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=8.3; b=10.7; c=4.076593068 and a=8.3; b=10.7; c=11.18876289319.

#1 Obtuse scalene triangle.

Sides: a = 8.3   b = 10.7   c = 4.076593068

Area: T = 15.28441880223
Perimeter: p = 23.076593068
Semiperimeter: s = 11.538796534

Angle ∠ A = α = 44.5° = 44°30' = 0.77766715171 rad
Angle ∠ B = β = 115.3677049386° = 115°22'1″ = 2.01435348601 rad
Angle ∠ C = γ = 20.13329506144° = 20°7'59″ = 0.35113862764 rad

Height: ha = 3.68329368728
Height: hb = 2.85768575743
Height: hc = 7.5499729128

Median: ma = 6.95219137979
Median: mb = 3.75988702364
Median: mc = 9.35661048131

Inradius: r = 1.32546865952
Circumradius: R = 5.92108805068

Vertex coordinates: A[4.076593068; 0] B[0; 0] C[-3.55658491259; 7.5499729128]
Centroid: CG[0.1733360518; 2.54999097093]
Coordinates of the circumscribed circle: U[2.038796534; 5.55990937434]
Coordinates of the inscribed circle: I[0.838796534; 1.32546865952]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5° = 135°30' = 0.77766715171 rad
∠ B' = β' = 64.63329506144° = 64°37'59″ = 2.01435348601 rad
∠ C' = γ' = 159.8677049386° = 159°52'1″ = 0.35113862764 rad




How did we calculate this triangle?

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 = 8.3 ; ; b = 10.7 ; ; c = 4.08 ; ;

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

p = a+b+c = 8.3+10.7+4.08 = 23.08 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.08 }{ 2 } = 11.54 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.54 * (11.54-8.3)(11.54-10.7)(11.54-4.08) } ; ; T = sqrt{ 233.61 } = 15.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.28 }{ 8.3 } = 3.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.28 }{ 10.7 } = 2.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.28 }{ 4.08 } = 7.5 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8.3**2-10.7**2-4.08**2 }{ 2 * 10.7 * 4.08 } ) = 44° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.7**2-8.3**2-4.08**2 }{ 2 * 8.3 * 4.08 } ) = 115° 22'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.08**2-8.3**2-10.7**2 }{ 2 * 10.7 * 8.3 } ) = 20° 7'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.28 }{ 11.54 } = 1.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.3 }{ 2 * sin 44° 30' } = 5.92 ; ;





#2 Acute scalene triangle.

Sides: a = 8.3   b = 10.7   c = 11.18876289319

Area: T = 41.9522093287
Perimeter: p = 30.18876289319
Semiperimeter: s = 15.09438144659

Angle ∠ A = α = 44.5° = 44°30' = 0.77766715171 rad
Angle ∠ B = β = 64.63329506144° = 64°37'59″ = 1.12880577935 rad
Angle ∠ C = γ = 70.86770493856° = 70°52'1″ = 1.2376863343 rad

Height: ha = 10.10989381414
Height: hb = 7.84215127639
Height: hc = 7.5499729128

Median: ma = 10.12993642722
Median: mb = 8.2710672316
Median: mc = 7.77216947779

Inradius: r = 2.77994228809
Circumradius: R = 5.92108805068

Vertex coordinates: A[11.18876289319; 0] B[0; 0] C[3.55658491259; 7.5499729128]
Centroid: CG[4.91444926859; 2.54999097093]
Coordinates of the circumscribed circle: U[5.59438144659; 1.94106353846]
Coordinates of the inscribed circle: I[4.39438144659; 2.77994228809]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5° = 135°30' = 0.77766715171 rad
∠ B' = β' = 115.3677049386° = 115°22'1″ = 1.12880577935 rad
∠ C' = γ' = 109.1332950614° = 109°7'59″ = 1.2376863343 rad

Calculate another triangle

How did we calculate this triangle?

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 = 8.3 ; ; b = 10.7 ; ; c = 11.19 ; ;

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

p = a+b+c = 8.3+10.7+11.19 = 30.19 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.19 }{ 2 } = 15.09 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.09 * (15.09-8.3)(15.09-10.7)(15.09-11.19) } ; ; T = sqrt{ 1759.98 } = 41.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.95 }{ 8.3 } = 10.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.95 }{ 10.7 } = 7.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.95 }{ 11.19 } = 7.5 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8.3**2-10.7**2-11.19**2 }{ 2 * 10.7 * 11.19 } ) = 44° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.7**2-8.3**2-11.19**2 }{ 2 * 8.3 * 11.19 } ) = 64° 37'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.19**2-8.3**2-10.7**2 }{ 2 * 10.7 * 8.3 } ) = 70° 52'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.95 }{ 15.09 } = 2.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.3 }{ 2 * sin 44° 30' } = 5.92 ; ; : Nr. 1




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