Question

bianca calculated the height of the equilateral triangle with side lengths of 10.$\tan(30) = \frac{5}{h}$$h approx 8.7$then, she used the formula for area of a triangle to approximate its area, as shown below.$a = \frac{1}{2}bh$$= \frac{1}{2}(10)(8.7)$$approx 43.5$ units²calculate the area of the equilateral triangle using the formula for area of a regular polygon, and compare it to biancas answer.the apothem, rounded to the nearest tenth, is $\boldsymbol{smash{
ule{1cm}{0.15mm}}}$ units.the perimeter of the equilateral triangle is $\boldsymbol{smash{
ule{1cm}{0.15mm}}}$ units.therefore, the area of the equilateral triangle is $\boldsymbol{smash{
ule{1cm}{0.15mm}}}$, or approximately 43.5 units².the calculated areas are $\boldsymbol{smash{
ule{3cm}{0.15mm}}}$

Explanation:

Step1: Find the perimeter

An equilateral triangle has 3 equal sides of length 10.
$P = 3 \times 10 = 30$ units

Step2: Find the apothem

For a regular polygon, the apothem $a$ of an equilateral triangle can be found using the relation with side length $s$: $a = \frac{s}{2\tan(60^\circ)}$.
$a = \frac{10}{2\tan(60^\circ)} = \frac{5}{\sqrt{3}} \approx 2.9$ units

Step3: Calculate area via polygon formula

The area formula for a regular polygon is $A = \frac{1}{2} \times P \times a$.
$A = \frac{1}{2} \times 30 \times 2.9 = 43.5$ units²

Step4: Compare the two areas

Bianca's calculated area is also 43.5 units².

Answer:

The apothem, rounded to the nearest tenth, is 2.9 units.
The perimeter of the equilateral triangle is 30 units.
Therefore, the area of the equilateral triangle is 43.5, or approximately 43.5 units².
The calculated areas are equal.