Triangle calculator

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

You have entered side a, b and height hb.

Triangle has two solutions: a=0.013; b=0.016; c=0.02441867732 and a=0.013; b=0.016; c=0.01662788206.

#1 Obtuse scalene triangle.

Sides: a = 0.013   b = 0.016   c = 0.02441867732

Area: T = 09.6E-5
Perimeter: p = 0.05331867732
Semiperimeter: s = 0.02765933866

Angle ∠ A = α = 29.74548812969° = 29°44'42″ = 0.51991461142 rad
Angle ∠ B = β = 37.6355253755° = 37°38'7″ = 0.65768590928 rad
Angle ∠ C = γ = 112.6219864948° = 112°37'11″ = 1.96655874465 rad

Height: ha = 0.01547692308
Height: hb = 0.012
Height: hc = 0.0087938223

Median: ma = 0.01994486503
Median: mb = 0.0187691806
Median: mc = 0.00881394103

Inradius: r = 0.00436099201
Circumradius: R = 0.01331011688

Vertex coordinates: A[0.02441867732; 0] B[0; 0] C[0.0110294883; 0.0087938223]
Centroid: CG[0.01114938854; 0.00326460743]
Coordinates of the circumscribed circle: U[0.01220933866; -0.00550389111]
Coordinates of the inscribed circle: I[0.01105933866; 0.00436099201]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.2555118703° = 150°15'18″ = 0.51991461142 rad
∠ B' = β' = 142.3654746245° = 142°21'53″ = 0.65768590928 rad
∠ C' = γ' = 67.3880135052° = 67°22'49″ = 1.96655874465 rad


How did we calculate this triangle?

1. Input data entered: side a, b and height hb.

a = 0.013 ; ; b = 0.016 ; ; h_b = 0.012 ; ;

2. From side b we calculate T:

T = fraction{ b h_b }{ 2 } ; ; ; ; T = fraction{ 0.016 * 0.012 }{ 2 } = 9.6 * 10**{ -5 } ; ;
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 = 0.01 ; ; b = 0.02 ; ; c = 0.02 ; ;

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

p = a+b+c = 0.01+0.02+0.02 = 0.05 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 0.05 }{ 2 } = 0.03 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.03 * (0.03-0.01)(0.03-0.02)(0.03-0.02) } ; ; T = sqrt{ 0 } = 0 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0 }{ 0.01 } = 0.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0 }{ 0.02 } = 0.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0 }{ 0.02 } = 0.01 ; ;

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{ 0.02**2+0.02**2-0.01**2 }{ 2 * 0.02 * 0.02 } ) = 29° 44'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.01**2+0.02**2-0.02**2 }{ 2 * 0.01 * 0.02 } ) = 37° 38'7" ; ;
gamma = 180° - alpha - beta = 180° - 29° 44'42" - 37° 38'7" = 112° 37'11" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0 }{ 0.03 } = 0 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 0.01 }{ 2 * sin 29° 44'42" } = 0.01 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.02**2+2 * 0.02**2 - 0.01**2 } }{ 2 } = 0.019 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.02**2+2 * 0.01**2 - 0.02**2 } }{ 2 } = 0.018 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.02**2+2 * 0.01**2 - 0.02**2 } }{ 2 } = 0.008 ; ;



#2 Acute scalene triangle.

Sides: a = 0.013   b = 0.016   c = 0.01662788206

Area: T = 09.6E-5
Perimeter: p = 0.04552788206
Semiperimeter: s = 0.02326394103

Angle ∠ A = α = 47.4989552922° = 47°29'22″ = 0.82988490588 rad
Angle ∠ B = β = 65.1330312026° = 65°7'49″ = 1.13767383877 rad
Angle ∠ C = γ = 67.3880135052° = 67°22'49″ = 1.17660052071 rad

Height: ha = 0.01547692308
Height: hb = 0.012
Height: hc = 0.01217944662

Median: ma = 0.01547732867
Median: mb = 0.01223693169
Median: mc = 0.01220933866

Inradius: r = 0.00442403931
Circumradius: R = 0.00988176945

Vertex coordinates: A[0.01662788206; 0] B[0; 0] C[0.00554672265; 0.01217944662]
Centroid: CG[0.00772486824; 0.00439314887]
Coordinates of the circumscribed circle: U[0.00881394103; 0.0033391421]
Coordinates of the inscribed circle: I[0.00766394103; 0.00442403931]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.5110447078° = 132°30'38″ = 0.82988490588 rad
∠ B' = β' = 114.8769687974° = 114°52'11″ = 1.13767383877 rad
∠ C' = γ' = 112.6219864948° = 112°37'11″ = 1.17660052071 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and height hb.

a = 0.013 ; ; b = 0.016 ; ; h_b = 0.012 ; ; : Nr. 1

2. From side b we calculate T:

T = fraction{ b h_b }{ 2 } ; ; ; ; T = fraction{ 0.016 * 0.012 }{ 2 } = 9.6 * 10**{ -5 } ; ; : 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 = 0.01 ; ; b = 0.02 ; ; c = 0.02 ; ; : Nr. 1

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

p = a+b+c = 0.01+0.02+0.02 = 0.05 ; ; : Nr. 1

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 0.05 }{ 2 } = 0.02 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.02 * (0.02-0.01)(0.02-0.02)(0.02-0.02) } ; ; T = sqrt{ 0 } = 0 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0 }{ 0.01 } = 0.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0 }{ 0.02 } = 0.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0 }{ 0.02 } = 0.01 ; ; : Nr. 1

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{ 0.02**2+0.02**2-0.01**2 }{ 2 * 0.02 * 0.02 } ) = 47° 29'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.01**2+0.02**2-0.02**2 }{ 2 * 0.01 * 0.02 } ) = 65° 7'49" ; ;
gamma = 180° - alpha - beta = 180° - 47° 29'22" - 65° 7'49" = 67° 22'49" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0 }{ 0.02 } = 0 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 0.01 }{ 2 * sin 47° 29'22" } = 0.01 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.02**2+2 * 0.02**2 - 0.01**2 } }{ 2 } = 0.015 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.02**2+2 * 0.01**2 - 0.02**2 } }{ 2 } = 0.012 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.02**2+2 * 0.01**2 - 0.02**2 } }{ 2 } = 0.012 ; ;
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