Question

An equilateral triangle has an apothem measuring 2. 16 cm and a perimeter of 22. 45 cm. An equilateral triangle has an apothem with length 2. 16 centimeters and a perimeter of 22. 45 centimeters. What is the area of the equilateral triangle, rounded to the nearest tenth? 2. 7 cm2 4. 1 cm2 16. 2 cm2 24. 2 cm2.

Answer 1

The area of the considered equilateral triangle is given by: Option D: $24.2 \: \rm cm^2$

What is apothem?

For a regular polygon (a polygon with all sides of same measure), the line segment drawn from the center of the polygon to the mid point of a side of the polygon is called apothem.

Referring to the attached figure below, for the considered case,  we're given that:

• The perimeter of the considered equilateral triangle is 22.45 cm
• The length of the apothem = 2.16 cm

Since all sides of an equilateral triangle are same, let they be of 'x'  cm length for this case, then:

$Perimeter = x + x + x = 22.45\\\\x = \dfrac{22.45}{3} \approx 7.48 \: \rm cm$

Since the equilateral triangle has all same sides, all three triangles AOC, AOB, and BOC are congruent and have same area.

The area of ABC = Area of AOC + Area of AOB + Area of BOC = 3 × Area of AOC

The area of AOC = half of base times height. The height is length of the apothem since because of symmetry in right and left of the apothem, it is standing with equal angle on both sides of AC, thus, half of straight line angle, therefore of 180/2 = 90 degrees, thus, perpendicular.

Area AOC =  $\dfrac{1}{2} \times |AC| \times |OD| \approx \dfrac{1}{2}{ \times 7.48 \times 2.16 \approx 8.078\rm \: cm^2$

Thus, area of ABC = 3 × Area of AOC = $3 \times 8.078 \approx 24.2 \rm \: cm^2$

Thus, the area of the considered equilateral triangle is given by: Option D: $24.2 \: \rm cm^2$

Learn more about apothem here:

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