work back from area to find height
How does Hero’s formula for triangle area also
make it easy for you to find the triangle’s height?
Solution
Using the other area formula, area - 1/2 base x height.
So, you can use Hero to find area and then divide by half the base to get height.
Hero’s formula and “1/2 base x height” work together
If you know three sides of a scalene triangle, you can use Hero’s formula to
find the area, and then use the formula you already know to find the height.
a
b
b
h
c
Semip
erimeter
Area = s (s-a) (s - b) (s - c)
s = a + b + c
2
Area = 1/2 b x h
h = 2 x area
b
Rearrange
Take y
our ar
ea value
f
rom her
e…
…and put i
t int
o the
other ar
ea f
ormula t
o
get the height.
198 Chapter 4
triangle properties
Q:
So do I actually ever need to do that gnarly three
Q:
Can I use Hero’s formula for isoceles and right triangles,
simultaneous equations thing? too?
A:No. It would work though, so if you ever can’t remember A:For a right triangle, 1/2 base x height is always easier as
Hero’s formula, it could get you out of a jam! But yeah—forget it. the two sides on the right angle give you base and height. For an
Sorry we did that to you.... isoceles triangle it’s up to you which you find easier.
Will the bargain screen still be visible to the
people at the back, 25 meters away?
So—can we use
the bargain
screen or what?
25m
28
31
30
T
he v
enue f
or
your gig
you are here 4 199
a perfect fit
Will the bargain screen still be visible to the
people at the back, 25 meters away?
28
31
30
Hero’s formula:
Area = s (s-a) (s - b) (s - c)
s = a + b + c
2
s = (31 + 30 + 28) / 2 = 44.5
Area = 44.5 (44.5-31) (44.5 - 30) (44.5 - 28)
= 44.5 x 13.5 x 14.5 x 16.5
= 143729 = 379.1 sq meters
Using conventional, 1/2 base x height, Height = 2 x area/base
Height = 2 x 379.1 / 30 = 25.27 m Perfect !
Do this bi
t firs
t then use i
t
in the
ar
ea f
ormul
a.
Combine tools from your toolbox to
get the answer you need.
Use the relationship between sides
and height to estimate angles.
There’s sometimes more than one
way to solve a problem—pick the
way that seems like the least amount
of work!
Draw sketches or graphs, or use your
hands if you’re stuck remembering
what those side-height-base
relationships are.
200 Chapter 4
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