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.
A square is divided by one diagonal. What are the three angles of each triangle formed? (Answer to Question 1: 90°, 45°, 45°.)
The diagonals of square $WXYZ$ meet at $O$. State ∠WOX. (Answer to Question 2: 90°.)
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.
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