Pythagorean Theorem or Trigonometry? How to Choose

Q: Can you use the Pythagorean theorem without a right angle?

No. a 2 + b 2 = c 2 is true only when the angle between a and b is exactly 90°. For other triangles use the law of cosines, c 2 = a 2 + b 2 − 2 ab cos C , which reduces to the theorem when C = 90° .

Use the Pythagorean theorem when a right-triangle problem involves only sides: you know two sides and want the third. Use trigonometry (SOH CAH TOA) when an angle is involved, either given along with a side or asked for. If the triangle has no right angle, use the law of cosines or the law of sines.

Both tools are exact, both need a right triangle, and they often work together in the same problem. The skill is reading the question carefully enough to know which one to pick up first.

The quick rule

Before you write any formula, list what the problem gives you and what it asks for. Then match the pattern:

You know	You want	Use
two sides of a right triangle	the third side	Pythagorean theorem
one side and one acute angle	another side	sine, cosine or tangent
two sides of a right triangle	an acute angle	inverse trig: sin⁻¹, cos⁻¹, tan⁻¹
one acute angle	the other acute angle	subtract from 90°
sides or angles of a triangle with no right angle	anything	law of cosines or law of sines

Key ideaSides only: Pythagorean theorem. An angle anywhere in the problem, given or wanted: trigonometry.

There is one shortcut to keep in mind. If the angle is 30°, 45° or 60°, the side ratios of the 30-60-90 and 45-45-90 triangles give exact answers without a calculator. They are trigonometry in disguise, worked out once with the theorem.

Flowchart: which tool do I need?

Two questions settle most right-triangle problems: is there a right angle, and does an angle appear anywhere in the problem?

The first question matters more than it looks. The theorem and SOH CAH TOA are both statements about right triangles, so the right angle must be marked, stated, or proved, for example with the converse of the Pythagorean theorem. A triangle that merely looks square in a sketch is not enough.

If there is no right angle, you have two options. The law of cosines handles two sides with the angle between them, or three sides. The law of sines handles problems where you know an angle and the side opposite it. Sometimes a third option is easier: draw an altitude to split the shape into two right triangles, as in Example 5 below.

Trigonometry in one minute

Trigonometry names the three ratios of a right triangle's sides, measured from one acute angle θ:

“Opposite” and “adjacent” depend on which angle you are looking from; the hypotenuse is always the side across from the right angle.

To find a side, pick the ratio that links the side you want with the side you know, then solve. To find an angle, work out the ratio from two sides and apply the inverse function; for example, θ = tan⁻¹(opp over adj). Your calculator must be in degree mode.

The two tools are closely related. Because sinθ = a over c and cosθ = b over c, the theorem a² + b² = c² becomes sin²θ + cos²θ = 1, the Pythagorean identity. The difference is what they need from you: the theorem works with sides alone, while every trig ratio involves an angle.

Same triangle, different question

The best way to see the difference is to keep the triangle fixed and change the question.

The triangle never changes, only the question does. What you know and what you want decide the tool.

Real problems work the same way. Here are two situations, each solved once with the theorem and once with trigonometry, depending on what is given.

Example 1a: ladder, two sides known

A 13-foot ladder has its foot 5 feet from a wall. How high does it reach?

Two sides, want the third: Pythagorean theorem.

h= √13² − 5²= √169 − 25 = √144 = 12

The ladder reaches 12 feet.

Example 1b: ladder, angle known

The same 13-foot ladder makes a 67° angle with the ground. How high does it reach?

A side and an angle: trigonometry. The height is opposite the angle.

h= 13sin67°≈ 13(0.9205) ≈ 11.97

The ladder reaches about 11.97 feet.

Example 2a: ramp length

A ramp rises 2 feet over a horizontal run of 24 feet. How long is the ramp surface?

Two legs, want the hypotenuse: theorem.

L = √2² + 24² = √580 ≈ 24.08

The ramp is about 24.08 feet long.

Example 2b: ramp angle

For the same ramp, what angle does it make with the ground?

Two sides, want an angle: inverse trig.

θ = tan⁻¹(2 over 24) ≈ 4.76°

The ramp rises at about 4.76°.

Ladders appear so often in homework that they have their own tool, the ladder calculator, and the word problems page has more situations to practice reading.

Problems that need both

Harder problems chain the two tools: one step finds a value, and the next step uses it. A good habit is to keep unrounded values in your calculator and round only the final answer.

