An Equilateral Triangle that has an Area of 9√3 is Inscribed in a Circle GMAT Problem Solving

Question: An equilateral triangle that has an area of 9√3 is inscribed in a circle. What is the area of the circle?

1. 6π
2. 9π
3. 12π
4. 9π√3
5. 18π√3

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

The problem statement states that:
Given:

• An equilateral triangle that has an area of 9√3 is inscribed in a circle.

Find out:

• The area of the circle.

Area of the equilateral triangle = 9√3

As per the formula of the area of an equilateral triangle, we can say:
Area of equilateral triangle= \(a^2\) * √3/4; where “a” is the length of the side of the triangle.

Now, putting the value of the area that is given we can determine the length of the side of the triangle.

Area of the equilateral triangle = \(a^2\) * √3/4

9√3 = \(a^2\) * √3/4

\(a^2\) = 9√3 * 4/ √3

\(a^2\) = 36

a = 6

So, we know that all sides of an equilateral triangle are equal which means each side of the given triangle has a length of 6 units.

It is given that the equilateral triangle is inscribed within a circle.
As per the formula of the radius of the circumscribed circle, we can say:
Radius of circumscribed circle = a * √3/3

Putting the value of a = 6 in this formula, we get:
Radius of circumscribed circle = 6 * √3/3 = 2√3

Now, we know the radius of the circle and it can be used to determine the area of the circle.
Area of the circle = \(πr^2\); where “r” is the radius of the circumscribed circle

Therefore, the area of the circle = \(π [2\sqrt3]^2\)
Hence the area of the circle = 12π

Approach Solution 2:

The problem statement informs that:
Given:

• An equilateral triangle that has an area of 9√3 is inscribed in a circle.

Find out:

• The area of the circle.

The area of the equilateral triangle is 9√3.
Let’s assume that the side of the triangle is “a” then we get:
½ ∗ √3/2 ∗ a = 9√3

Solving this we get

a = 6

Since we know that all sides of an equilateral triangle are equal which means each side of the given triangle has a length of 6 units.

Now, in the figure, we need to consider the shaded triangle. Clearly, we can notice that the area of this triangle is a third of the area of the whole triangle.

So we have ½ ∗ h ∗ a/2 = ⅓ ∗ 9√3, therefore from here we get h=√3
This implies that the radius, as we can see from the picture = 3√3 −√3 =2√3

Hence area of the circle = π∗ (2√3)^2
= 12π

Approach Solution 3:

The problem statement indicates that:
Given:

• An equilateral triangle that has an area of 9√3 is inscribed in a circle.

Find out:

• The area of the circle.

We know that the formula of the area of an equilateral triangle = √3/4 (side)²
Therefore we can say, 9√3 = √3/4 (side)²
=> 9 = 1/4 (side)²
=> (side)² = 9(4)
=> (side)² = 36
=> Taking square root, we get: Side = 6 cm
Therefore, the length of each side of an equilateral triangle is 6 cm.

Since we know the formula of the radius of circumscribed circle = side * √3/3
Therefore, we can say, the radius of the circle = 6* √3/3 = 2√3.

As per the formula of the area of the circle, we can say,
The area of the circle = \(π(radius)^2\);
Therefore, the area of the circle =\( π [2\sqrt3]^2\)
Hence the area of the circle = 12π

“An equilateral triangle that has an area of 9√3”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “501 GMAT Questions”. GMAT Problem Solving questions facilitate the candidates to assess every requirement of the question to solve numerical problems. GMAT Quant practice papers assist the candidates to get familiar with several sorts of questions that will enable them to improve their mathematical learning.

