55. The area of an equilateral triangle with sides of length a is:
(A) (√3/4)a2
(B) (√3/2)a2
(C) (1/2)a2
(D) (1/√2)a2
Find the Area of an Equilateral Triangle With Side Length a
Understanding the Given Equilateral Triangle Problem
This question asks for the area of an equilateral triangle whose side length is a. An equilateral triangle is a special type of triangle in which all three sides are equal and all three interior angles are equal to 60°.
If each side of the equilateral triangle has length a, then:
Side = a
To calculate the area of any triangle, we can use the basic formula:
Area = (1/2) × Base × Height
Therefore, the main task is to determine the height of the equilateral triangle in terms of its side length a. Once the height is known, it can be substituted into the standard triangle area formula.
Drawing the Altitude of the Equilateral Triangle
Consider an equilateral triangle with each side equal to a. Draw a perpendicular from the top vertex to the opposite side. This perpendicular represents the height or altitude of the triangle.
In an equilateral triangle, the altitude also bisects the opposite side into two equal parts. Therefore, the base of length a is divided into two segments, each of length:
a/2
The altitude divides the original equilateral triangle into two congruent right-angled triangles. In each right-angled triangle:
Hypotenuse = a
Base = a/2
Height = h
We can now use the Pythagorean theorem to calculate the height h.
Finding the Height Using the Pythagorean Theorem
According to the Pythagorean theorem, in a right-angled triangle:
(Hypotenuse)2 = (Base)2 + (Height)2
Substituting the known values:
a2 = (a/2)2 + h2
Since:
(a/2)2 = a2/4
we obtain:
a2 = a2/4 + h2
Subtracting a²/4 from both sides:
h2 = a2 − a2/4
Taking the common denominator:
h2 = (4a2 − a2)/4
Therefore:
h2 = 3a2/4
Taking the positive square root, because height is a positive length:
h = √(3a2/4)
Hence:
h = (√3/2)a
Therefore, the height of an equilateral triangle with side length a is:
Height = (√3/2)a
Calculating the Area of the Equilateral Triangle
The general formula for the area of a triangle is:
Area = (1/2) × Base × Height
For the given equilateral triangle:
Base = a
and:
Height = (√3/2)a
Substituting these values into the area formula:
Area = (1/2) × a × (√3/2)a
Multiplying the numerical factors:
(1/2) × (√3/2) = √3/4
and multiplying the variables:
a × a = a2
Therefore:
Area = (√3/4)a2
Hence, the area of an equilateral triangle with side length a is:
(√3/4)a2
Alternative Solution Using the Standard Formula
The standard formula for the area of an equilateral triangle with side length a is:
Area = (√3/4)a2
Since the question directly gives the side length as a, we can substitute it into the standard formula without any additional calculation.
Therefore:
Area = (√3/4)a2
This directly gives Option (A) as the correct answer.
Alternative Solution Using the Sine Area Formula
The area of a triangle can also be calculated when two sides and the included angle are known. The formula is:
Area = (1/2)ab sin C
In an equilateral triangle, any two sides have length a, and the angle between them is 60°. Therefore:
Area = (1/2) × a × a × sin 60°
Using:
sin 60° = √3/2
we obtain:
Area = (1/2) × a2 × (√3/2)
Therefore:
Area = (√3/4)a2
This method also confirms the same result.
Why the Area Contains a²
Area is a two-dimensional quantity, so it must always be expressed in square units. Since the only length given in the problem is the side a, the area must be proportional to a².
If the side length of an equilateral triangle is doubled, its area becomes four times the original area because:
(2a)2 = 4a2
This explains why every dimensionally reasonable option in the question contains a². The main difference between the options is the numerical coefficient multiplying a².
Analysis of All the Given Options
Option (A): (√3/4)a2
This option is correct. The height of an equilateral triangle with side length a is (√3/2)a. Using the formula Area = (1/2) × Base × Height, we get:
Area = (1/2) × a × (√3/2)a = (√3/4)a2
Therefore, Option (A) is correct.
Option (B): (√3/2)a2
This option is incorrect. The factor √3/2 appears in the formula for the height of the equilateral triangle:
Height = (√3/2)a
However, the area formula also contains the factor 1/2. Therefore, the final coefficient becomes √3/4, not √3/2.
Option (C): (1/2)a2
This option is incorrect. The formula (1/2) × Base × Height cannot be simplified to (1/2)a² because the height of an equilateral triangle is not equal to its side length a. The correct height is (√3/2)a.
Option (D): (1/√2)a2
This option is incorrect. The factor 1/√2 does not arise from the geometry of an equilateral triangle. The correct height is obtained using a 30°–60°–90° right triangle or the Pythagorean theorem, leading to the coefficient √3/4 in the area formula.
Final Answer
The height of an equilateral triangle with side length a is:
h = (√3/2)a
Using:
Area = (1/2) × Base × Height
we get:
Area = (1/2) × a × (√3/2)a
Therefore:
Area = (√3/4)a2
Correct Option: (A) (√3/4)a2
