Triangle calculator SSA

Triangle has two solutions with side c=74.49548974278 and with side c=25.50551025722

#1 Acute scalene triangle.

The lengths of the sides of the triangle:
a = 100
b = 90
c = 74.49548974278

Area: T = 3225.72436812409
Perimeter: p = 264.49548974278
Semiperimeter: s = 132.24774487139

Angle ∠ A = α = 74.20768309517° = 74°12'25″ = 1.29551535276 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 45.79331690483° = 45°47'35″ = 0.79992415748 rad

Altitude (height) to the side a: h_a = 64.51444736248
Altitude (height) to the side b: h_b = 71.6832748472
Altitude (height) to the side c: h_c = 86.60325403784

Median: m_a = 65.76327924543
Median: m_b = 75.82770721536
Median: m_c = 87.53664356386

Inradius: r = 24.39215758876
Circumradius: R = 51.96215242271

Vertex coordinates: A[74.49548974278; 0] B[0; 0] C[50; 86.60325403784]
Centroid: CG[41.49882991426; 28.86875134595]
Coordinates of the circumscribed circle: U[37.24774487139; 36.23302023774]
Coordinates of the inscribed circle: I[42.24774487139; 24.39215758876]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 105.79331690483° = 105°47'35″ = 1.29551535276 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 134.20768309517° = 134°12'25″ = 0.79992415748 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 = 100 b = 90 β = 60 ° b^{2} = a^{2} + c^{2} - 2 a c cos β 9 0^{2} = 10 0^{2} + c^{2} - 2 \cdot 100 \cdot c \cdot cos 60 ° c^{2} - 100 c + 1900 = 0 p = 1; q = - 100; r = 1900 D = q^{2} - 4 p r = 10 0^{2} - 4 \cdot 1 \cdot 1900 = 2400 D > 0 c_{1, 2} = \frac{- q \pm D}{2 p} = \frac{1 0 0 \pm 2 4 0 0}{2} = 50 \pm 600 c_{1, 2} = 50 \pm 24.494897 c_{1} = 74.494897428 c_{2} = 25.505102572 c > 0 c = 74.49

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 = 100 b = 90 c = 74.49

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

p = a + b + c = 100 + 90 + 74.49 = 264.49

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 = \frac{p}{2} = \frac{2 6 4 . 4 9}{2} = 132.25

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 (s - a) (s - b) (s - c) T = 132.25 (132.25 - 100) (132.25 - 90) (132.25 - 74.49) T = 10405293.27 = 3225.72

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 = \frac{a h _{a}}{2} h_{a} = \frac{2 T}{a} = \frac{2 \cdot 3 2 2 5 . 7 2}{1 0 0} = 64.51 h_{b} = \frac{2 T}{b} = \frac{2 \cdot 3 2 2 5 . 7 2}{9 0} = 71.68 h_{c} = \frac{2 T}{c} = \frac{2 \cdot 3 2 2 5 . 7 2}{7 4 . 4 9} = 86.6

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 an inverse cosine called arccosine to determine the angle from the cosine value.

a^{2} = b^{2} + c^{2} - 2 b c cos α α = arccos (\frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}) = arccos (\frac{9 0 ^{2} + 7 4 . 4 9 ^{2} - 1 0 0 ^{2}}{2 \cdot 9 0 \cdot 7 4 . 4 9}) = 74 ° 1 2^{'} 25 " b^{2} = a^{2} + c^{2} - 2 a c cos β β = arccos (\frac{a ^{2} + c ^{2} - b ^{2}}{2 a c}) = arccos (\frac{1 0 0 ^{2} + 7 4 . 4 9 ^{2} - 9 0 ^{2}}{2 \cdot 1 0 0 \cdot 7 4 . 4 9}) = 60 ° γ = 180 ° - α - β = 180 ° - 74 ° 1 2^{'} 25 " - 60 ° = 45 ° 4 7^{'} 35 "

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 a triangle's inradius and semiperimeter (half the perimeter) is its area.

T = r s r = \frac{T}{s} = \frac{3 2 2 5 . 7 2}{1 3 2 . 2 5} = 24.39

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 = \frac{a b c}{4 r s} = \frac{1 0 0 \cdot 9 0 \cdot 7 4 . 4 9}{4 \cdot 2 4 . 3 9 2 \cdot 1 3 2 . 2 4 7} = 51.96

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 a median's length from its side's lengths.

m_{a} = \frac{2 b ^{2} + 2 c ^{2} - a ^{2}}{2} = \frac{2 \cdot 9 0 ^{2} + 2 \cdot 7 4 . 4 9 ^{2} - 1 0 0 ^{2}}{2} = 65.763 m_{b} = \frac{2 c ^{2} + 2 a ^{2} - b ^{2}}{2} = \frac{2 \cdot 7 4 . 4 9 ^{2} + 2 \cdot 1 0 0 ^{2} - 9 0 ^{2}}{2} = 75.827 m_{c} = \frac{2 a ^{2} + 2 b ^{2} - c ^{2}}{2} = \frac{2 \cdot 1 0 0 ^{2} + 2 \cdot 9 0 ^{2} - 7 4 . 4 9 ^{2}}{2} = 87.536

#2 Obtuse scalene triangle.

The lengths of the sides of the triangle:
a = 100
b = 90
c = 25.50551025722

Area: T = 1104.40333376813
Perimeter: p = 215.50551025722
Semiperimeter: s = 107.75325512861

Angle ∠ A = α = 105.79331690483° = 105°47'35″ = 1.8466439126 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 14.20768309517° = 14°12'25″ = 0.24879559764 rad

Altitude (height) to the side a: h_a = 22.08880667536
Altitude (height) to the side b: h_b = 24.54222963929
Altitude (height) to the side c: h_c = 86.60325403784

Median: m_a = 43.30442160604
Median: m_b = 57.4487847032
Median: m_c = 94.27328616077

Inradius: r = 10.24994402638
Circumradius: R = 51.96215242271

Vertex coordinates: A[25.50551025722; 0] B[0; 0] C[50; 86.60325403784]
Centroid: CG[25.16883675241; 28.86875134595]
Coordinates of the circumscribed circle: U[12.75325512861; 50.37223380011]
Coordinates of the inscribed circle: I[17.75325512861; 10.24994402638]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 74.20768309517° = 74°12'25″ = 1.8466439126 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 165.79331690483° = 165°47'35″ = 0.24879559764 rad

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