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Rectangle calculator

Five known-condition modes for sides, perimeter, area, diagonal, circumradius and diagonal angle, with a clickable figure

Formula

P=2(a+b),\quad S=ab,\quad d=\sqrt{a^2+b^2}

P = 2(a + b), S = ab, d = sqrt(a^2 + b^2)

Enter length a and width b to get perimeter, area and diagonal at once.

Presets

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

Sides

Angles

Diagonal and angles

Relationd5

d=\sqrt{3^2+4^2}=5

d = √(a² + b²) = √(3² + 4²) = 5; the two diagonals are equal and bisect each other at the centre O

Area S12

Perimeter P14

Diagonal d5

Circumradius R2.5

a3b4d5∠A90°∠B90°∠C90°∠D90°P14S12R2.5α53.13β36.87

Steps

7 / 7

1Given
$a=3,\ b=4,\ \angle A=\angle B=\angle C=\angle D=90^\circ$
List what is known first; every angle of a rectangle is 90°
2Perimeter
$P=2(a+b)=2\times(3+4)=14$
P = 2(a + b) = 2 × (3 + 4) = 14
3Area
$S=ab=3\times4=12$
S = a × b = 3 × 4 = 12
4Pythagoras
$d=\sqrt{a^2+b^2}=\sqrt{3^2+4^2}=\sqrt{25}=5$
d² = a² + b² = 9 + 16 = 25, so d = √25 = 5
5Find angle
$\tan\alpha=\frac{b}{a}=\frac{4}{3}=1.3333\Rightarrow\alpha\approx53.13^\circ$
tan α = b ÷ a = 4 ÷ 3 = 1.3333; take the inverse tangent to get α ≈ 53.13°
6Angle sum
$\beta=90^\circ-53.13^\circ=36.87^\circ$
The diagonal makes complementary angles with the two sides, α + β = 90°
7Result
$a=3,\ b=4,\ d=5,\ P=14,\ S=12,\ R=2.5$
a = 3, b = 4, d = 5; P = 14, S = 12, R = 2.5; α = 53.13°, β = 36.87°

Concept

A rectangle is a parallelogram with four right angles: opposite sides are equal and parallel, and the diagonals are equal and bisect each other.

Perimeter is the total length of the four sides; area is length times width, i.e. a unit squares in each of b rows.

Going further

Why the diagonals are equal: Both diagonals are hypotenuses of a right triangle with legs a and b, so by Pythagoras they must be equal.
Can perimeter and area be swapped: No. With the same 20 cm perimeter, 1x9 gives 9 cm2 while 5x5 gives 25 cm2.
Golden ratio: When a : b is about 1.618 the rectangle looks most pleasing; this golden rectangle appears in paper sizes, architecture and photography.

Common mistakes

Perimeter uses cm, area uses cm2 - never mix the units.
With a known perimeter, a + b = P/2; do not divide P by 4.
The diagonal is always longer than either side; d < a means the input is wrong.

Page keywordsrectangle calculatorrectangle arearectangle perimeterdiagonal of a rectanglerectangle sidesrectangle sides from perimeter and areadiagonal of a rectangle formulacircumradius of a rectangleangle between diagonal and side

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