Right angle

A right angle is an angle of exactly 90°, a quarter of a full turn. It is the square corner you see on a sheet of paper, a door frame or a book. Two lines that meet at a right angle are called perpendicular, and a triangle with a right angle is called a right triangle. In diagrams, a right angle is marked with a small square in the corner.

What is a right angle?

A full turn is 360°. Split it into four equal parts and each part is 90°: a right angle. So a right angle is a quarter turn, like turning from facing north to facing east.

• Two right angles make a straight angle of 180°, a straight line.
• Four right angles make a full turn of 360°, all the way around a point.
• In radians, a right angle is π/2, about 1.571 radians.

The right angle symbol

Instead of writing 90°, diagrams mark a right angle with a small square drawn in the corner where the two lines meet. If you see that square, the angle is exactly 90°, even if the drawing is not to scale. In writing, the symbol ⊥ means "is perpendicular to", so AB ⊥ CD says the lines AB and CD meet at a right angle.

Right angle vs acute and obtuse angles

The right angle is the dividing line for naming angles, which is why it matters so much. Any angle smaller than one is an acute angle. In the US, students learn to draw and recognize right, acute and obtuse angles in grade 4.

Right angle examples in real life

Right angles are everywhere people build things, because square corners stack, line up and fit together:

• The corners of a sheet of paper, a phone screen, a window and a door.
• Where a wall meets the floor, and where two walls meet.
• The hands of a clock at 3:00 and at 9:00.
• The plus sign +, the letters L and T, and the lines on graph paper.
• The corners of a basketball court or a soccer field.

How to check for a right angle

The paper corner test

Fit the corner of a sheet of paper or a card into the angle. If both edges line up exactly with both sides, it is a right angle. If there is a gap, it is obtuse. If the corner will not fit, it is acute.

With a protractor

Put the protractor's center on the vertex and its zero line along one side. A right angle reads exactly 90 on either scale.

The 3-4-5 rule

Builders use this to check that a corner is square without any special tools. Measure 3 units along one side from the corner and 4 units along the other side, then measure straight across between the two marks. If that distance is exactly 5 units, the corner is a right angle.

It works with any unit, and with any multiple: 6, 8 and 10 feet, or 30, 40 and 50 inches. Bigger multiples are more accurate for large jobs like laying out a patio or the walls of a shed. The rule comes from the converse of the Pythagorean theorem: 3² + 4² = 9 + 16 = 25 = 5².

Perpendicular lines

Two lines that cross or meet at a right angle are perpendicular. When two lines cross, they make four angles, and if one of them is a right angle, all four are.

On a coordinate graph, perpendicular lines have slopes that multiply to −1. The line y = 2x + 1 has slope 2, and the line y = −½x + 4 has slope −½. Since 2 × (−½) = −1, the two lines meet at a right angle. The x-axis and y-axis are perpendicular too, which is why graph paper is made of squares.

Right triangles

A right triangle (called a right-angled triangle in the UK) has one right angle. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides, which form the right angle, are called the legs.

• A triangle can have only one right angle. Two right angles would already use up 180°, leaving nothing for the third angle.
• The other two angles are acute and add up to 90°, so they are complementary angles. If one is 35°, the other is 55°.
• A right triangle can be scalene or isosceles. The 3-4-5 triangle is scalene. The 45°-45°-90° triangle is isosceles.

In trigonometry, the legs get new names depending on which acute angle you are looking from: the opposite side faces the angle and the adjacent side is next to it. Try naming them in this question from the Trigonometry course.

The Pythagorean theorem

In every right triangle, the square of the hypotenuse equals the sum of the squares of the legs:

Students in the US explain and apply this in grade 8. It also works backward: if the sides of a triangle fit a² + b² = c², the triangle has a right angle. That is the converse, and it is why the 3-4-5 rule works.

Example 1: find the hypotenuse

A right triangle has legs of 5 cm and 12 cm. How long is the hypotenuse?

The hypotenuse is 13 cm.

Example 2: is it a right triangle?

• 9, 12, 15: 9² + 12² = 81 + 144 = 225, and 15² = 225. They match, so yes, it has a right angle.
• 7, 9, 12: 7² + 9² = 49 + 81 = 130, but 12² = 144. They do not match, so no. Since 144 is bigger than 130, the largest angle is obtuse.

Example 3: a right angle split in two

A ray splits a right angle into two angles of 2x° and 3x°. Find both angles.

The angles are 2(18) = 36° and 3(18) = 54°. You can solve any right triangle from two measurements in the Trigonometry calculator.

Test yourself before the common mistakes.

Right angles in shapes

Common mistakes

• Assuming an angle is 90° because it looks square. Only trust the small square symbol or a measurement.
• Calling 90° acute or obtuse. It is neither: it is exactly a right angle.
• Using a leg as the hypotenuse. The hypotenuse is always opposite the right angle and is always the longest side.
• Adding the sides instead of the squares. 3 + 4 is not 5, but 3² + 4² is 5².
• Thinking "right" means facing right. A right angle can point in any direction. "Right" here comes from the Latin rectus, meaning upright.

Practice with a short timed paper

Right angles and right triangles appear all through geometry, trigonometry and the SAT. The daily Pre-Algebra & Algebra 1 and Geometry, Algebra 2 & SAT Math papers are short timed practice tests of 8 questions, with a new one every day.

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