Calculate the Area of an Equilateral Triangle with 6-Unit Sides

Question

What is the area of the triangle in the drawing?

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Video Solution

Solution Steps

00:00 Calculate the area of the triangle
00:06 Mark the given information, the triangle is isosceles
00:09 Draw height in the triangle
00:16 Equal sides because it's an equilateral triangle
00:21 The height in an isosceles triangle is also the median
00:30 From this we deduce the size of half the side
00:38 Use the Pythagorean theorem in triangle ADB
00:48 Substitute values according to the given data and solve to find AD
01:01 Isolate AD
01:18 This is the height value AD
01:25 Use the formula for calculating triangle area
01:28 (height times side) divided by 2
01:34 Substitute values according to the given data and calculate the area
01:39 And this is the solution to the problem

Step-by-Step Solution

There are two ways to solve the exercise:

It is possible to drop a height from one of the vertices, as we know

In an equilateral triangle, the height intersects the base,

This creates a right triangle whose two sides are 6 and 3,

Using the Pythagorean theoremA2+B2=C2 A^2+B^2=C^2

We can find the length of the missing side.

32+X2=62 3^2+X^2=6^2

We convert the formula

6232=X2 6^2-3^2=X^2

369=27 36-9=27

Therefore, the height of the triangle is equal to:27 \sqrt{27}

From here we calculate with the usual formula for the area of a triangle.

6×272=15.588 \frac{6\times\sqrt{27}}{2}=15.588

Option B for the solution is through the formula for the area of an equilateral triangle:

S=3×X24 S=\frac{\sqrt{3}\times X^2}{4}

Where X is one of the sides.

3×624=62.3534=15.588 \frac{\sqrt{3}\times6^2}{4}=\frac{62.353}{4}=15.588

Answer

15.588