One Diagonal Proves Every Quadrilateral Adds to 360

Because a single diagonal splits it into two triangles, and each triangle's angles total $180°$. Two triangles give $2\times 180=360°$.

That one line of reasoning works for every quadrilateral, however irregular. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: the angle sum property, the properties of a rectangle, the properties of a square, and calculating unknown measures.

A rectangle is a quadrilateral with all four angles equal to $90°$. Its properties are:

- Opposite sides are equal and parallel
- All four angles are $90°$
- The diagonals are equal in length
- The diagonals bisect each other, so each is cut into two equal halves

Because each angle is $90°$, a diagonal creates a right-angled triangle, and Pythagoras gives its length.

Worked example. A rectangle measures $8$ cm by $6$ cm. Its diagonal is

$$\sqrt{8^2+6^2}=\sqrt{64+36}=\sqrt{100}=10\text{ cm}$$

Both diagonals are $10$ cm, and since they bisect each other, each half is $5$ cm.

Worked example in reverse. A rectangle has a diagonal of $13$ cm and one side of $5$ cm, so the other side is

$$\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12\text{ cm}$$

A door frame or a notebook page is a rectangle, and measuring both diagonals is exactly how a carpenter checks that a frame is square rather than leaning.

The property to be careful with is that the diagonals bisect each other but do not cross at right angles. They are equal and they share a midpoint, yet the angle between them is generally not $90°$ — that extra property belongs to the square, which is the next section.

A square is a rectangle with all four sides equal, so it inherits every rectangle property and adds more.

From being a rectangle:

- Opposite sides equal and parallel
- All angles $90°$
- Diagonals equal
- Diagonals bisect each other

And in addition:

- All four sides are equal
- The diagonals **bisect each other at $90°$
- Each diagonal bisects the angles** of the square, cutting each $90°$ corner into two angles of $45°$

So a diagonal of a square creates an isosceles right-angled triangle, with two angles of $45°$.

Worked example. A square has side $6$ cm. Its diagonal is

$$\sqrt{6^2+6^2}=\sqrt{72}\approx 8.49\text{ cm}$$

In general the diagonal of a square of side $s$ is $s\sqrt{2}$, since $\sqrt{s^2+s^2}=s\sqrt{2}$.

Worked example with angles. In a square $PQRS$, the diagonal $PR$ makes an angle with side $PQ$ of

$$\frac{90}{2}=45°$$

and where the two diagonals meet, each of the four angles is $90°$.

A carrom board and a chessboard are squares, and folding a square sheet of paper corner to corner produces the $45°$ angles directly.

The angle-bisecting property is the one that separates a square from a rectangle, and it explains why the $45°$ appears. In a rectangle a diagonal makes unequal angles with the two sides, because those sides have different lengths — only when the sides are equal does the diagonal split the corner evenly.

Identify which property gives you the missing piece, then set up an equation.

Using equal diagonals. In rectangle $ABCD$ the diagonals meet at $O$, with $AO=(3x-4)$ cm and $OC=(x+6)$ cm. Since the diagonals bisect each other, $AO=OC$:

$$3x-4=x+6\Rightarrow 2x=10\Rightarrow x=5$$

So $AO=11$ cm, and the whole diagonal $AC=22$ cm.

Using equal opposite sides. In a rectangle, $AB=(2y+3)$ cm and $DC=15$ cm. Opposite sides are equal, so

$$2y+3=15\Rightarrow y=6$$

Using the diagonal of a square. A square has a diagonal of $10$ cm. Since $2s^2=100$:

$$s^2=50\Rightarrow s=\sqrt{50}\approx 7.07\text{ cm}$$

Using angles at the diagonal crossing. In a square, the diagonals meet at $90°$, so if one angle there is given as $(5x)°$, then $5x=90$ and $x=18$.

Perimeter and area. A rectangle $8$ cm by $6$ cm has perimeter $2(8+6)=28$ cm and area $8\times 6=48$ cm². A square of side $6$ cm has perimeter $24$ cm and area $36$ cm².

The step that decides these questions is naming the property first. Writing *diagonals of a rectangle bisect each other, so $AO=OC$* before the equation both earns the mark and tells you which two expressions to equate.