Right Triangle Side Calculator – Pythagorean Theorem & Trigonometry

The calculator above finds the missing side of a right triangle in two ways. The Know 2 Sides (Pythagoras) tab uses the Pythagorean theorem: enter both legs to get the hypotenuse, or enter the hypotenuse and one leg to get the other leg. The Know 1 Side & 1 Angle (Trigonometry) tab uses sine, cosine and tangent to find all three sides from one side and one acute angle. Every result includes a to-scale drawing of the triangle, both acute angles in decimal degrees and degrees-minutes, the altitude, the projections, the area and the perimeter, plus a step-by-step solution you can check against your homework, copy, or print on A4 paper.

📐 The right triangle and its basic elements

A right triangle is a triangle with one 90° angle. The two sides that form the right angle are the legs, and the side opposite the right angle is the hypotenuse. The hypotenuse is always the longest side, and the other two angles are always acute and add up to 90° (they are complementary).

The calculator and this article use one consistent convention: triangle ABC has its right angle at C; the hypotenuse is c = AB; the legs are a = BC and b = CA; the acute angle α sits at vertex A and the acute angle β at vertex B. So leg a is opposite α and leg b is adjacent to α. The altitude from C to the hypotenuse is h = CH, which splits the hypotenuse into the projections a′ = BH and b′ = HA.

🔺 The Pythagorean theorem

The Pythagorean theorem is the most famous result in plane geometry: in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.

Rearranging gives the three forms you use to find a missing side:

To find the hypotenuse you add the two squares; to find a leg you subtract the square of the known leg from the square of the hypotenuse. Because a length is always positive, only the positive square root is taken.

Converse of the Pythagorean theorem: if the three sides of a triangle satisfy c² = a² + b², the triangle has a right angle at the vertex opposite side c. This is the quickest way to check whether a triangle is right-angled when only its side lengths are known. For example, a triangle with sides 7, 24 and 25 is a right triangle because 7² + 24² = 625 = 25².

If you just need a quick check of the theorem, you can also use the Pythagorean theorem calculator on ATMath.

🔢 Common Pythagorean triples

A Pythagorean triple is a set of three positive whole numbers (a, b, c) with a² + b² = c². Knowing a few by heart lets you spot answers instantly on multiple-choice tests.

Every primitive triple comes from Euclid’s formula with two coprime whole numbers a = m² − n², b = 2mn, c = m² + n². Taking m = 2, n = 1 gives 3-4-5; taking m = 3, n = 2 gives 5-12-13.

📏 Metric relations between the sides and the altitude

Drawing the altitude CH to the hypotenuse splits triangle ABC into two smaller triangles that are similar to it. These similar triangles give five important metric relations:

Adding the first two relations gives a² + b² = c(a′ + b′) = c², which is one proof of the Pythagorean theorem. The calculator above works out h, a′ and b′ after every solve, so you can use it to check metric-relation exercises too.

📊 Trigonometric ratios of an acute angle

When an acute angle is known, the sides of a right triangle are linked by four trigonometric ratios. For angle α at A, the opposite side is a, the adjacent side is b and the hypotenuse is c:

The classic mnemonic is SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Rearranging these ratios gives the side-angle relations that the trigonometry tab of the calculator uses directly:

In words: a leg equals the hypotenuse times the sine of the opposite angle or the cosine of the adjacent angle; one leg equals the other leg times the tangent of the opposite angle or the cotangent of the adjacent angle.

⭐ Trigonometric values of special angles

A right triangle with a 30° (or 60°) angle is half of an equilateral triangle: its sides are in the ratio 1 : √3 : 2 and the side opposite 30° is half the hypotenuse. A right triangle with a 45° angle is an isosceles right triangle with sides in the ratio 1 : 1 : √2. When you enter one of these angles, the calculator shows the exact surd values instead of decimals.

💡 Worked examples

Example 1. Finding the hypotenuse

A right triangle has legs of 5 cm and 12 cm. Find the hypotenuse.

Example 2. Finding a leg

A right triangle has a hypotenuse of 10 cm and one leg of 6 cm. Find the other leg.

Example 3. An irrational answer

The legs are 4 cm and 6 cm. The hypotenuse is:

The calculator shows both the simplified surd 2√13 and the decimal value, so it fits questions that ask for an “exact answer” as well as those that ask you to “round to two decimal places”.

Example 4. Metric relations

Triangle ABC has its right angle at C, a = 6 cm and b = 8 cm. Find the hypotenuse, the altitude h and both projections.

