Triangle calculator

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

You have entered side a, b and angle α.

Triangle has two solutions: a=984.2; b=4000; c=3241.957731666 and a=984.2; b=4000; c=4636.505470744.

#1 Obtuse scalene triangle.

Sides: a = 984.2   b = 4000   c = 3241.957731666

Area: T = 1125919.966022
Perimeter: p = 8226.157731666
Semiperimeter: s = 4113.079865833

Angle ∠ A = α = 10° = 0.17545329252 rad
Angle ∠ B = β = 135.1110361827° = 135°6'37″ = 2.35881206674 rad
Angle ∠ C = γ = 34.89896381728° = 34°53'23″ = 0.60989390609 rad

Height: ha = 2287.999016505
Height: hb = 562.9659980111
Height: hc = 694.5932710668

Median: ma = 3607.351099644
Median: mb = 1318.889909371
Median: mc = 2420.07329347

Inradius: r = 273.7411412152
Circumradius: R = 2833.891095476

Vertex coordinates: A[3241.957731666; 0] B[0; 0] C[-697.2743695393; 694.5932710668]
Centroid: CG[848.2287873754; 231.5310903556]
Coordinates of the circumscribed circle: U[1620.979865833; 2324.514417132]
Coordinates of the inscribed circle: I[113.0798658328; 273.7411412152]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170° = 0.17545329252 rad
∠ B' = β' = 44.89896381728° = 44°53'23″ = 2.35881206674 rad
∠ C' = γ' = 145.1110361827° = 145°6'37″ = 0.60989390609 rad


How did we calculate this triangle?

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

a = 984.2 ; ; b = 4000 ; ; alpha = 10° ; ;

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 984.2**2 = 4000**2 + c**2 - 2 * 4000 * c * cos 10° ; ; ; ; ; ; c**2 -7878.462c +15031350.36 =0 ; ; p=1; q=-7878.462; r=15031350.36 ; ; D = q**2 - 4pr = 7878.462**2 - 4 * 1 * 15031350.36 = 1944762.42515 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7878.46 ± sqrt{ 1944762.43 } }{ 2 } ; ;
c_{1,2} = 3939.23101205 ± 697.273695393 ; ; c_{1} = 4636.50470744 ; ; c_{2} = 3241.95731666 ; ; ; ; text{ Factored form: } ; ; (c -4636.50470744) (c -3241.95731666) = 0 ; ; ; ; c > 0 ; ; ; ; c = 4636.505 ; ;
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 = 984.2 ; ; b = 4000 ; ; c = 3241.96 ; ;

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

p = a+b+c = 984.2+4000+3241.96 = 8226.16 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8226.16 }{ 2 } = 4113.08 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4113.08 * (4113.08-984.2)(4113.08-4000)(4113.08-3241.96) } ; ; T = sqrt{ 1.268 * 10**{ 12 } } = 1125919.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1125919.96 }{ 984.2 } = 2287.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1125919.96 }{ 4000 } = 562.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1125919.96 }{ 3241.96 } = 694.59 ; ;

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{ 4000**2+3241.96**2-984.2**2 }{ 2 * 4000 * 3241.96 } ) = 10° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 984.2**2+3241.96**2-4000**2 }{ 2 * 984.2 * 3241.96 } ) = 135° 6'37" ; ;
 gamma = 180° - alpha - beta = 180° - 10° - 135° 6'37" = 34° 53'23" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1125919.96 }{ 4113.08 } = 273.74 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 984.2 }{ 2 * sin 10° } = 2833.89 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4000**2+2 * 3241.96**2 - 984.2**2 } }{ 2 } = 3607.351 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3241.96**2+2 * 984.2**2 - 4000**2 } }{ 2 } = 1318.889 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4000**2+2 * 984.2**2 - 3241.96**2 } }{ 2 } = 2420.073 ; ;



#2 Obtuse scalene triangle.

Sides: a = 984.2   b = 4000   c = 4636.505470744

Area: T = 1610241.186638
Perimeter: p = 9620.705470744
Semiperimeter: s = 4810.352235372

Angle ∠ A = α = 10° = 0.17545329252 rad
Angle ∠ B = β = 44.89896381728° = 44°53'23″ = 0.78334719861 rad
Angle ∠ C = γ = 125.1110361827° = 125°6'37″ = 2.18435877422 rad

Height: ha = 3272.183286199
Height: hb = 805.1210593192
Height: hc = 694.5932710668

Median: ma = 4301.910952265
Median: mb = 2689.408751302
Median: mc = 1763.528795398

Inradius: r = 334.7454955874
Circumradius: R = 2833.891095476

Vertex coordinates: A[4636.505470744; 0] B[0; 0] C[697.2743695393; 694.5932710668]
Centroid: CG[1777.926613428; 231.5310903556]
Coordinates of the circumscribed circle: U[2318.252235372; -1629.921146066]
Coordinates of the inscribed circle: I[810.3522353721; 334.7454955874]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170° = 0.17545329252 rad
∠ B' = β' = 135.1110361827° = 135°6'37″ = 0.78334719861 rad
∠ C' = γ' = 54.89896381728° = 54°53'23″ = 2.18435877422 rad

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How did we calculate this triangle?

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

a = 984.2 ; ; b = 4000 ; ; alpha = 10° ; ; : Nr. 1

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 984.2**2 = 4000**2 + c**2 - 2 * 4000 * c * cos 10° ; ; ; ; ; ; c**2 -7878.462c +15031350.36 =0 ; ; p=1; q=-7878.462; r=15031350.36 ; ; D = q**2 - 4pr = 7878.462**2 - 4 * 1 * 15031350.36 = 1944762.42515 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7878.46 ± sqrt{ 1944762.43 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 3939.23101205 ± 697.273695393 ; ; c_{1} = 4636.50470744 ; ; c_{2} = 3241.95731666 ; ; ; ; text{ Factored form: } ; ; (c -4636.50470744) (c -3241.95731666) = 0 ; ; ; ; c > 0 ; ; ; ; c = 4636.505 ; ; : Nr. 1
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 = 984.2 ; ; b = 4000 ; ; c = 4636.5 ; ;

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

p = a+b+c = 984.2+4000+4636.5 = 9620.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9620.7 }{ 2 } = 4810.35 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4810.35 * (4810.35-984.2)(4810.35-4000)(4810.35-4636.5) } ; ; T = sqrt{ 2.593 * 10**{ 12 } } = 1610241.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1610241.19 }{ 984.2 } = 3272.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1610241.19 }{ 4000 } = 805.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1610241.19 }{ 4636.5 } = 694.59 ; ;

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{ 4000**2+4636.5**2-984.2**2 }{ 2 * 4000 * 4636.5 } ) = 10° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 984.2**2+4636.5**2-4000**2 }{ 2 * 984.2 * 4636.5 } ) = 44° 53'23" ; ;
 gamma = 180° - alpha - beta = 180° - 10° - 44° 53'23" = 125° 6'37" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1610241.19 }{ 4810.35 } = 334.74 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 984.2 }{ 2 * sin 10° } = 2833.89 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4000**2+2 * 4636.5**2 - 984.2**2 } }{ 2 } = 4301.91 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4636.5**2+2 * 984.2**2 - 4000**2 } }{ 2 } = 2689.408 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4000**2+2 * 984.2**2 - 4636.5**2 } }{ 2 } = 1763.528 ; ;
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