Example 3: find the legs, then check Level 2

A right triangle has a hypotenuse of 10 cm and an acute angle of 35°. Find both legs.

a= 10sin35° ≈ 5.7358b= 10cos35° ≈ 8.1915

Now check with the theorem: 5.7358² + 8.1915² ≈ 100.00 = 10². The check works because the values were kept to four decimal places. With the rounded legs 5.74 and 8.19 you get 100.02, which is still 100 up to rounding.

The legs are about 5.74 cm and 8.19 cm.

Example 4: height and line of sight Level 2

You stand 30 m from the base of a flagpole and look up at its top at an angle of elevation of 40°. How tall is the pole, and how far is it from you to the top? (Ignore the height of your eyes.)

Step 1 uses trigonometry, because an angle is given:

h = 30tan40° ≈ 25.17

Step 2 can use the theorem, because two sides are now known:

d = √30² + 25.17² ≈ 39.16

The pole is about 25.17 m tall and the top is about 39.16 m away. Using d = 30 over cos40° gives the same 39.16 m, a useful check.

Example 5: angles of an isosceles triangle Level 3

An isosceles triangle has two sides of 10 and a base of 12. Find its height, its angles and its area.

There is no right angle, but the altitude from the top creates two right triangles, each with hypotenuse 10 and base 6.

The altitude turns one non-right triangle into two right triangles. The theorem gives the height; trigonometry gives the angles.

h= √10² − 6² = √64 = 8base angle= cos⁻¹(6 over 10) ≈ 53.13°top angle= 180° − 2(53.13°) = 73.74°area= 1 over 2 · 12 · 8 = 48

The height is 8, the base angles are about 53.13°, the top angle is about 73.74° and the area is 48 square units. More on this technique: the height of a triangle.

Example 6: distance and direction Level 2

A hiker walks 3 km east and then 4 km north. How far is she from her starting point, and in what direction?

d= √3² + 4² = 5θ= tan⁻¹(4 over 3) ≈ 53.13° north of east

She is 5 km away, about 53.13° north of east. As a compass bearing, measured clockwise from north, that is 90° − 53.13° = 36.87°. Displacement problems like this are the bread and butter of navigation.

Example 7: no right angle at all Level 3

Two sides of a triangle are 7 and 9, and the angle between them is 65°. Find the third side.

No right angle, two sides and the included angle: law of cosines.

c²= 7² + 9² − 2(7)(9)cos65° ≈ 130 − 53.25 = 76.75c≈ 8.76

The third side is about 8.76. The Pythagorean theorem would have given √130 ≈ 11.40, which is wrong here.

Common mix-ups

Using a² + b² = c² without a right angle. The theorem fails for every other triangle, as Example 7 shows. Check for the right-angle mark first.
Using SOH CAH TOA in a non-right triangle. The ratios are defined from a right triangle. For other triangles use the laws of sines and cosines, or draw an altitude.
Trying the theorem with only one side. The theorem needs two sides. With one side and an angle, you must use trigonometry first.
Mixing up opposite and adjacent. They depend on which angle you are using. The hypotenuse touches the angle too, but it is never called the adjacent side.
Radian mode. sin67 in radians is about −0.86, which would give a negative height. If a length comes out negative, check the calculator mode.

For more slips to avoid with the theorem itself, see common mistakes. When you just want the answer checked, the right triangle calculator accepts any mix of sides and angles, and the Pythagorean theorem calculator handles the sides-only case.

Frequently asked questions

Can you use the Pythagorean theorem without a right angle?

No. a² + b² = c² is true only when the angle between a and b is exactly 90°. For other triangles use the law of cosines, c² = a² + b² − 2abcosC, which reduces to the theorem when C = 90°.

Do I need trigonometry to find an angle in a right triangle?

In general, yes: you calculate a ratio of two sides and apply sin⁻¹, cos⁻¹ or tan⁻¹. The exceptions are special triangles you can recognize, such as a right triangle whose short leg is half the hypotenuse, which always has a 30° angle.

When should I use SOH CAH TOA instead of the Pythagorean theorem?

Whenever an angle other than the right angle is part of the problem. If you are given an acute angle and one side, or asked to find an angle, the theorem cannot help on its own because it contains no angles.

What if I only know the angles of a triangle?

Then you cannot find the side lengths. Triangles with the same angles are similar: they have the same shape but can be any size. You need at least one side length to fix the scale.

If both methods work, which should I choose?

Choose the one that uses the numbers you were given rather than numbers you calculated. Given values are exact, while calculated ones may already be rounded. Use the other method as a check if you have time.

Keep exploring

Pythagorean Theorem or Trigonometry: Which Should You Use?