Check: h² = 23.04 = 3.6 × 6.4, which matches h² = a′·b′.

Example 5. Hypotenuse and one acute angle

A right triangle has a hypotenuse of 12 cm and angle α = 40°. Find both legs.

The other angle is β = 90° − 40° = 50°.

Example 6. A ladder against a wall

A 5 m ladder leans against a wall with its foot 1.5 m from the wall. How high does it reach, and what angle does it make with the ground?

Safety guidelines recommend an angle of about 75° with the ground, so at 72.54° the foot of the ladder should move a little closer to the wall.

🏗️ Right triangles in everyday life

• Construction: builders use the “3-4-5 rule” to square up foundations: measure 3 m along one side and 4 m along the other; if the diagonal between the two marks is exactly 5 m, the corner is a right angle.
• Indirect height measurement: a tree that casts an 8 m shadow when the sun is 35° above the horizon is about 8·tan 35° ≈ 5.60 m tall.
• TV and phone screens: a “55-inch” screen is measured along the diagonal. With a 16 : 9 ratio, the screen is about 121.76 cm wide and 68.49 cm tall, worked out with the Pythagorean theorem.
• Wheelchair ramps: the standard 1 : 12 slope means tan α = 1/12, a ramp angle of about 4.76°.
• Maps and navigation: the straight-line distance between two points on a grid is the hypotenuse of a right triangle whose legs are the differences in the x and y coordinates.

📝 A 4-step method for right triangle problems

1. Draw and label. Find the right-angle vertex, then mark the hypotenuse (opposite the right angle) and the two legs.
2. Identify the data. Two sides known: use the Pythagorean theorem. One side and one acute angle known: use trigonometric ratios.
3. Pick the right formula. For trigonometry, decide whether the known and unknown sides are opposite, adjacent or hypotenuse relative to the given angle, then choose sin, cos or tan.
4. Calculate and check. The hypotenuse must be the longest side, the two acute angles must add up to 90°, and a² + b² = c² must hold.

⚠️ Common mistakes

1. Adding the sides directly. Legs of 6 and 8 do not give a hypotenuse of 14 but √(36 + 64) = 10.
2. Adding instead of subtracting for a leg. With c = 13 and a = 5, b² = 169 − 25 = 144, not 169 + 25.
3. Mixing up the hypotenuse. The hypotenuse is always opposite the right angle and is the longest side. A “hypotenuse” shorter than a leg means the data are wrong.
4. Confusing opposite and adjacent. “Opposite” and “adjacent” always depend on the angle you are using; switch to the other angle and the roles of the legs swap.
5. Calculator in radian mode. If the angle is in degrees, your calculator must be in DEG mode, otherwise sin 30 returns −0.988 instead of 0.5.
6. Rounding too early. Keep intermediate results (or surds) unrounded until the final step to avoid accumulated error.
7. Mixed units. Convert m, cm and mm to a single unit before squaring and adding.

❓ Frequently asked questions (FAQ)

What is the formula for the hypotenuse of a right triangle?

By the Pythagorean theorem, the hypotenuse c = √(a² + b²), where a and b are the legs. For legs of 6 cm and 8 cm, the hypotenuse is √(36 + 64) = √100 = 10 cm.

How do I find a leg when I know the hypotenuse and the other leg?

Subtract the square of the known leg from the square of the hypotenuse and take the square root: b = √(c² − a²). For c = 13 cm and a = 5 cm, b = √(169 − 25) = √144 = 12 cm.

How do I find the sides from one side and one acute angle?

Use the trigonometric ratios: opposite side a = c × sin α, adjacent side b = c × cos α, and a = b × tan α. For a 10 cm hypotenuse and a 30° angle, the opposite side is 10 × sin 30° = 5 cm and the adjacent side is 10 × cos 30° ≈ 8.66 cm.

How can I tell whether a triangle is a right triangle?

Use the converse of the Pythagorean theorem: if the square of the longest side equals the sum of the squares of the other two, the triangle has a right angle opposite the longest side. Since 7² + 24² = 625 = 25², the 7-24-25 triangle is a right triangle.

What is the formula for the altitude to the hypotenuse?

The altitude h = a × b / c, the product of the legs divided by the hypotenuse. Also h² = a′ × b′, where a′ and b′ are the projections of the legs on the hypotenuse. For the 6-8-10 triangle, h = 48 / 10 = 4.8 cm.