Square Area Calculator

A square’s area can be found from whichever measurement you already have on hand — not just the side length. This calculator solves for area, perimeter, and diagonal together from any one of five starting points: side length, diagonal, perimeter, apothem (inradius), or even the area itself worked backward. A separate mode also finds the area, side lengths, and perimeter of a four-sided shape directly from its four corner coordinates.

Enter whichever value you know, and the calculator shows the area alongside the perimeter and diagonal — every mode returns all three together, not just the one number you asked for.

Square Area Formula

Area = Side²

Where:
Side = the length of any one edge of the square, since all four edges are equal

The perimeter and diagonal follow directly from the side length too: Perimeter = 4 × Side, and Diagonal = Side × √2 (from the Pythagorean theorem, since a square’s diagonal is the hypotenuse of a right triangle formed by two adjacent sides).

By Side Length, Diagonal, or Perimeter

Since side, diagonal, and perimeter are all mathematically tied to each other, this calculator lets you enter any one of them and solves for the rest — along with the area — automatically.

By Side Length

Area = Side², Perimeter = 4 × Side, Diagonal = Side × √2

Worked Example

Side = 6 ft: Area = 6² = 36 square feet, Perimeter = 4 × 6 = 24 feet, Diagonal = 6 × √2 ≈ 8.49 feet.

By Diagonal

Side = Diagonal ÷ √2, then Area = Side² and Perimeter = 4 × Side

Worked Example

Diagonal = 10 ft: Side = 10 ÷ √2 ≈ 7.07 ft. Area ≈ 7.07² ≈ 50 square feet, Perimeter ≈ 4 × 7.07 ≈ 28.28 feet.

By Perimeter

Side = Perimeter ÷ 4, then Area = Side² and Diagonal = Side × √2

Worked Example

Perimeter = 24 ft: Side = 24 ÷ 4 = 6 ft, Area = 6² = 36 square feet, Diagonal ≈ 8.49 feet — the same square as the Side Length example above, arrived at from a different starting measurement.

By Apothem (Inradius)

Side = 2 × Apothem, then Area = Side² and Perimeter = 4 × Side

The apothem (also called the inradius) is the distance from the center of the square to the midpoint of any side — exactly half the side length, since the incircle touching all four sides has a radius equal to half the square’s width.

Worked Example

Apothem = 3 ft: Side = 2 × 3 = 6 ft, Area = 36 square feet, Perimeter = 24 feet, Diagonal ≈ 8.49 feet.

By Area — Finding the Side Length (Reverse)

Side = √Area, then Perimeter = 4 × Side and Diagonal = Side × √2

This reverses the usual direction: instead of computing area from a measurement, it works backward from a known area to find the side length, perimeter, and diagonal — useful when you know how much floor or land area you need and want to know what size square would cover it.

Worked Example

Area = 49 sq ft: Side = √49 = 7 ft, Perimeter = 4 × 7 = 28 feet, Diagonal = 7 × √2 ≈ 9.90 feet.

By Coordinates (Four Corner Points)

Area = 0.5 × |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|, the shoelace formula, applied to the four entered points in order

This mode finds the area, all four side lengths, and the perimeter directly from four (x, y) corner points — a genuine gap in most other square-area calculators, which stop at side, diagonal, and perimeter inputs. Since the shoelace formula works for any simple quadrilateral, not just a perfect square, this mode reports whatever the four points actually form; it doesn’t assume or verify that the result is a true square with four equal sides and right angles.

Worked Example

Points (0, 0), (4, 0), (4, 4), and (0, 4): Area = 0.5 × |(0×0−4×0) + (4×4−4×0) + (4×4−0×4) + (0×0−0×4)| = 0.5 × |0 + 16 + 16 + 0| = 16 square units. All four side lengths come out to 4 units each, giving a perimeter of 16 units — confirming these four points do form a true square.

Frequently Asked Questions

How do I find a square’s area from its perimeter?

Divide the perimeter by 4 to get the side length, then square that result: Area = (Perimeter ÷ 4)². A 24-ft perimeter gives a 6-ft side and a 36-square-foot area.

How do I find a square’s area from its diagonal?

Divide the diagonal by √2 to get the side length, then square it: Area = (Diagonal ÷ √2)². A 10-ft diagonal gives a side of about 7.07 ft and an area of about 50 square feet.

What’s the difference between apothem and side length?

The apothem is the distance from the center of the square to the midpoint of a side — exactly half the side length. A square with a 3-ft apothem has a 6-ft side, since Side = 2 × Apothem.

Does the Coordinates mode work if my four points don’t form a perfect square?

Yes — the shoelace formula calculates the area of whatever simple quadrilateral the four points form, and the calculator reports the actual side lengths so you can check whether they’re a true square (four equal sides) or an irregular four-sided shape.