Knowing how to find the missing side length of a triangle begins with identifying the information already given. The best method depends on whether the triangle is right-angled, whether an included angle is known, and which sides or angles are available. By matching the given information to the correct formula, a missing side can usually be found with algebra, the Pythagorean theorem, trigonometric ratios, or the laws of sines and cosines.
Introduction: Start by Identifying the Triangle
A triangle has three sides and three interior angles. To calculate an unknown side, at least some information about the other sides or angles must be known. Before selecting a formula, write down:
- The lengths of every known side
- The measures of every known angle
- Whether one angle is a right angle, measuring 90°
- Which side must be found
- The required degree of accuracy and unit of measurement
The arrangement of the information matters as much as the information itself. To give you an idea, knowing two sides is not enough unless their positions or an included angle are also known.
Method 1: Use the Pythagorean Theorem for a Right Triangle
If the triangle contains a 90° angle, the two sides forming that angle are called legs, while the side opposite the right angle is the hypotenuse. The hypotenuse is always the longest side.
The Pythagorean theorem states:
[ a^2+b^2=c^2 ]
Here, (a) and (b) represent the legs, while (c) represents the hypotenuse.
Finding the Hypotenuse
When both legs are known, square each leg, add the results, and take the square root:
[ c=\sqrt{a^2+b^2} ]
Take this: if the legs measure 8 cm and 15 cm:
[ c=\sqrt{8^2+15^2} ]
[ c=\sqrt{64+225}=\sqrt{289}=17 ]
The missing hypotenuse is 17 cm.
Finding a Missing Leg
If the hypotenuse and one leg are known, rearrange the formula:
[ a=\sqrt{c^2-b^2} ]
Suppose the hypotenuse is 13 m and one leg is 5 m:
[ a=\sqrt{13^2-5^2} ]
[ a=\sqrt{169-25}=\sqrt{144}=12 ]
The missing leg is 12 m.
A common mistake is subtracting the squares in the wrong order. Because a real side length cannot have an imaginary value in ordinary geometry, subtract the known leg squared from the hypotenuse squared The details matter here..
Method 2: Use Trigonometric Ratios
Right-triangle trigonometry is useful when one acute angle and one side are known. The three primary ratios are:
[ \sin \theta=\frac{\text{opposite}}{\text{hypotenuse}} ]
[ \cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}} ]
[ \tan \theta=\frac{\text{opposite}}{\text{adjacent}} ]
The words opposite, adjacent, and hypotenuse describe each side’s position relative to the chosen angle. The hypotenuse remains opposite the right angle, while the opposite and adjacent sides may change if a different acute angle is selected.
A convenient memory aid is SOH CAH TOA:
- SOH: Sine equals opposite divided by hypotenuse
- CAH: Cosine equals adjacent divided by hypotenuse
- TOA: Tangent equals opposite divided by adjacent
Example Using Sine
Suppose a right triangle has a 32° angle, a hypotenuse of 14 cm, and an unknown side opposite the 32° angle. Let the unknown side be (x):
[ \sin 32^\circ=\frac{x}{14} ]
Multiply both sides by 14:
[ x=14\sin 32^\circ\approx7.42 ]
The missing side is approximately 7.42 cm.
Choosing the Correct Ratio
Follow these steps:
- Mark the known angle.
- Label the known side as opposite, adjacent, or hypotenuse.
- Label the unknown side in the same way.
- Choose the ratio containing both sides.
- Substitute the values and solve algebraically.
Here's one way to look at it: if the known and unknown sides are the opposite and adjacent sides, tangent is the appropriate ratio. If the hypotenuse is involved, use sine or cosine.
Method 3: Use Special Right-Triangle Ratios
Some right triangles have predictable side relationships.
45°-45°-90° Triangles
The two legs are equal, and the hypotenuse equals a leg multiplied by (\sqrt{2}):
[ x:x:x\sqrt{2} ]
If the hypotenuse is 10, each leg is:
[ \frac{10}{\sqrt{2}}=5\sqrt{2} ]
30°-60°-90° Triangles
The sides follow this ratio:
[ x:x\sqrt{3}:2x ]
- (x) is the side opposite 30
