Question
The area of an equilateral triangle is $\sqrt{3}$. What is the perimeter of the triangle?
The correct answer is
6
Finding Equilateral Triangle Perimeter
The problem asks for the perimeter of an equilateral triangle given its area.
- Area Formula: The area ($A$) of an equilateral triangle with side length ($s$) is given by the formula: $A = \frac{s^2 \sqrt{3}}{4}$
- Given Area: We are given that the area $A = \sqrt{3}$.
- Calculate Side Length ($s$): Substitute the given area into the formula and solve for $s$: $ \sqrt{3} = \frac{s^2 \sqrt{3}}{4} $ Divide both sides by $\sqrt{3}$: $ 1 = \frac{s^2}{4} $ Multiply both sides by 4: $ s^2 = 4 $ Take the square root of both sides (since side length must be positive): $ s = \sqrt{4} = 2 $ So, the side length of the equilateral triangle is 2.
- Calculate Perimeter: The perimeter ($P$) of an equilateral triangle is 3 times its side length ($s$): $ P = 3s $ Substitute the calculated side length ($s=2$): $ P = 3 \times 2 = 6 $
Therefore, the perimeter of the equilateral triangle is 6.
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Important Questions from Mensuration and Geometry
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
- A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in $m^2$ is ______
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