We know this because each side of the triangle consists of two radii of each circle (i.e. If we look closely at the diagram, we see that an equilateral triangle is involved. What signals does the area of the shaded region give us? Think about it before reading on… However, if we use signals the problem is giving us, we can get to the answer more quickly than we might initially think. This is a complex problem that seems intimidating at first. If the area of the shaded region is 64 √3 – 32π, what is the radius of each circle? The figure shown above consists of three identical circles that are tangent to each other. Now that we’ve seen the relationship between equilateral and 30-60-90 triangles, let’s see how it plays out in an official GMAT problem: This is how we finally get the universal formula for an equilateral triangle: In this case, the base is s, while the height is s√3/2. Now, we’re very close to deriving the area of the triangle, which is simply base*height/2. If we fill in all of the appropriate lengths, we would get the following: Remember that the ratio of side lengths is 1 : √3 : 2. Before viewing the diagram below, take a moment to consider what the height of the triangle would be. Not only that, but we can then use s to denote the side length of the equilateral triangle and map out each segment of the 30-60-90 right triangles. Take a moment to consider what this produces and what the implications are.Īs you might have guessed, this line segment produces two 30-60-90 right triangles: Because the line segment down the middle acts as an angle bisector, the 60 degree angle at the top vertex becomes two 30-degree angles. Now, what happens when we take such a triangle and split it down the middle? And given that the angles in a triangle must sum to 180 degrees, each angle must be 60 degrees: And the √3 term in the area is a big clue.įirst, it helps to remember that an equilateral triangle has all equal angles as well as all equal sides. There is a formula for the area of an equilateral triangle as it relates to the length of its side s, and it is as follows:īut more likely than not for the GMAT, you’ll need to understand how this formula is derived. As promised, we will now connect the 30-60-90 triangle to the equilateral triangle, specifically its area.
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