Head First 2D Geometry

quadrilaterals

Use diagonals to find the area of the kite

Just like the parallelogram, there’s a shortcut we can take to find the area

of a kite. All we need to do is multiply the lengths of the two diagonals

together, and divide by 2:

Kite Area = 1/2 (diagonal 1 × diagonal 2)

Diagonal 1

Diagona

l 2

Diagona

l 2 is the base

1/2 Diagona

l 1

(the height)

How much should you charge for maintaining this lawn? We’ve

added in some extra measurements to help you.

Don’t forget that you’ll still need to edge round the flower bed,

even though you aren’t mowing it.

30 meters

20 meters

10.25 meters

3.75 meters

4.5 meters

13 meters

W

atch out f

or the flowers when

you’r

e edging, they’re exp

ensiv

e!.

you are here 4 257

sharpen solution

How much should you charge for maintaining the lawn? We’ve

added in some extra measurements to help you.

Don’t forget that you’ll still need to edge round the flower bed,

even though you aren’t mowing it.

30 meters

20 meters

10.25 meters

3.75 meters

4.5 meters

13 meters

Area of rectangle = 20 x 30

= 600 meters

2

Perimeter of rectangle = (2 x 20) + (2 x 30)

= 40 + 60

= 100 meters

Area of kite = 1/2 (4.5 x 13)

= 1/2 (58.5)

= 29.25 meters

2

Perimeter of kite = (2 x 3.75) + (2 x 10.25)

= 7.5 + 20.5

= 28 meters

Total area to maintain = 600 - 29.25

= 570.75 meters

2

Total perimeter to maintain = 100 + 28

= 128 meters

Total charge = $0.05 x 570.75 + $0.10 x 128

= $28.54 + $12.80

= $41.34

Round t

o 2

decima

l pl

ac

es.

258 Chapter 6

quadrilaterals

Kites Up Close

Diagonals

The diagonals of a kite are perpendicular,

but that’s not all there is to say about them.

For starters, one diagonal bisects the other,

meaning it chops the other diagonal in half.

It also bisects the pair of opposite angles, and

if you look at the remaining pair of angles,

they’re congruent, too. So there’s a lot you

can know about a kite without having to do

any calculations!

Area and perimeter

As you discovered earlier, you find the area of a kite by

multiplying together the lengths of the two diagonals and

dividing by two. To find the perimeter, remember that

there are two pairs of congruent sides so you only have to

add the two different sides together and multiply by two.

T

his diagona

l bisects

the other diagonal.

T

he diagona

l bisects

these angles, too.

T

hese angles ar

e congruent.

Q:

So the diagonals of a kite

are perpendicular. What about the

diagonals of a parallelogram, are they

perpendicular, too?

A:In general, parallelograms don’t have

perpendicular diagonals.

The diagonals of a parallelogram are

still important though. If a shape is a

parallelogram, then its diagonals bisect

each other. Try adding diagonals to the

parallelograms earlier in the chapter and

you’ll see what we mean.

Q:

Could I have calculated the area

of the kite by splitting it into simpler

shapes like before?

A:You could, but it would have taken

you much longer to calculate. All you really

need to do is multiply the two diagonals

together and divide the result by 2.

Q:

The kites we’ve looked at in this

chapter look symmetrical. Is that a

coincidence?

A:No, not at all. Every kite is

symmetrical along one diagonal.

Q:

Can a shape be both a

parallelogram and a kite?

A:Yes it can. A shape is a parallelogram

and a kite if it fits the description of both.

In other words, it must have two pairs of

separate adjacent congruent sides, and

also the opposing sides must be parallel.

This means that all four sides must be

congruent.

An example of a shape that is both a

parallelogram and a kite is a square. All four

sides are congruent, and opposite sides are

parallel. We’ll get to that in a little bit.…

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