Angles of a Square - Interior, Diagonal, and Sum of Angles

What Are The Angles Of A Square?

A square is a quadrilateral with four equal sides and four equal interior angles, each measuring exactly 90° (a right angle). Because all four are equal and every quadrilateral's interior angles sum to 360°, each angle must be $360° \div 4 = 90°$. The angle notation used throughout is ∠, so the four corner angles of square $ABCD$ are ∠A, ∠B, ∠C, and ∠D.

Two kinds of angle live inside a square, and confusing them is the most common error on this topic: the interior (corner) angles at the vertices, and the diagonal angles formed where the two diagonals cross. Keep them separate - the corners are 90° because of the shape's definition, while the diagonal angles are 90° for a different reason, shown below.

Why Every Angle Is 90° And They Sum to 360°

The 360° total is not a fact to memorise; it follows from triangles, a result set out clearly in Cuemath's guide to the angles of a square. Draw one diagonal of any quadrilateral and it splits the shape into two triangles. Since the interior angles of a triangle always sum to 180°, two triangles give:

$$180° + 180° = 360°$$

So the interior angles of every quadrilateral sum to 360°. For the general result, see angles of a quadrilateral. A square is the special case where all four angles are equal:

$$\text{each angle} = \frac{360°}{4} = 90°$$

That is why a square's corners are right angles: the equal-angle condition plus the fixed 360° total forces each one to 90°. This is the same right-angle property shared with a rectangle - the difference is only that a square also has equal sides. Compare the two in angles of a rectangle.

Diagonal Angles: Why They Meet At 90°

The two diagonals of a square are equal in length, they bisect each other, and - unlike a general rectangle - they cross at a right angle. They also bisect each corner angle.

Consider square $ABCD$ with diagonals meeting at centre $O$. Each diagonal splits a 90° corner into two equal parts:

$$\frac{90°}{2} = 45°$$

So ∠OAB $= 45°$ and ∠OBA $= 45°$. In triangle $OAB$, the three angles must total 180°, giving:

$$\angle AOB = 180° - 45° - 45° = 90°$$

That is why the diagonals cross at 90°. This right-angle crossing is what a general rectangle does not have - a rectangle's diagonals are equal and bisect each other, but they meet at an angle other than 90° unless the rectangle happens to be a square.

Examples Of Angles Of A Square

Each example moves from a direct fact to a fuller reasoning task. The question is in bold; the working is not.

Example 1

What is the measure of one interior angle of a square?

All four interior angles are equal and sum to 360°.

$$\text{one angle} = \frac{360°}{4} = 90°$$

Each interior angle is 90°.

Example 2

A diagonal of square $PQRS$ is drawn from $P$ to $R$. What angle does it make with side $PQ$?

A diagonal bisects the 90° corner at $P$.

$$\frac{90°}{2} = 45°$$

The diagonal makes a 45° angle with side $PQ$.

Example 3: The intuitive guess that goes wrong

A student is asked for the angle at which a square's diagonals cross. They reason: "The corners are 90°, and the diagonals just connect corners, so the crossing angle must also be 45° like the split corners."

Following that reasoning gives 45°. But test it against the triangle at the centre. The diagonals split each corner into two 45° angles, so in triangle $OAB$ the base angles are each 45°.

$$\angle AOB = 180° - 45° - 45° = 90°$$

The crossing angle is 90°, not 45°. The mistake was confusing the split corner angle (45°) with the crossing angle (90°). They are different angles in the same figure.

Example 4

Prove that the four angles where the diagonals meet are all 90°.

The diagonals meet at $O$, forming four angles around a point that sum to 360°. From Example 3, ∠AOB = 90°. The angle vertically opposite it, ∠COD, is also 90°. The remaining two angles, ∠BOC and ∠AOD, together make $360° - 90° - 90° = 180°$, and being equal, each is 90°. So all four central angles are 90°.

Example 5

In square $ABCD$, a diagonal from $A$ meets $C$. Triangle $ABC$ is formed. Find all three of its angles.

