An equilateral triangle has an area of 16\sqrt{3}\text{ cm}^2. If a square has the same pe… — JAMB UTME

An equilateral triangle has an area of $16\sqrt{3}\text{ cm}^2$ . If a square has the same perimeter as this equilateral triangle, what is the area of the square?

$36\text{ cm}^2$ Answer
B
$48\text{ cm}^2$
C
$64\text{ cm}^2$
D
$144\text{ cm}^2$

Answer

The area of the square is

36\text{ cm}^2

Using the area formula for an equilateral triangle

\text{Area} = \frac{\sqrt{3}}{4}s^2 = 16\sqrt{3}

, we determine the side length

s = 8\text{ cm}

. The perimeter of the triangle is

3 \times 8\text{ cm} = 24\text{ cm}

. Since the square has an equal perimeter of

24\text{ cm}

, each side of the square measures

24\text{ cm} / 4 = 6\text{ cm}

. The area of the square is therefore

6^2 = 36\text{ cm}^2

Step-by-Step Solution

Find the side length of the equilateral triangle.

Side length

s = 8\text{ cm}

The area of an equilateral triangle is given by

\text{Area} = \frac{\sqrt{3}}{4}s^2

. Setting

\frac{\sqrt{3}}{4}s^2 = 16\sqrt{3}

yields

s^2 = 64

, so

s = 8\text{ cm}

Calculate the perimeter of the equilateral triangle.

Perimeter

P = 24\text{ cm}

An equilateral triangle has 3 equal sides, so

P = 3 \times 8 = 24\text{ cm}

Determine the side length of the square.

Square side length

a = 6\text{ cm}

The square and triangle have equal perimeters (

24\text{ cm}

). Since a square has 4 equal sides,

a = \frac{24}{4} = 6\text{ cm}

Calculate the area of the square.

Area

= 36\text{ cm}^2

The area of a square is

a^2 = 6^2 = 36\text{ cm}^2

Key Concept

Perimeter and Area of Equilateral Triangles and Squares

Estimated Time:1m 30s

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