Head First 2D Geometry

factoring in perimeter

Calculate the perimeters of the two lawns. How does this account

for the difference in time spent on each lawn? What do you think

Edward should do?

45 meters

16 meters

34 meters

Parallelogram perimeter = 45 + 34 + 45 + 34

= 158 meters

36 meters

20 meters

Rectangle perimeter = 36 + 20 + 36 + 20

= 112 meters

It takes longer to maintain the parallelogram lawn because the perimeter is so much greater

and therefore takes longer to edge. Edward’s charges need to change so that lawn perimeter is

considered, too.

Same shape, different perimeters

Your friends ran into a similar problem back in Chapter 4 when they

were picking out a venue for the rock festival—the biggest field didn’t

necessarily have the most perimeter. For our lawn calculations, even if we

know the area of a particular lawn, we can’t make any assumptions about

its perimeter or how long it will take to edge the lawn. And currently,

Edward’s charges are only based on lawn area.

Same Area ≠ Same Perimeter

250 Chapter 6

quadrilaterals

Q:

So it’s not just triangles that have

perimeter?

A: No, not at all. The perimeter of a

shape is basically the edge around the

outside of a shape, whether it’s a triangle,

rectangle, parallelogram, or some other

shape.

Q:

Are there any shortcuts we can

take in calculating the perimeter of a

particular shape?

A:It all depends on the shape.

You calculate the perimeter of a shape by

adding the lengths of all its sides together.

You can take a few shortcuts if some of

these sides are the same length.

As an example, a square has four sides that

are all the same length, so the perimeter of

a square is just 4 times the length of one

side.

Q:

What about for a parallelogram?

A:A parallelogram has two pairs of

congruent sides, so there are two unique

side lengths. You can find the perimeter of

a parallelogram by adding these two unique

side lengths together, and then multiplying

by 2.

Q:

What if two shapes have the same

area and the same perimeter? Does this

mean they’re the same shape?

A:Not automatically, no, you need

more information. Many different shapes

can have the same area and the same

perimeter.

Q:

More information? Like what?

A:Well, a good starting point is if you

know what sort of shapes you’re dealing

with. This will tell you how many sides your

shapes have, and how many of these are

congruent.

As an example, if you’re told you have two

squares and they both have the same area,

the squares must be congruent. This is

because you calculate the area of a square

by calculating (side length)

2

. The only way

the squares can have the same area is if

their sides are the same length.

In general, if you know what type of shape

you’re dealing with, you can use the

properties of the shape to help you deduce

further facts.

 A parallelogram is a four-sided shape  Consecutive angles of a

whose opposite sides are parallel to parallelogram are supplementary.

each other.

 To calculate parallelogram area,

 Opposite sides of a parallelogram multiply the width by the height:

are congruent.

 To calculate parallelogram perimeter,

 Opposite angles of a parallelogram add together the length of each side.

are congruent.

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