Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=94.51550067534 and with side c=46.55334537968

#1 Acute scalene triangle.

Sides: a = 120   b = 100   c = 94.51550067534

Area: T = 4587.85548012158
Perimeter: p = 314.51550067534
Semiperimeter: s = 157.25875033767

Angle ∠ A = α = 76.12548052335° = 76°7'29″ = 1.32986284938 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 49.87551947665° = 49°52'31″ = 0.87704863637 rad

Height: ha = 76.46442466869
Height: hb = 91.75770960243
Height: hc = 97.0822039325

Median: ma = 76.59333629683
Median: mb = 95.74220662551
Median: mc = 99.83435032672

Inradius: r = 29.17441551449
Circumradius: R = 61.8033398875

Vertex coordinates: A[94.51550067534; 0] B[0; 0] C[70.53442302751; 97.0822039325]
Centroid: CG[55.01664123428; 32.3610679775]
Coordinates of the circumscribed circle: U[47.25875033767; 39.82994926795]
Coordinates of the inscribed circle: I[57.25875033767; 29.17441551449]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 103.87551947665° = 103°52'31″ = 1.32986284938 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 130.12548052335° = 130°7'29″ = 0.87704863637 rad

How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other triangle characteristics, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines

a = 120 \ \\ b = 100 \ \\ β = 54\degree \ \\ \ \\ b^2 = a^2 + c^2 - 2ac \cos β \ \\ 100^2 = 120^2 + c^2 -2 \cdot \ 120 \cdot \ c \cdot \ \cos 54\degree \ \\ \ \\ $mXz_{9}pQ_{8}Cc \ \\ \ \\ c>0

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides SSS.
a=120 b=100 c=94.52

2. The triangle perimeter is the sum of the lengths of its three sides

p=a+b+c=120+100+94.52=314.52

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles to be given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=2p=2314.52=157.26

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides is known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T=s(sa)(sb)(sc) T=157.26(157.26120)(157.26100)(157.2694.52) T=21048411.68=4587.85

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The triangle area is half of the product of the base's length and height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T=2aha  ha=a2 T=1202 4587.85=76.46 hb=b2 T=1002 4587.85=91.76 hc=c2 T=94.522 4587.85=97.08

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding a triangle's angles when we know all three sides. The cosine rule, also known as the Law of Cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines extrapolates the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines because the cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from the cosine value.

a2=b2+c22bccosα  α=arccos(2bcb2+c2a2)=arccos(2 100 94.521002+94.5221202)=76°729"  b2=a2+c22accosβ β=arccos(2aca2+c2b2)=arccos(2 120 94.521202+94.5221002)=54° γ=180°αβ=180°76°729"54°=49°5231"

7. Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

T=rs r=sT=157.264587.85=29.17

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R=4 rsabc=4 29.174 157.258120 100 94.52=61.8

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio of 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

ma=22b2+2c2a2=22 1002+2 94.5221202=76.593 mb=22c2+2a2b2=22 94.522+2 12021002=95.742 mc=22a2+2b2c2=22 1202+2 100294.522=99.834


#2 Obtuse scalene triangle.

Sides: a = 120   b = 100   c = 46.55334537968

Area: T = 2259.75221161093
Perimeter: p = 266.55334537968
Semiperimeter: s = 133.27767268984

Angle ∠ A = α = 103.87551947665° = 103°52'31″ = 1.81329641598 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 22.12548052335° = 22°7'29″ = 0.38661506977 rad

Height: ha = 37.66325352685
Height: hb = 45.19550423222
Height: hc = 97.0822039325

Median: ma = 49.83658508526
Median: mb = 76.05500626575
Median: mc = 107.97331169546

Inradius: r = 16.95553392306
Circumradius: R = 61.8033398875

Vertex coordinates: A[46.55334537968; 0] B[0; 0] C[70.53442302751; 97.0822039325]
Centroid: CG[39.0299228024; 32.3610679775]
Coordinates of the circumscribed circle: U[23.27767268984; 57.25325466455]
Coordinates of the inscribed circle: I[33.27767268984; 16.95553392306]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 76.12548052335° = 76°7'29″ = 1.81329641598 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 157.87551947665° = 157°52'31″ = 0.38661506977 rad

Calculate another triangle

How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other triangle characteristics, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines

a = 120 \ \\ b = 100 \ \\ β = 54\degree \ \\ \ \\ b^2 = a^2 + c^2 - 2ac \cos β \ \\ 100^2 = 120^2 + c^2 -2 \cdot \ 120 \cdot \ c \cdot \ \cos 54\degree \ \\ \ \\ $mXz_{9}pQ_{8}Cc \ \\ \ \\ c>0

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides SSS.
a=120 b=100 c=46.55

2. The triangle perimeter is the sum of the lengths of its three sides

p=a+b+c=120+100+46.55=266.55

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles to be given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=2p=2266.55=133.28

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides is known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T=s(sa)(sb)(sc) T=133.28(133.28120)(133.28100)(133.2846.55) T=5106479.63=2259.75

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The triangle area is half of the product of the base's length and height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T=2aha  ha=a2 T=1202 2259.75=37.66 hb=b2 T=1002 2259.75=45.2 hc=c2 T=46.552 2259.75=97.08

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding a triangle's angles when we know all three sides. The cosine rule, also known as the Law of Cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines extrapolates the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines because the cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from the cosine value.

a2=b2+c22bccosα  α=arccos(2bcb2+c2a2)=arccos(2 100 46.551002+46.5521202)=103°5231"  b2=a2+c22accosβ β=arccos(2aca2+c2b2)=arccos(2 120 46.551202+46.5521002)=54° γ=180°αβ=180°103°5231"54°=22°729"

7. Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

T=rs r=sT=133.282259.75=16.96

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R=4 rsabc=4 16.955 133.277120 100 46.55=61.8

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio of 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

ma=22b2+2c2a2=22 1002+2 46.5521202=49.836 mb=22c2+2a2b2=22 46.552+2 12021002=76.05 mc=22a2+2b2c2=22 1202+2 100246.552=107.973

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