Square Formulas (Area, Perimeter, Diagonal)

A square is a quadrilateral with four equal sides and four right angles. Its area is the side length squared, A = s²; its perimeter is four times a side, P = 4s; and its diagonal is d = s√2.

Square formulas

Area: $A = s^2$
Perimeter: $P = 4s$
Diagonal: $d = s\sqrt{2}$

where s = side length, A = area, P = perimeter, d = diagonal

Area

A = s^2

Worked example

A square with side 4 has area 16 square units.

A = s^2 = 4^2 = 16

Perimeter

P = 4s

Worked example

The same square has perimeter 16 units.

P = 4s = 4\cdot 4 = 16

Diagonal

d = s\sqrt{2}

Interactive square explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Side length s

Area A 16units²

Perimeter P 16units

Diagonal d 5.66units

Step-by-step substitution

A = s² = 4² = 16
P = 4s = 4 · 4 = 16
d = s√2 = 4 · 1.4142 = 5.66

Classroom use: square formulas

Who it's for: Grades 3–8 · Elementary and middle-school students meeting area and perimeter for the first time, and any student reviewing the basics before a geometry unit.

Learning objectives

After using this page, students will be able to:

Calculate the area of a square from its side length using A = s².
Calculate the perimeter of a square using P = 4s and explain why the two results carry different units.
Work backwards from a known area to recover the side length.

Suggested classroom activity

Project the explorer and drag the side slider from 1 to 10, asking students to predict the area before each release. The square-vs-linear growth becomes visible immediately, which sets up the units discussion (cm vs cm²) far better than a static table does.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`3.MD.C.7`	Relate area to multiplication and addition.
`3.MD.D.8`	Solve real-world problems involving perimeters of polygons.
`6.G.A.1`	Find the area of triangles, special quadrilaterals, and polygons by composing or decomposing shapes.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Common mistakes

Using 4s (perimeter) when the question asks for area s².
Forgetting the √2 factor on the diagonal — the diagonal is longer than a side.

Related & next steps

Add square to your formula sheet →

Frequently asked questions

What is the formula for the area of a square?

The area of a square is A = s², the side length multiplied by itself. For a square with side 5, the area is 5² = 25 square units.

How do you find the diagonal of a square?

The diagonal of a square is d = s√2, because the diagonal splits the square into two 45-45-90 right triangles. A square with side 6 has a diagonal of 6√2 ≈ 8.49.

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