Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=3.92223491461; b=49; c=48 and a=24.73300779167; b=49; c=48.

#1 Obtuse scalene triangle.

Sides: a = 3.92223491461   b = 49   c = 48

Area: T = 91.8998548003
Perimeter: p = 100.9222349146
Semiperimeter: s = 50.46111745731

Angle ∠ A = α = 4.48219495841° = 4°28'55″ = 0.07882247772 rad
Angle ∠ B = β = 102.5188050416° = 102°31'5″ = 1.78992775225 rad
Angle ∠ C = γ = 73° = 1.2744090354 rad

Height: ha = 46.85989330422
Height: hb = 3.7510961143
Height: hc = 3.82991061668

Median: ma = 48.46329115334
Median: mb = 23.65325349891
Median: mc = 25.14334367462

Inradius: r = 1.82111733829
Circumradius: R = 25.09766021557

Vertex coordinates: A[48; 0] B[0; 0] C[-0.85501580956; 3.82991061668]
Centroid: CG[15.71766139681; 1.27663687223]
Coordinates of the circumscribed circle: U[24; 7.3387536355]
Coordinates of the inscribed circle: I[1.46111745731; 1.82111733829]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 175.5188050416° = 175°31'5″ = 0.07882247772 rad
∠ B' = β' = 77.48219495841° = 77°28'55″ = 1.78992775225 rad
∠ C' = γ' = 107° = 1.2744090354 rad




How did we calculate this triangle?

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

b = 49 ; ; c = 48 ; ; gamma = 73° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 48**2 = 49**2 + a**2 - 2 * 49 * a * cos 73° ; ; ; ; ; ; a**2 -28.652a +97 =0 ; ; p=1; q=-28.652; r=97 ; ; D = q**2 - 4pr = 28.652**2 - 4 * 1 * 97 = 432.961576591 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 28.65 ± sqrt{ 432.96 } }{ 2 } ; ; a_{1,2} = 14.32621353 ± 10.4038643853 ; ; a_{1} = 24.7300779153 ; ; a_{2} = 3.9223491447 ; ; ; ; text{ Factored form: } ; ; (a -24.7300779153) (a -3.9223491447) = 0 ; ; ; ; a > 0 ; ; ; ; a = 24.73 ; ;


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 = 3.92 ; ; b = 49 ; ; c = 48 ; ;

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

p = a+b+c = 3.92+49+48 = 100.92 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 100.92 }{ 2 } = 50.46 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 50.46 * (50.46-3.92)(50.46-49)(50.46-48) } ; ; T = sqrt{ 8445.34 } = 91.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 91.9 }{ 3.92 } = 46.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 91.9 }{ 49 } = 3.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 91.9 }{ 48 } = 3.83 ; ;

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{ 49**2+48**2-3.92**2 }{ 2 * 49 * 48 } ) = 4° 28'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.92**2+48**2-49**2 }{ 2 * 3.92 * 48 } ) = 102° 31'5" ; ; gamma = 180° - alpha - beta = 180° - 4° 28'55" - 102° 31'5" = 73° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 91.9 }{ 50.46 } = 1.82 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.92 }{ 2 * sin 4° 28'55" } = 25.1 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 48**2 - 3.92**2 } }{ 2 } = 48.463 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 3.92**2 - 49**2 } }{ 2 } = 23.653 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 3.92**2 - 48**2 } }{ 2 } = 25.143 ; ;







#2 Acute scalene triangle.

Sides: a = 24.73300779167   b = 49   c = 48

Area: T = 579.4132532614
Perimeter: p = 121.7330077917
Semiperimeter: s = 60.86550389584

Angle ∠ A = α = 29.51880504159° = 29°31'5″ = 0.51551871685 rad
Angle ∠ B = β = 77.48219495841° = 77°28'55″ = 1.35223151311 rad
Angle ∠ C = γ = 73° = 1.2744090354 rad

Height: ha = 46.85989330422
Height: hb = 23.64994911271
Height: hc = 24.14221888589

Median: ma = 46.98999553471
Median: mb = 29.28437561949
Median: mc = 30.50106291227

Inradius: r = 9.52196280579
Circumradius: R = 25.09766021557

Vertex coordinates: A[48; 0] B[0; 0] C[5.36601745184; 24.14221888589]
Centroid: CG[17.78767248395; 8.04773962863]
Coordinates of the circumscribed circle: U[24; 7.3387536355]
Coordinates of the inscribed circle: I[11.86550389584; 9.52196280579]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.4821949584° = 150°28'55″ = 0.51551871685 rad
∠ B' = β' = 102.5188050416° = 102°31'5″ = 1.35223151311 rad
∠ C' = γ' = 107° = 1.2744090354 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 49 ; ; c = 48 ; ; gamma = 73° ; ; : Nr. 1

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 48**2 = 49**2 + a**2 - 2 * 49 * a * cos 73° ; ; ; ; ; ; a**2 -28.652a +97 =0 ; ; p=1; q=-28.652; r=97 ; ; D = q**2 - 4pr = 28.652**2 - 4 * 1 * 97 = 432.961576591 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 28.65 ± sqrt{ 432.96 } }{ 2 } ; ; a_{1,2} = 14.32621353 ± 10.4038643853 ; ; a_{1} = 24.7300779153 ; ; a_{2} = 3.9223491447 ; ; ; ; text{ Factored form: } ; ; (a -24.7300779153) (a -3.9223491447) = 0 ; ; ; ; a > 0 ; ; ; ; a = 24.73 ; ; : 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 = 24.73 ; ; b = 49 ; ; c = 48 ; ;

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

p = a+b+c = 24.73+49+48 = 121.73 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 121.73 }{ 2 } = 60.87 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.87 * (60.87-24.73)(60.87-49)(60.87-48) } ; ; T = sqrt{ 335718.88 } = 579.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 579.41 }{ 24.73 } = 46.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 579.41 }{ 49 } = 23.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 579.41 }{ 48 } = 24.14 ; ;

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{ 49**2+48**2-24.73**2 }{ 2 * 49 * 48 } ) = 29° 31'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.73**2+48**2-49**2 }{ 2 * 24.73 * 48 } ) = 77° 28'55" ; ; gamma = 180° - alpha - beta = 180° - 29° 31'5" - 77° 28'55" = 73° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 579.41 }{ 60.87 } = 9.52 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.73 }{ 2 * sin 29° 31'5" } = 25.1 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 48**2 - 24.73**2 } }{ 2 } = 46.9 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 24.73**2 - 49**2 } }{ 2 } = 29.284 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 24.73**2 - 48**2 } }{ 2 } = 30.501 ; ;
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