.44 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 820**2+2 * 1090**2 - 1000**2 } }{ 2 } = 824.773 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1090**2+2 * 1000**2 - 820**2 } }{ 2 } = 962.263 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 820**2+2 * 1000**2 - 1090**2 } }{ 2 } = 734.285 ; ;
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