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

Q: Why can't I just use side × side for the area?

Because the area formula needs the height, not another side! In an equilateral triangle, the height is shorter than the sides. Using side × side gives you 36, which is way too big.

Q: How do I remember the special equilateral triangle formula?

The formula $ \frac{\sqrt{3} \times s^2}{4} $ comes from always having the same height-to-side ratio. You can always derive it using the Pythagorean theorem if you forget!

Q: Which method should I use - Pythagorean theorem or the special formula?

Both work perfectly! The Pythagorean method helps you understand why the formula works. The special formula is faster once you memorize it. Use whichever feels more comfortable!

Q: What if I get a decimal that doesn't match the answer choices exactly?

That's normal with square roots! $ \sqrt{27} \approx 5.196 $, so your final answer will be approximate. Always check if you need to round to match the given choices.

Q: Can I use this method for other types of triangles?

No! The special formula $ \frac{\sqrt{3} \times s^2}{4} $ only works for equilateral triangles. For other triangles, you must find the actual height using coordinate geometry or trigonometry.

Triangle Area with Special Formula Applications

What is the area of the triangle in the drawing?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the area of the triangle in the drawing?

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 theorem $A^2+B^2=C^2$

We can find the length of the missing side.

$3^2+X^2=6^2$

We convert the formula

$6^2-3^2=X^2$

$36-9=27$

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

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

$\frac{6\times\sqrt{27}}{2}=15.588$

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

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

Where X is one of the sides.

$\frac{\sqrt{3}\times6^2}{4}=\frac{62.353}{4}=15.588$

Final Answer

15.588

Key Points to Remember

Essential concepts to master this topic

Equilateral Triangle Rule: All sides equal, all angles 60°, height creates right triangle
Height Formula: Use Pythagorean theorem: $h^2 + 3^2 = 6^2$ gives $h = \sqrt{27}$
Area Check: Both methods give 15.588: $\frac{6 \times \sqrt{27}}{2} = \frac{\sqrt{3} \times 36}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong triangle area formula
Don't use Area = 1/2 × base × side = 1/2 × 6 × 6 = 18! This treats the side as height, which is wrong for equilateral triangles. Always find the actual height first using the Pythagorean theorem or use the special equilateral triangle formula.

Practice Quiz

Test your knowledge with interactive questions

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

FAQ

Everything you need to know about this question

Why can't I just use side × side for the area?

Because the area formula needs the height, not another side! In an equilateral triangle, the height is shorter than the sides. Using side × side gives you 36, which is way too big.

How do I remember the special equilateral triangle formula?

The formula $\frac{\sqrt{3} \times s^2}{4}$ comes from always having the same height-to-side ratio. You can always derive it using the Pythagorean theorem if you forget!

Why does the height split the base in half?

In an equilateral triangle, the height from any vertex to the opposite side creates two identical right triangles. This is because of the triangle's perfect symmetry.

Which method should I use - Pythagorean theorem or the special formula?

Both work perfectly! The Pythagorean method helps you understand why the formula works. The special formula is faster once you memorize it. Use whichever feels more comfortable!

What if I get a decimal that doesn't match the answer choices exactly?

That's normal with square roots! $\sqrt{27} \approx 5.196$ , so your final answer will be approximate. Always check if you need to round to match the given choices.

Can I use this method for other types of triangles?

No! The special formula $\frac{\sqrt{3} \times s^2}{4}$ only works for equilateral triangles. For other triangles, you must find the actual height using coordinate geometry or trigonometry.

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

Triangle Area with Special Formula Applications

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just use side × side for the area?

How do I remember the special equilateral triangle formula?

Why does the height split the base in half?

Which method should I use - Pythagorean theorem or the special formula?

What if I get a decimal that doesn't match the answer choices exactly?

Can I use this method for other types of triangles?

🌟 Unlock Your Math Potential

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