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Toolbox · Square calculator

Square calculator

Side, diagonal, area or perimeter as input; get side, diagonal, perimeter, area, circumradius and inradius

Formula

P=4a,\quad S=a^2,\quad d=a\sqrt{2}

P = 4a, S = a^2, d = a sqrt2

Enter side a; everything else follows.

Presets

Tap an edge, angle or region to see how it is computed

Side and diagonal

Angles

Perimeter, area, circles

Relationd7.071

d=a\sqrt{2}=5\times1.4142=7.071

Area S25

Perimeter P20

Diagonal d7.071

Circumradius R3.536

a5d7.071∠A90°∠B90°∠C90°∠D90°P20S25R3.536r2.5

Steps

6 / 6

1Given
$a=5,\ \angle A=\angle B=\angle C=\angle D=90^\circ$
Collect the given quantities first.
2Perimeter
$P=4a=4\times5=20$
Add all sides to get the perimeter.
3Area
$S=a^2=5^2=25$
Apply the area formula; the unit is squared.
4Pythagoras
$d=\sqrt{a^2+a^2}=a\sqrt{2}=5\times1.4142=7.071$
In a right triangle the squares of the legs add up to the square of the hypotenuse.
5Circumcircle
$R=\frac{d}{2}=3.536,\quad r=\frac{a}{2}=2.5$
Circumradius from R = abc/(4S) or half the diagonal.
6Result
$a=5,\ d=7.071,\ P=20,\ S=25$
Collect all computed quantities.

Concept

A square is both a rectangle and a rhombus: four equal sides, four right angles, perpendicular equal diagonals.

Area can be a^2 from the side or d^2/2 from the diagonal - handy when only the diagonal is given.

Going further

Why S = d^2/2: The diagonals cut the square into four congruent isosceles right triangles of area d^2/8 each.
Incircle and circumcircle: Inradius r = a/2 and circumradius R = a sqrt2/2, so r : R is always 1 : sqrt2.
Square and irrationals: A unit square has diagonal sqrt2, one of the first irrational numbers ever discovered.

Common mistakes

Area is in cm2, perimeter in cm - keep them apart.
From a perimeter, divide by 4 first, then square.
The diagonal is sqrt2 times the side, not twice the side.

Page keywordssquare calculatorarea of a squaresquare diagonalperimeter of a squarehow to find the diagonal of a squareside length from areasquare circumradiusperimeter of square formulasquare vs rhombus

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