Suggested GMAT Problem Solving Samples

• The first five numbers in a regular sequence are 4, 10, 22, 46, and 94 GMAT Problem Solving
• A Bar is Creating a New Signature Drink. There are Five possible Alcohol GMAT Problem Solving
• A Can Complete a Certain Job in 12 Days. B is 60% More Efficient than GMAT Problem Solving
• An Alloy of Gold,Silver and Bronze Contain 90% Bronze, 7% Gold and GMAT Problem Solving
• When Positive Integer n is Divided by 5, the Remainder is 1 GMAT Problem Solving
• A is Thrice as Efficient as B and Hence Takes 12 Days Less to Complete GMAT Problem Solving
• A Tank is Filled by Three Pipes with Uniform Flow. The First Two Pipes GMAT Problem Solving
• Each Person in a Group of 110 Investors Has Investments in Either GMAT Problem Solving
• Jim Travels the First 3 Hours of His Journey at 60 mph Speed GMAT Problem Solving
• A Committee Of 7 Members Is To Be Formed To Put Up The Christmas GMAT Problem Solving
• In A Class Of 60 Students, 23 Play Hockey, 15 Play Basketball GMAT Problem Solving
• The Area Of An Equilateral Triangle With Side Length X Is The Same GMAT Problem Solving
• Which Of The Following Has A Decimal Equivalent That Is A GMAT Problem Solving
• A Number when Divided by a Divisor Leaves a Remainder of 24 GMAT Problem Solving
• How many Integers Less than 1000 have no Factors (other than 1) GMAT Problem Solving
• How Many Prime Numbers are there Between 50 and 70? GMAT Problem Solving
• If a circle passes through points (1, 2) (2, 5), and (5, 4) GMAT Problem Solving
• If the Perimeter of Square Region S and the Perimeter of Rectangular GMAT Problem Solving
• If x^9= 9^9^9, What is the Value of x? GMAT Problem Solving
• In a Competition, a School Awarded Medals in Different Categories GMAT Problem Solving
• In how many ways can 5 apples (identical) be distributed among 4 children? GMAT problem solving
• What is Greatest Positive Integer n such that 2^n is a Factor of 12^10? GMAT Problem Solving
• All the five digit numbers in which each successive digit exceeds its GMAT Problem Solving
• A box contains two white balls, three black balls and four red balls GMAT Problem Solving
• Consider An Obtuse-Angled Triangles With Sides 8 Cm, 15 Cm GMAT Problem Solving
• I Travel The First Part Of My Journey At 40 Miles Per Hour GMAT Problem Solving
• Profit Earned By Selling An Article At 1060 Is 20 % More Than The Loss GMAT Problem Solving
• A Perfect Square is a Number that Becomes an Integer when Square GMAT Problem Solving
• If x + y = 2 and x^2 + y^2 = 2, What is the Value of xy? GMAT Problem Solving
• The Owner of A Local Jewellery Store Hired 3 Watchmen to Guard his Diamonds GMAT Problem Solving
• What is the smallest integer n for which 25^n>5^12 GMAT Problem Solving
• Ten Coins are Tossed Simultaneously. In how many of the Outcomes GMAT Problem Solving
• A Pool can be Filled in 4 Hours and Drained in 5 Hours GMAT Problem Solving
• If x^2 − 2 < 0, which of the following specifies all the possible GMAT Problem Solving
• The Mass Of 1 Cubic Meter Of A Substance Is 800 Kilograms GMAT Problem Solving
• A Dairyman Pays Rs. 6.40 Per Liter Of Milk GMAT Problem Solving
• A Train Approaches A Tunnel AB. Inside The Tunnel Is A Cat Located GMAT Problem Solving
• John is 20 years older than Brian. 12 years ago, John was twice as old GMAT Problem Solving
• A Train Overtakes Two Persons Who Are Walking In The Same Direction GMAT Problem Solving
• If 1/x − 1/x+1= 1/x+4, Then x could be GMAT Problem Solving
• If x = - |x|, Then Which One of the Following Statements GMAT Problem Solving
• A Bag Contains 4 Red and 3 Black Balls. A Second Bag Contains 2 Red GMAT Problem Solving
• A Mixture of 40 Litres of Milk and Water, Contains 20% of Water GMAT Problem Solving
• A Candidate is Required to Answer 6 Out of 10 Questions Divided into 2 Groups GMAT Problem Solving
• Company XYZ had $498.2 Million in Profits for the Year GMAT Problem Solving
• If -2 < a < 11 and 3 < b < 12, Then Which of the Following is NOT Always True GMAT Problem Solving
• Karan and Arjun run a 100-metre Race, Where Karan Beats Arjun by 10 metres GMAT Problem Solving
• The Figure below Shows an Equilateral Triangle ABC Tangent GMAT Problem Solving
• Three groups of children contain respectively 3 girls and 1 boy GMAT Problem Solving
• If x=10^{10},\frac{x^2+2x+7}{3x^2-10x+200} is closest to GMAT Problem Solving