CDS Mathematics · Quadrilaterals

Rhombus and Kite

A rhombus's diagonals bisect each other at right angles; a kite's diagonals are also perpendicular, so both have area half the product of the diagonals.

All Quadrilaterals notes CDS Mathematics strategy

Why this matters

Eight PYQs, none HARD. Every one uses the right triangle made by half of each diagonal and a side: side² = (d₁/2)² + (d₂/2)², so d₁² + d₂² = 4 × side². Three are data-sufficiency items, and in two of them the question's own data decide the answer before either statement is read.

Concept 1 of 1: Diagonals, side and area

The two diagonals cross at right angles and halve each other, splitting the rhombus into four congruent right triangles. Each has the half-diagonals as legs and a side as hypotenuse.

Definition

Rhombus: diagonals are perpendicular bisectors of each other. $s^2 = \left(\dfrac{d_1}{2}\right)^2 + \left(\dfrac{d_2}{2}\right)^2$ , so $d_1^2 + d_2^2 = 4s^2$ .
Area $= \dfrac12 d_1 d_2$ (also $s^2\sin\theta$ ).
A diagonal equal to the side: the rhombus is two equilateral triangles, with diagonals $s$ and $s\sqrt3$ .
Kite ( $AB = BC$ , $CD = DA$ ): the diagonals are perpendicular, so area $= \dfrac12 d_1 d_2$ too.

Rhombus

d_1^2 + d_2^2 = 4s^2, \qquad S = \tfrac12 d_1 d_2

Worked example

A rhombus has area

120

^2

and one diagonal

10

cm. Find its perimeter.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q86Easy

Example 1 · Quadrilaterals · Rhombus and Kite

The area of a rhombus is 96 square cm and one of its diagonals is of length 12 cm. What is the perimeter of the rhombus ?

Half-diagonals are the legs

The right triangle uses HALF of each diagonal. Diagonals of

10

and

24

give a side of

13

, not

26

The side alone can be enough

Given the side,

d_1^2 + d_2^2 = 4s^2

is fixed for EVERY rhombus. In data-sufficiency items that means no statement is needed.

Drill 7 more on diagonals, side and area

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Diagonals, side and area
Rhombus
$d_1^2 + d_2^2 = 4s^2, \qquad S = \tfrac12 d_1 d_2$

Watch out for (2)

Half-diagonals are the legs
→ Diagonals, side and area
The side alone can be enough
→ Diagonals, side and area

Test yourself on Quadrilaterals

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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