How to calculate the side length of a triangle
In mathematics and geometry, calculating the side lengths of a triangle is a common problem. Depending on the known conditions, the formulas and methods for calculating the length of the sides of a triangle are also different. This article will introduce in detail how to calculate the side length of a triangle based on different known conditions, and attach specific formulas and cases.
1. Known two sides and the included angle (cosine theorem)
When two sides of a triangle and their angle are known, the length of the third side can be calculated using the law of cosines. The formula of the cosine theorem is as follows:
| Known conditions | formula |
|---|---|
| Sides a and b, angle C | c² = a² + b² - 2ab * cos(C) |
For example, it is known that the two sides a=5, b=7, and the angle C=60 degrees, then the length of the third side c can be calculated by the following steps:
c² = 5² + 7² - 2*5*7*cos(60°) = 25 + 49 - 70*0.5 = 74 - 35 = 39
c = √39 ≈ 6.245
2. Two angles and one side are known (sine theorem)
If you know the angles and one side of a triangle, you can use the law of sines to calculate the lengths of the other two sides. The formula for the sine theorem is as follows:
| Known conditions | formula |
|---|---|
| Two angles A and B, side a | b = (a * sin(B)) / sin(A) |
| Two angles A and C, side a | c = (a * sin(C)) / sin(A) |
For example, it is known that angle A=30 degrees, angle B=60 degrees, and side a=4, then the length of side b can be calculated by the following steps:
b = (4 * sin(60°)) / sin(30°) = (4 * √3/2) / (1/2) = 4√3 ≈ 6.928
3. Right triangle (Pythagorean theorem)
For a right triangle, if the lengths of two sides are known, the length of the third side can be calculated using the Pythagorean Theorem. The formula of the Pythagorean Theorem is as follows:
| Known conditions | formula |
|---|---|
| Right angle sides a and b | Hypotenuse c = √(a² + b²) |
| Right angle side a, hypotenuse c | Right angle side b = √(c² - a²) |
For example, it is known that the right-angled side a=3 and the right-angled side b=4, then the length of the hypotenuse c is:
c = √(3² + 4²) = √(9 + 16) = √25 = 5
4. Three known sides (Heron’s formula)
If the lengths of the sides of a triangle are known, the area of the triangle can be calculated using Heron's formula, but the side lengths themselves need to be determined by other methods. Helen's formula is as follows:
| Known conditions | formula |
|---|---|
| Three sides a, b, c | s = (a + b + c) / 2 Area = √(s(s - a)(s - b)(s - c)) |
For example, it is known that the three sides a=5, b=6, and c=7, then the area can be calculated by the following steps:
s = (5 + 6 + 7) / 2 = 9
Area = √(9 * 4 * 3 * 2) = √216 ≈ 14.697
Summary
According to different known conditions, the methods of calculating the length of the sides of a triangle are also different. Here is a summary of each situation:
| Known conditions | Applicable method |
|---|---|
| Both sides and angle | cosine theorem |
| Two corners and one side | Sine theorem |
| Two sides of a right triangle | Pythagorean theorem |
| three sides | Heron's formula (for area) |
I hope that through the introduction of this article, you can master the various methods of calculating the side lengths of triangles and use them flexibly in practical applications.
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