Question
A traffic signal board, indicating $SCHOOL AHEAD$, is an equilateral triangle with side a. Find the area of the signal board, using Heron’s formula. If its perimeter is $180 \ cm$, what will be the area of the signal board?
✓
Answer
A traffic signal board is an equilateral triangle with side a.
Perimeter of the signal board,
$2s = a + a + a$
$\Rightarrow \mathrm{s}=\frac{3}{2} a$
Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{\frac{3 a}{2}\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)}$
$=\sqrt{\frac{3 a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}=\sqrt{\frac{3 a^{4}}{16}}=\frac{\sqrt{3}}{4} a^{2}$ sq. units
Now, if perimeter $= 180 cm$
$3a = 180$
$\Rightarrow a = 60 cm$
$\therefore \text { Area of signal board }= \frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times(60)^{2}=900 \sqrt{3} \mathrm{cm}^{2}$
So, area of the signal board is $900 \sqrt{3} \mathrm{cm}^{2}$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that the diameter of a circle perpendicular to one of the two parallel chords of a circle is perpendicular to the other and bisects it.
In the given figure, $AB$ is a diameter of the circle such that $\angle\text{A}=35^\circ$ and $\angle\text{Q}=25^\circ,$ find $\angle\text{PBR.}$
→If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.
Find the area of a trapezium whose parallel sides are $11\ m$ and $25\ m$ long, and the nonparallel sides are $15\ m$ and $13\ m$ long.
→ If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
Prove: Two distinct lines cannot have more than one point in common.
Two sides of a triangular field are $85\ m$ and $154\ m$ in length and its perimeter is $324\ m$. Find:
$i.$ The area of the field.
$ii.$ The length of the perpendicular from the opposite vertex on the side measuring $154\ m.$
→$i.$ The area of the field.
$ii.$ The length of the perpendicular from the opposite vertex on the side measuring $154\ m.$
Two lines $AB$ and $CD$ intersect at a point $O,$ such that $\angle\text{BOC}+\angle\text{AOD}=280^\circ,$ as shown in the figure. Find all the four angles. 
→Find the measure of each angle of a parallelogram, if one of its angles is $30^\circ $ less than twice the smallest angle.
→Calculate the value of $x$ in the given figure.
→