Pythagorean Theorem Word Problems With Answers

Pythagorean theorem word problems are translation problems before they are calculation problems. The story rarely hands you a labeled right triangle. Your first job is to find two perpendicular directions, draw the triangle, and decide whether the missing length is a leg or the hypotenuse.

For a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$$a^2+b^2=c^2$$

The hypotenuse is always opposite the right angle and is always the longest side. That one identification determines whether you add two squares or subtract one square from another.

How to recognize a Pythagorean word problem

Look for a situation that contains a right angle, even if the words never say “right triangle.” Common clues include:

• a vertical wall meeting level ground;
• the length and width of a rectangle;
• movement north or south followed by movement east or west;
• a square or rectangular field and its diagonal;
• a pole, cable or ramp perpendicular to the ground.

A diagonal alone is not enough. The theorem applies only when the three relevant lengths form a right triangle. The Khan Academy Pythagorean theorem unit also separates ordinary applications from multi-step problems, which is a useful reminder to identify the triangle before calculating.

Use this five-step routine:

1. Sketch the situation. A rough drawing is enough.
2. Mark the right angle. If you cannot justify one, do not use the theorem yet.
3. Label the known sides and the unknown. Include units.
4. Identify the hypotenuse. It lies across from the right angle.
5. Write the equation before substituting. Solve, attach units and check whether the result is reasonable.

Which form of the formula should you use?

Never write $c^2-a^2-b^2$ and guess which side is which. Labeling the hypotenuse first prevents the most common setup error.

Worked example 1: ladder against a wall

Problem: A $13\text{ ft}$ ladder leans against a vertical wall. Its base is $5\text{ ft}$ from the wall. How high up the wall does the ladder reach?

The wall and ground are perpendicular, so they are the legs. The ladder is opposite the right angle, so it is the hypotenuse.

Let $h$ be the height:

$$h^2+5^2=13^2$$ $$h^2+25=169$$ $$h^2=144 \quad\Rightarrow\quad h=12$$

Answer: The ladder reaches $12\text{ ft}$ up the wall.

Check: $12<13$, so the leg is shorter than the hypotenuse. The values also form the familiar $5$–$12$–$13$ right triangle.

Worked example 2: television screen diagonal

Problem: A screen is $48\text{ in}$ wide and $27\text{ in}$ high. What is its diagonal size to the nearest tenth of an inch?

The width and height meet at a right angle. The advertised screen size is the diagonal, so we are finding the hypotenuse $d$.

$$d^2=48^2+27^2$$ $$d^2=2304+729=3033$$ $$d=\sqrt{3033}\approx55.1$$

Answer: The screen diagonal is approximately $55.1\text{ in}$.

The exact value is $\sqrt{3033}\text{ in}$. Rounding belongs at the end, after the square root—not after squaring the side lengths.

Worked example 3: shortest path across a field

Problem: A rectangular sports field is $60\text{ m}$ long and $80\text{ m}$ wide. How much shorter is a straight diagonal walk than walking along two sides from one corner to the opposite corner?

First find the diagonal $d$:

$$d^2=60^2+80^2=3600+6400=10000$$ $$d=100\text{ m}$$

Walking along two sides takes:

$$60+80=140\text{ m}$$

The saving is:

$$140-100=40\text{ m}$$

Answer: The diagonal route is $40\text{ m}$ shorter.

This is a two-part question. Finding the diagonal is not the final answer because the story asks for the difference between two routes.

Worked example 4: north-and-east navigation

Problem: A hiker walks $7\text{ km}$ east and then $24\text{ km}$ north. How far is the hiker from the starting point in a straight line?

East and north are perpendicular directions, so the two stages form the legs. The straight-line displacement is the hypotenuse $s$.

$$s^2=7^2+24^2=49+576=625$$ $$s=25\text{ km}$$

Answer: The hiker is $25\text{ km}$ from the starting point.

Do not add $7+24$. That gives the distance walked, $31\text{ km}$, not the straight-line distance from start to finish.

Worked example 5: finding a missing leg

Problem: A support cable is $20\text{ m}$ long and runs from the top of a vertical pole to a point $12\text{ m}$ from the pole’s base. How tall is the pole?

The cable is the hypotenuse. Let $h$ be the pole’s height:

$$h^2+12^2=20^2$$ $$h^2=400-144=256$$ $$h=16\text{ m}$$

Answer: The pole is $16\text{ m}$ tall.

Notice the subtraction. When the missing side is a leg, subtract the known leg’s square from the hypotenuse’s square.

When the Pythagorean theorem does not apply

Do not use $a^2+b^2=c^2$ merely because you see three lengths. You need a right angle. If a triangle is not right, another method—such as the law of cosines—may be required.

The converse can test whether three side lengths form a right triangle. Put the longest length in the $c$ position and compare:

• if $a^2+b^2=c^2$, the triangle is right;
• if $a^2+b^2>c^2$, it is acute;
• if $a^2+b^2<c^2$, it is obtuse.

That is a classification test, not permission to call the longest side a hypotenuse before you establish a right angle.

Common mistakes and quick checks

Calling the unknown side $c$

$c$ is the hypotenuse, not automatically the unknown. An unknown leg should still be labeled $a$, $b$ or another variable.

Forgetting the final square root

If $d^2=3033$, the diagonal is $\sqrt{3033}$, not $3033$. Check the unit too: the side length is in inches, not square inches.

Rounding too early

Keep the square root in the calculator until the last step. Early rounding can noticeably change a multi-step answer.

Answering the calculation instead of the question

A problem may ask for time saved, remaining cable, perimeter or the difference between two routes. Reread the final sentence after you find the missing side.

Trusting a diagram’s scale

Geometry diagrams are often not drawn to scale. Use the stated lengths and right-angle marks, not visual measurement. The same rule matters in triangle congruence proofs and circle theorem problems.

Try these three problems

1. A rectangular garden is $9\text{ m}$ by $12\text{ m}$. Find its diagonal.
2. A $17\text{ ft}$ ramp rises $8\text{ ft}$. How long is its horizontal run?
3. A boat travels $15\text{ km}$ south and $20\text{ km}$ west. How far is it from its starting point?

Answers

1. $\sqrt{9^2+12^2}=15\text{ m}$.
2. $\sqrt{17^2-8^2}=15\text{ ft}$.
3. $\sqrt{15^2+20^2}=25\text{ km}$.

For a problem that begins as a photo or screenshot, SolveTap’s free geometry solver from a picture can identify the right triangle and explain the setup. Use the explanation to check which side is the hypotenuse, then solve the next example without assistance.

Frequently asked questions

How do I know whether to add or subtract?

Add when the missing side is the hypotenuse: $c=\sqrt{a^2+b^2}$. Subtract when the hypotenuse and one leg are known and the other leg is missing: $a=\sqrt{c^2-b^2}$.

Can the Pythagorean theorem find an angle?

No. It relates the three side lengths of a right triangle. Use sine, cosine or tangent to find an unknown acute angle once the needed side lengths are known.

Does the theorem work for every triangle?

No. It applies only to right triangles. The law of cosines is the more general side-and-angle relationship for non-right triangles.

Should I give an exact answer or a decimal?

Follow the question. If no rounding instruction is given, an exact radical such as $\sqrt{41}$ is often preferred. For a real-world measurement, a sensible decimal with units is usually expected.