∠B is a full corner of the square, so ∠B = 90°. The diagonal bisects corners $A$ and $C$, so ∠BAC = 45° and ∠BCA = 45°.

$$90° + 45° + 45° = 180°$$

The triangle has angles 90°, 45°, and 45° - a right isosceles triangle. This is why a square's diagonal always creates two 45-45-90 triangles. The equal-side reasoning here connects to similar triangles.

Example 6

A square floor tile is rotated 45° about its centre. What is the sum of its interior angles after rotation?

Rotation does not change any interior angle - it only turns the whole shape. Each corner is still 90°, so the sum is unchanged:

$$4 \times 90° = 360°$$

The interior-angle sum stays 360° regardless of orientation.

Where the Square's Angles Earn Their Keep

The square's angle facts are the quiet reason so much of the built world lines up.

• Tiling and flooring. Four squares meet at a point because $4 \times 90° = 360°$ exactly - the tiles fill the space with no gap and no overlap. Change the angle even slightly and the floor either buckles or leaves seams. This is the same 360°-at-a-corner idea that limits how shapes fit together.

• Construction and framing. Carpenters check a frame is "square" by measuring the diagonals: in a true square (or rectangle), the diagonals are equal. If they differ, a corner has drifted off 90° and the frame is a leaning parallelogram.

• Screens and pixels. Digital displays are grids of square (or near-square) cells precisely because right-angle corners tile perfectly and address cleanly by row and column. The way regular shapes fill a plane without gaps is the study of tessellation.

The deeper "why" is that 90° is the angle that makes four copies close a full turn. That single fact, that four right angles complete 360°, is what lets squares tile a plane, a property explored further in the Wikipedia article on the square.

The Mistakes Students Make Most Often Working With Square Angles

Mistake 1: Confusing corner angles with diagonal-crossing angles

Where it slips in: When a problem draws the diagonals, the reader treats the 45° split-corner and the 90° crossing as interchangeable.

Don't do this: Report the diagonal-crossing angle as 45° because that is the corner-split value.

The correct way: Name the angle precisely. Corner angle = 90°; corner split by a diagonal = 45°; diagonals crossing at the centre = 90°. Draw and label triangle $OAB$ to keep them straight.

Mistake 2: Assuming a rectangle's diagonals also cross at 90°

Where it slips in: Extending the square's diagonal property to every rectangle.

Don't do this: Claim any rectangle's diagonals meet at 90°.

The correct way: In a rectangle the diagonals are equal and bisect each other, but they meet at 90° only when the rectangle is a square. The right-angle crossing depends on equal sides. See properties of a rectangle for the contrast.

Mistake 3: Thinking the interior-angle sum changes with size or rotation

Where it slips in: The second-guesser assumes a bigger square, or a tilted one, has a different angle total.

Don't do this: Recompute the sum for each square as if size or orientation mattered.

The correct way: The interior-angle sum of any square is always 360°, and each angle is always 90°, no matter the side length or rotation. Size changes area, not angles.

Conclusion

• Every interior angle of a square is 90°, and the four angles sum to 360°.

• The 360° total comes from splitting the square into two triangles, each contributing 180°.

• The diagonals cross at 90° and bisect each corner into two 45° angles, forming 45-45-90 triangles.

• The most common mistake is confusing the 45° split-corner with the 90° diagonal-crossing angle.

• A rectangle shares the 90° corners but only crosses its diagonals at 90° when it is a square.

A Practical Next Step

Practise these problems to solidify your understanding, and check each answer as you go.

1. A square is divided by one diagonal. What are the three angles of each triangle formed? (Answer to Question 1: 90°, 45°, 45°.)

2. The diagonals of square $WXYZ$ meet at $O$. State ∠WOX. (Answer to Question 2: 90°.)

3. What is the sum of the interior angles of a square rotated 30°? (Answer to Question 3: 360° — rotation does not change angles.)

To take the angles of a square further with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor for quadrilateral properties, or browse math classes online. Want to work these through with a live trainer? Try a free class.

Read More

• Square in Geometry — the complete guide to the square's sides, area, and diagonals.

• Is a Square a Rectangle — how the two shapes relate through their angles.

• Difference Between a Square and a Rectangle — where their properties diverge.

• Difference Between Square and Rhombus — angle and diagonal contrasts with the rhombus.

• Diagonal of Square Formula — the length of the diagonal that bisects each corner.