What is the side of an equilateral triangle whose area is 9 root 3 cm square?

The equilateral triangle calculator will help you with calculations of standard triangle parameters. Whether you are looking for the equilateral triangle area, its height, perimeter, circumradius, or inradius, this great tool is a safe bet. Scroll down to read more about useful formulas (such as for the height of an equilateral triangle) and to get to know what is an equilateral triangle.

The equilateral triangle, also called a regular triangle, is a triangle with all three sides equal. What are the other important properties of that specific regular shape?

• All three internal angles are congruent to each other, and all of them are equal to 60°.
• The altitudes, the angle bisectors, the perpendicular bisectors, and the medians coincide.

The equilateral triangle is a special case of an isosceles triangle having not just two but all three sides equal.

The formula for a regular triangle area is equal to squared side times square root of 3 divided by 4:

area = (a² × √3)/ 4

and the equation for the height of an equilateral triangle looks as follows:

h = a × √3 / 2, where a is a side of the triangle.

But do you know where the formulas come from? You can find them in at least two ways: deriving from Pythagorean theorem or using trigonometry.

1. Using Pythagorean theorem

• The basic formula for triangle area is side a (base) times the height h, divided by 2:

  area = (a × h) / 2

• Height of the equilateral triangle is splitting the equilateral triangle into two right triangles. One leg of that right triangle is equal to height, another leg is half of the side, and the hypotenuse is the equilateral triangle side.

  (a/2)² + h² = a²

  After simple transformations, we get a formula for the height of the equilateral triangle:

  h = a × √3 / 2

• Substituting h into the first area formula, we obtain the equation for the equilateral triangle area:

  area = a² × √3 / 4

2. Using trigonometry

• Let's start from the trigonometric triangle area formula:

  area = (1/2) × a × b × sin(γ), where γ is the angle between sides

• We remember that all sides and all angles are equal in the equilateral triangle, so the formula simplifies to:

  area = 0.5 × a × a × sin(60°)

• What is more, we know that the sine of 60° is √3/2, so the formula for equilateral triangle area is:

  area = (1/2) × a² × (√3 / 2) = a² × √3 / 4

  Height of the equilateral comes from sine definition:

  h / a = sin(60°) so h = a × sin(60°) = a × √3 / 2

You can easily find the perimeter of an equilateral triangle by adding all triangles sides together. The regular triangle has all sides equal, so the formula for the perimeter is:

perimeter = 3 × a

How to find the radius of the circle circumscribing the three vertices and the inscribed circle radius?

circumcircle_radius = 2 × h / 3 = a × √3 / 3

incircle_radius = h / 3 = a × √3 / 6

Let's take the example from everyday life: we want to find all the parameters of the yield sign.

1. Type the given value into the correct box. Assume we have a sign with a 36-in side length.

2. The equilateral triangle calculator finds the other values in no time. Now we know that:

   • Yield sign height is 31.2 in;
   • Its area equals 561 in²;
   • Perimeter: 108 in;
   • Circumcircle radius is 20.8 in; and
   • Incircle radius 10.4 in.

3. Check out our tool flexibility. Refresh the calculator, and type in the other parameter, e.g., perimeter. It's working this way as well, isn't that cool?

To find the area of an equilateral triangle, follow the given instructions:

1. Take the square root of 3 and divide it by 4.

2. Multiply the square of the side with the result from step 1.

3. Congratulations! You have calculated the area of an equilateral triangle.

To find the height of an equilateral triangle, proceed as follows:

1. Take the square root of 3 and divide it by 2.

2. Multiply the result from step 1 with the length of the side.

3. You will get the height of the equilateral triangle.

The perimeter of the given triangle is 24 cm. To calculate the perimeter of an equilateral triangle, we need to multiply its side length by 3. The length of each side of the given triangle is 8 cm. Hence its perimeter will be 3 × 8 cm = 24 cm.

No, a right triangle can't be an equilateral triangle. One of the angles in a right triangle is 90°. Since the sum of all the interior angles in a triangle is 180°, the other two angles in a right triangle are always less than 90°.

According to the definition of equilateral triangles, all the internal angles are equal. Hence, a right triangle can never be equilateral.

Text Solution

8 cm36 cm4 cm6 cm

Answer : D

Solution : (d) `"Given, area of an equilateral triangle"=9sqrt3cm^(2)` <br> `because"Area of an equilateral triangle"=(sqrt3)/(4)("Side")^(2)` <br> `rArr(sqrt3)/(4)("Side")^(2)=9sqrt3` <br> `rArr("Side")^(2)=36` <br> `therefore "Side" =6 cm` <br> `["Taking positive square root becuase side is always positive"]` <br> `"Hence, the length of an equilateral triangle is "6cm.`

The length of each side of an equilateral triangle having an area of 9 √3 cm 2 isA. 36 cmB. 4 cmC. 6 cmD. 8 cm

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