Head First 2D Geometry

Jus

t try

drawing a

triangle tha

t

br

eak

s this

rule!

regular polygons

You’ve found the formula for the area of an equilateral triangle

Recognizing special shapes is a great shortcut. Being able to spot

that a quadrilateral is a square, or that a triangle is an equilateral

triangle can save you a lot of work by letting you use extra tools

and formulas which apply to that special shape specifically.

Any old triangle Equilateral triangle

Sides

Angles

Area

Are all 60º

Add up to 180º

1/2 base × height

= 3 t

2

or Hero’s formula

4

Longest must be shorter than

other two sides added together

Are equal lengths

Use your new equilateral triangle area formula to find the butt

count of the triangular-shaped hot tub.

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sharpen solution

Use your new equilateral triangle area formula to find the butt-

count of the triangular shaped hot tub.

3t

2

4

3t

2

4

Area =

Perimeter = 3t

= 3 x 3.72

= 11.17m

Butts = 11.17 = 22.3 butts

0.5

6 =

24 = t

2

3

3.72m = t

Length of

each side

Q:

Why bother learning another

formula? Won’t I be covered if I just

know Hero’s formula and the “1/2 base

× height” version?

A:Any time you use Hero’s formula

to work on a equilateral triangle you’re

gonna wind up doing all the math we just

did anyway. If you can learn the equilateral

triangle area formula then you won’t need to

do all that algebra again!

Q:

But what if I wanted to use “1/2

base × height.” I thought in geometry it

didn’t matter how you work stuff out, the

answer should be the same?

A:That’s true, and starting from 1/2

base × height, you’d have ended up with

the exact same formula. You just have to do

more work to get there. First you’d need to

use the Pythagorean Theorem to find the

sides in terms of the height, and then you’d

need to do some simultaneous equations to

put it all together. It would definitely work but

there’s a lot more to go wrong.

Q:

OK, so this formula only works

for equilateral triangles? What about

triangles that are nearly equilateral?

A:You could use it to approximate area

for a triangle with sides of similar length, but

it wouldn’t be accurate. It’s a good way to

check your answer in an exam though. Pick

the middle side length, work the equilateral

triangle area formula and it should come out

in the same ball park. If your two answers

are 9 and 10, you’re all good; if they’re 9

and 50, you know you need to check your

work.

294 Chapter 7

Frank

Jill

Joe

regular polygons

Dude, what did you

say the circle tub

butt-count was again?

Jill: Uh, I think it was like 19 point something.

Frank: No, that’s the square isn’t it?

Jill: Oh, yeah. Sorry. It’s on this other piece of paper.

Frank: And did you keep track of what the

dimensions were for the carpenters?

Joe: Yeah, well, uh, not exactly, but I did scribble it

down in my notes somewhere so I must have it.…

Hmm. This is getting messy fast. Do you think there’s a more

efficient way to keep track of the calculations for each hot tub?

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