Diagonal of a Rectangle Calculator
Calculate the diagonal of a rectangle seven different ways: from length and width, from one side and the diagonal, from one side and area, from perimeter and area, from the circumcircle radius, or from a side and the angle between diagonals. The result panel also shows area, perimeter, angle between diagonals, and circumscribed-circle radius.
Formula
Worked example
For a rectangle 8 by 6 cm: diagonal = sqrt(8² + 6²) = sqrt(64 + 36) = sqrt(100) = 10 cm. Area = 8 x 6 = 48 cm². Perimeter = 2 x (8 + 6) = 28 cm. Circumcircle radius = 10 / 2 = 5 cm. Angle between diagonals = 2 x arctan(6/8) = 2 x 36.87 deg = 73.74 deg. This is the classic 6-8-10 right triangle (scaled 3-4-5).
How the diagonal of a rectangle is calculated
A diagonal of a rectangle connects two opposite corners and cuts the rectangle into two congruent right triangles. In each triangle the length and width are the two legs, and the diagonal is the hypotenuse. The Pythagorean theorem therefore gives the diagonal directly: d = sqrt(l^2 + w^2). Because addition does not care about order, swapping which side you call the length and which the width never changes the result, so both diagonals of any rectangle are always equal.
Seven ways to solve for the diagonal
You do not always have both sides. This calculator handles seven common knowledge configurations. If you know one side and the diagonal you can reverse-solve the missing side: w = sqrt(d^2 - l^2). If you know one side and the area you get the other side by division (w = A / l), then apply the theorem. If you know only the perimeter P and area A, the shortcut formula d = (1/2) sqrt(P^2 - 8A) gives the diagonal without needing the individual sides. If you have the circumscribed-circle radius r, the diagonal is simply d = 2r, since every rectangle fits perfectly inside a circle whose diameter equals the diagonal (Thales theorem). Finally, if you know a side and the acute angle where the two diagonals cross, the other side follows from w = l x tan(alpha / 2).
Area, perimeter, circumcircle, and angle between diagonals
This calculator always returns four companion quantities when both sides are known. The area is length times width (A = l x w). The perimeter is the total edge length (P = 2l + 2w). The circumcircle radius is half the diagonal (r = d / 2); every rectangle has exactly one circumscribed circle and its center sits at the intersection of the two diagonals. The angle between the diagonals is twice the arctangent of the ratio of the sides: alpha = 2 arctan(w / l). For a square this angle is exactly 90 degrees; for very elongated rectangles it approaches 0 or 180 degrees. Knowing this angle is useful in joinery, picture framing, and checking whether a four-sided frame is truly rectangular.
Practical applications and the 3-4-5 check
Builders and carpenters use the diagonal as a squareness check: measure both diagonals of a frame or slab, and if they are equal the four corners are right angles. The classic 3-4-5 triangle (or its multiples: 6-8-10, 9-12-15) is widely used because the whole-number sides make layout quick and the diagonal is guaranteed to be a whole number. Tile setters use the diagonal of each tile to ensure grout lines stay straight. Screen sizes (TVs, monitors, phones) are always quoted as the diagonal measurement, so this calculator lets you recover the actual width and height from the diagonal and aspect ratio. Entering the diagonal and one side (ld or wd mode) does exactly that.
Common rectangle diagonals with full properties
| Length | Width | Diagonal | Area | Perimeter | Circ. radius | Angle (deg) |
|---|---|---|---|---|---|---|
| 3 | 4 | 5 | 12 | 14 | 2.5 | 106.26 |
| 6 | 8 | 10 | 48 | 28 | 5 | 73.74 |
| 5 | 5 | 7.071 | 25 | 20 | 3.536 | 90.00 |
| 12 | 5 | 13 | 60 | 34 | 6.5 | 45.24 |
| 10 | 4 | 10.770 | 40 | 28 | 5.385 | 43.60 |
| 16 | 9 | 18.358 | 144 | 50 | 9.179 | 60.26 |
| 1 | 1 | 1.414 | 1 | 4 | 0.707 | 90.00 |
All values computed from d = sqrt(l² + w²). Angle is between the two diagonals at their crossing point.
Frequently asked questions
How do you find the diagonal of a rectangle?
Use the Pythagorean theorem: d = sqrt(l^2 + w^2). Square the length, square the width, add the two results, then take the square root. For a rectangle 8 by 6, the diagonal is sqrt(64 + 36) = sqrt(100) = 10. Both diagonals of a rectangle are equal.
Can I find the diagonal if I only know the area and perimeter?
Yes. Use the formula d = (1/2) * sqrt(P^2 - 8A), where P is the perimeter and A is the area. This works because the sides satisfy l + w = P/2 and l * w = A, from which you can reconstruct l^2 + w^2 = (l+w)^2 - 2lw = (P/2)^2 - 2A, so d = sqrt((P/2)^2 - 2A). Select "Perimeter and area" in the solve-from menu to use this mode.
What is the relationship between the diagonal and the circumscribed circle?
Every rectangle has a unique circumscribed circle (the smallest circle that passes through all four corners). The diameter of this circle equals the diagonal of the rectangle, so the radius is simply r = d / 2. This follows from Thales theorem: any angle inscribed in a semicircle that subtends the diameter is a right angle, and each corner of the rectangle subtends the diagonal, which is the diameter.
What is the angle between the diagonals of a rectangle?
The acute angle alpha where the two diagonals cross satisfies alpha = 2 * arctan(w / l), where w is the shorter side and l is the longer side. For a square (l = w) this is exactly 90 deg. For a 6 by 8 rectangle, alpha = 2 * arctan(6/8) = 73.74 deg. The two diagonals also create a supplementary angle of 180 - 73.74 = 106.26 deg on the other side of the crossing point.
Are both diagonals of a rectangle the same length?
Yes. A rectangle has two diagonals, and they are always equal because each is the hypotenuse of a right triangle with the same two legs. They also bisect each other, meeting at the center of the rectangle and dividing it into four triangles of equal area.
What is the diagonal of a square?
A square is a rectangle with equal sides, so its diagonal is d = sqrt(s^2 + s^2) = s * sqrt(2), about 1.4142 times the side length. For a square with 5-unit sides the diagonal is 5 * 1.4142 = 7.071 units. The diagonals of a square are perpendicular, crossing at exactly 90 deg.
How do TV and monitor screen sizes use the diagonal?
Display sizes are always quoted as the diagonal measurement. To find the actual width and height you need the aspect ratio (e.g. 16:9). Set the longer side to 16 and the shorter side to 9 in this calculator to see the diagonal for that unit of measure, then scale by the real diagonal. Alternatively use the reverse-solve mode (length and diagonal) by entering one side and the quoted diagonal to recover the other dimension.