Triangle calculator

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

You have entered side a, angle β and angle γ.

Right scalene triangle.

Sides: a = 100   b = 1.74655064928   c = 100.0155232804

Area: T = 87.27553246411
Perimeter: p = 201.7610739297
Semiperimeter: s = 100.8880369649

Angle ∠ A = α = 89° = 1.55333430343 rad
Angle ∠ B = β = 1° = 0.01774532925 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.74655064928
Height: hb = 100
Height: hc = 1.74552406437

Median: ma = 50.03304586519
Median: mb = 100.0043808419
Median: mc = 50.00876164022

Inradius: r = 0.86551368442
Circumradius: R = 50.00876164022

Vertex coordinates: A[100.0155232804; 0] B[0; 0] C[99.98547695156; 1.74552406437]
Centroid: CG[66.667666744; 0.58217468812]
Coordinates of the circumscribed circle: U[50.00876164022; -0]
Coordinates of the inscribed circle: I[99.13548631558; 0.86551368442]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 91° = 1.55333430343 rad
∠ B' = β' = 179° = 0.01774532925 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

a = 100 ; ; beta = 1° ; ; gamma = 90° ; ;

2. From angle β and angle γ we calculate angle α:

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 1 ° - 90 ° = 89 ° ; ;

3. From angle β, angle α and side a we calculate side b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 100 * fraction{ sin 1° }{ sin 89° } = 1.75 ; ;

4. From angle γ, angle α and side a we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 100 * fraction{ sin 90° }{ sin 89° } = 100.02 ; ;
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 = 100 ; ; b = 1.75 ; ; c = 100.02 ; ;

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

p = a+b+c = 100+1.75+100.02 = 201.76 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.76 }{ 2 } = 100.88 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.88 * (100.88-100)(100.88-1.75)(100.88-100.02) } ; ; T = sqrt{ 7616.98 } = 87.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 87.28 }{ 100 } = 1.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 87.28 }{ 1.75 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 87.28 }{ 100.02 } = 1.75 ; ;

9. 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{ 1.75**2+100.02**2-100**2 }{ 2 * 1.75 * 100.02 } ) = 89° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+100.02**2-1.75**2 }{ 2 * 100 * 100.02 } ) = 1° ; ; gamma = 180° - alpha - beta = 180° - 89° - 1° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 87.28 }{ 100.88 } = 0.87 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 89° } = 50.01 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.75**2+2 * 100.02**2 - 100**2 } }{ 2 } = 50.03 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100.02**2+2 * 100**2 - 1.75**2 } }{ 2 } = 100.004 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.75**2+2 * 100**2 - 100.02**2 } }{ 2 } = 50.008 ; ;
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