A square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.
A 5 by 5 square holds 25 unit squares, so its area is 52 = 25. The diagonal is longer than a side: 5√2 ≈ 7.07.
Formal definition and notation
A square is a quadrilateral ABCD with AB = BC = CD = DA and ∠ A = ∠ B = ∠ C = ∠ D = 90°. It is the regular quadrilateral. Its diagonals are equal, perpendicular, bisect each other and bisect the corner angles into 45° angles, and each has lengths√2.
A rhombus has four equal sides but its angles need not be 90°. A square is a rhombus with right angles.
Squarevs.Square number
A square number is the result of multiplying a whole number by itself, like 72 = 49. It is named after the square shape: 49 dots make a 7 by 7 square.
Quick check: 3 questions
Question 1 of 3
A square has a perimeter of 28 cm. What is its area?
Show answer and explanation
Answer: B. Side: 28 ÷ 4 = 7. Area: 72 = 49. Choice A squares the perimeter, C repeats it, and D halves it.
Question 2 of 3
A rhombus has one 90° angle. What is it?
Show answer and explanation
Answer: C. If one angle of a rhombus is 90°, all four are, and a rhombus already has four equal sides, so it is a square.
Question 3 of 3
What is the length of the diagonal of a square with side 4?
Show answer and explanation
Answer: A.d = √42 + 42 = √32 = 4√2 ≈ 5.66. B adds two sides, C is the area, and D assumes the diagonal equals a side.
Square: frequently asked questions
Is a square a rectangle?
Yes. A rectangle is any quadrilateral with four right angles, and a square has four right angles. The square is the special rectangle whose length and width are equal. So every square is a rectangle, but a 6 by 4 rectangle is not a square, because its sides are not all equal.
Why is x squared called squared?
Because x2 is the area of a square with side x. A square with side 3 is covered by 3 rows of 3 unit squares, which is 9, so 32 = 9. In the same way, x3 is called x cubed because it is the volume of a cube with edge x.
QuadrilateralA quadrilateral is a polygon with four straight sides, four vertices and four angles. Its four interior angles always add to 360 degrees. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are special quadrilaterals, each named for extra properties of its sides or angles.Geometry · Grade 2, Grade 3, Grade 5RhombusA rhombus is a quadrilateral with four sides of equal length. It is a special parallelogram, so its opposite sides are parallel and its opposite angles are equal. Its diagonals cross at right angles and cut each other in half, and its area is half the product of the diagonals.Geometry · Grade 3, Grade 4, Grade 5PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2ParallelogramA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Geometry · Grade 3, Grade 5, Grade 6Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4SymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometryInchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8CentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1Square NumberA square number is the result of multiplying a whole number by itself. The first square numbers are 1, 4, 9, 16, 25 and 36, because 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. That many dots fill a square exactly, like 16 dots in a 4 by 4 square.Operations · Grade 3, Grade 4, Grade 5Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Regular PolygonA regular polygon is a polygon whose sides are all the same length and whose angles are all the same size. The equilateral triangle, the square and the stop-sign octagon are regular. Each interior angle of a regular n-gon is (n − 2) × 180°n, and each exterior angle is 360°n.Geometry · Grade 5, Grade 8, GeometryExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8