Find the area of an equilateral triangle with sidelength 6 units. [Hint: Show that an altitude bisects the opposite side. Use this to find the height.]

Question

Homi · Accepted Answer

We will find the height of the triangle first, then use it to calculate the area.

Step 1 — Find the height of the triangle

Let us consider an equilateral triangle, ABC, with all sides equal to 6 units. We draw an altitude from vertex A to the opposite side BC, and let it meet BC at point D. An altitude is a line segment from a vertex perpendicular to the opposite side. In an equilateral triangle, this altitude also divides the opposite side into two equal parts. So, BD will be equal to DC.

$BD = \frac{1}{2} \times BC$

$BD = \frac{1}{2} \times 6 \text{ units}$

$BD = 3 \text{ units}$

Now, we have a right-angled triangle, ADC. The sides of this right-angled triangle are AD (the height, let's call it $h$ ), DC (which is 3 units), and AC (the hypotenuse, which is 6 units). We can use the Pythagorean theorem to find the height $h$ . The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

$AD^2 + DC^2 = AC^2$

$h^2 + 3^2 = 6^2$

$h^2 + 9 = 36$

$h^2 = 36 - 9$

$h^2 = 27$

$h = \sqrt{27}$

$h = \sqrt{9 \times 3}$

$h = 3 \times \sqrt{3} \text{ units}$

$\boxed{h = 3\sqrt{3} \text{ units}}$

Diagram 1

Step 2 — Calculate the area of the triangle

The area of any triangle is given by the formula: half times base times height. For triangle ABC, the base is BC, which is 6 units. The height is AD, which we found to be $3\sqrt{3}$ units.

$\text{Area of triangle ABC} = \frac{1}{2} \times \text{base} \times \text{height}$

$\text{Area} = \frac{1}{2} \times BC \times AD$

$\text{Area} = \frac{1}{2} \times 6 \times 3\sqrt{3}$

$\text{Area} = 3 \times 3\sqrt{3}$

$\boxed{\text{Area} = 9\sqrt{3} \text{ square units}}$

Answer

(i) The height of the equilateral triangle is $3\sqrt{3}$ units. (ii) The area of the equilateral triangle is $9\sqrt{3}$ square units.

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