Book A Free Math Class # Angles of a Square - Interior, Diagonal, and Sum of Angles [Geometry](/content/tag/geometry/index.html) TL;DR Every interior angle of a square measures exactly 90°, so the four angles sum to 360°. Its two diagonals cross at the centre at 90° and split each 90° corner into two 45° angles. This article proves each of these facts, works through examples, and separates the square's angle behaviour from that of a general rectangle or quadrilateral. BT Bhanzu Team Last updated on July 22, 20268 min read ## 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°÷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](https://www.cuemath.com/geometry/angles-of-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](/content/math/geometry/angles-of-quadrilateral/index.html). A square is the special case where all four angles are equal: each angle = 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](/content/math/geometry/angles-of-rectangle/index.html). ## 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: 90°2 = 45° So ∠OAB = 45° and ∠OBA = 45°. In triangle OAB, the three angles must total 180°, giving: ∠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°. one angle = 360°4 = 90° Each interior angle is **90°**. ### Example 2 **A diagonal of square PQR is drawn from P to R. What angle does it make with side PQ?** A diagonal bisects the 90° corner at P. 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°. ∠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](/content/math/geometry/similar-triangles/index.html). ### 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: 4×90°=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 4×90°=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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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](/content/math/geometry/properties-of-rectangle/index.html) 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.