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Home < Algebra Starter Lessons < 8 Unifying Theme For Algebra << 5 Triangle Area Formula Backwards
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Triangle Area Formula Forwards and Backwards
Forward Use
Example. Find the area of a triangle with base 12 by and height 5
Solution
Draw a triangle.
Data identification step. The triangle has base length $B = 12 $
and height $H = 5$. This data could be marked on the diagram instead of
being written below it.
Formula Evaluation Step. Triangle area
\begin{eqnarray*} A &=& \frac 12 B \times H \\ &=&\frac
12 (12)(5) \\ &=& 6 \times 5 \\ & =& 30 \end{eqnarray*}
Remark. Many different and unsimilar triangles have base 12 and
hieght 5 km. The sketch above just picks one. The above calculations
show all such triangles would have the same area.
Backward Use - Numerically
Example 2. Find the height of triangle with base 11 and area 81
Arithmetic Solution
We use the letters L, W and A because of previous use of the
The triangle area calculation formula
\[ A = \frac12 BH \]
We are given values A =81 and H = 11. Substitution in the formula
yields
\[ 81 = \frac12 11 H \] Multiply both sides by 2 to obtain \[ 2
\times 81 = 11H \] Multiply both sides by $\frac 1{11}$ to find \[ \frac{
2 \times 81}{11} = H \] The latter gives \[H = \frac{162}{11} = 14 +
\frac 8{11} \]
Check: \begin{eqnarray*} A &=& \frac12 BH \\ &=&
\frac12 \cdot 11 \cdot \frac{162}{11} \\ &=& \frac12 \times 162
\\ &=& 81 \end{eqnarray*}
Exercise: Find the height of triangle with height 4 and area 17
numerically.
Backward Use - Algebraically
Example 2 - Generalization. When the base and area of a area are
given or known, how can one find its height.
Algebraic or Literal Solution
Draw a Triangle.
The triangle area formula says area
\[ \frac12 BH = A \]
In it, base B and area A are supposedly known or given. Multiply both
sides by two to clear the fraction. That implies
\[ BH = 2 A \]
That implies give
\[ \frac {BH}B = \frac{2A}B \]
and hence \[ H = \frac{2A}B\]
That should be the answer. It should satisfy \[ \frac12 BH = A \]
Check: the right hand side \begin{eqnarray*} \frac12 BH &=&
\frac12 \cdot B \cdot \frac{2A}{B} \\ &=& \frac12 \times 2A
\\ &=& A \end{eqnarray*}
Now the heigth formula \[ H = \frac{2A}B\] can be used to find H
whenever the triangle area A and base B are given. Likewise, the
formula \[B = \frac{2A}H\] can be derived and then used to find H
whenever the triangle area A and height B are given. Thus algebraic
solutions that follow the pattern of a numerical solution may give
general formulas for a solution. The
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Work Booklets for ages 3+ to 13 Use these or others to check
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the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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