Unit 11 Homework 4 Area Of Regular Figures Answers: Exact Answer & Steps

Struggling with Unit 11 Homework 4? Here's What You Need to Know About Finding the Area of Regular Figures

You're staring at homework problem #4, the one with the hexagon, and you're not sure where to even start. Maybe you've got the formula written down somewhere but it doesn't match the shapes in your book. Or perhaps you got an answer but it just feels wrong — and you can't figure out why.

Here's the thing: finding the area of regular figures isn't as hard as it looks once you understand the logic behind the formulas. Most students get stuck not because the math is difficult, but because they're applying the wrong approach to the wrong type of shape.

Let me break it down so you can actually finish that homework with confidence — and know that your answers are right.

What Is Area of Regular Figures?

When your textbook talks about "regular figures," it means polygons where all the sides are the same length and all the angles are equal. A regular triangle (equilateral triangle), a square, a regular pentagon, a regular hexagon — these are all regular figures.

The key word is regular. That's what makes these problems different from finding the area of random polygons. Because the shapes are symmetrical, we can use specific formulas that take advantage of that symmetry.

Your Unit 11 Homework 4 is probably asking you to find the area of one or more of these shapes:

Regular triangles (3 sides)
Regular quadrilaterals — squares (4 sides)
Regular pentagons (5 sides)
Regular hexagons (6 sides)

Each one has its own area formula, and some of them require a little extra work with right triangles and the Pythagorean theorem.

Why Does This Matter?

Beyond getting the homework done — why should you care about finding the area of regular figures?

For one, this skill shows up constantly in real life. Architects use regular polygons. Even video game designers use these shapes when they're building digital worlds. Engineers do. Understanding the math behind them gives you a foundation for all kinds of careers and hobbies.

But here's the more immediate reason: this unit is building on concepts you'll need for later math. The way you break a regular hexagon into triangles? That's the same thinking you use in trigonometry. Day to day, the Pythagorean theorem showing up in your area calculations? You'll see that again and again.

So yes, this homework matters. And yes, you can figure it out.

How to Find the Area of Regular Figures

Let's get into the actual methods. I'll walk through the most common regular figures you'll encounter.

Finding the Area of a Regular Triangle (Equilateral Triangle)

For an equilateral triangle where all sides have length s, the area formula is:

Area = (s² × √3) / 4

Here's how it works. Worth adding: you can't just use the standard "base times height divided by 2" formula because you don't always have the height given to you. With a regular triangle, you have to find the height first — and that involves splitting the triangle into two right triangles.

The height of an equilateral triangle is (s × √3) / 2. Once you have that, you can plug it into the standard area formula:

Area = ½ × base × height
Area = ½ × s × (s√3 / 2)
Area = (s² × √3) / 4

Example: If each side of your triangle is 6 cm, the area would be (36 × 1.732) / 4 = 62.35 / 4 = 15.59 cm² (rounded).

Finding the Area of a Regular Quadrilateral (Square)

This one's straightforward. If you know the side length s, the area is simply:

Area = s²

If you're given the diagonal instead, you can find the side length using the relationship between the sides and diagonal in a square: d = s√2, so s = d / √2. Then square that to get your area.

Finding the Area of a Regular Pentagon

A regular pentagon has five equal sides. To find its area, you typically need to know either the side length or the apothem (the distance from the center to the midpoint of any side).

If you know the side length s and the apothem a, the formula is:

Area = ½ × perimeter × apothem
Area = ½ × (5s) × a
Area = (5sa) / 2

Often in homework, you'll be given the apothem. If you're only given the side length, you'll need to use trigonometry to find the apothem first — and that's where many students get stuck.

Finding the Area of a Regular Hexagon

A regular hexagon is actually easier than it looks. Here's the secret: you can split any regular hexagon into six equilateral triangles.

Want to learn more? We recommend words that ends with ng and x 4 x 4 answer for further reading.

If the hexagon has side length s, each of those triangles has area (s² × √3) / 4. Multiply by 6, and you get:

Area = (6 × s² × √3) / 4
Area = (3s² × √3) / 2

That's the quick formula. Alternatively, if your homework gives you the apothem instead, use the perimeter method: Area = ½ × perimeter × apothem.

Common Mistakes Students Make

Let me save you some frustration. Here are the errors I see most often:

Using the wrong formula for the wrong shape. Students sometimes try to apply the hexagon formula to a pentagon, or use the square formula for any four-sided shape (it only works for squares, not rectangles or parallelograms). Double-check what type of regular figure you're working with.

Forgetting to square the side length. This sounds obvious, but under time pressure, students sometimes multiply the side length by the coefficient without first squaring it. The formulas require s², not s.

Rounding too early. If you're working with √3 (about 1.732), don't round it to 1.7 until the very end of your calculation. Rounding early will give you an imprecise answer.

Confusing the apothem with the radius. The apothem goes from the center to the edge (like the radius of a circle inscribed in the polygon). The radius of the polygon goes from the center to a vertex. These are different lengths, and using the wrong one will give you the wrong answer.

Practical Tips for Tackling This Homework

Here's what actually works:

Identify the shape first. Before you grab any formula, look at what you're working with. Count the sides. Is it a triangle, square, pentagon, or hexagon? That determines everything else.
Write down everything you know. Side length? Apothem? Radius? Put all your given values on the paper where you can see them.
Choose the right formula. Match the shape to the formula. If you have a pentagon and the apothem, use the perimeter-apothem method. If you have a hexagon and the side length, use the six-triangles method.
Check your units. Your answer should be in square units (cm², in², etc.). If you're getting a linear answer, something went wrong.
Estimate to catch big errors. If you have a hexagon with side length 10, your answer should be in the ballpark of 260 square units. If you get 26, you probably forgot to multiply by something. A quick estimate can catch mistakes before you turn in your work.

FAQ

What's the formula for the area of any regular polygon?

The general formula that works for any regular polygon is: Area = ½ × perimeter × apothem. This works for triangles, squares, pentagons, hexagons, and any other regular polygon — as long as you know the perimeter and the apothem.

Do I need to memorize all the specific formulas?

It helps to remember the equilateral triangle formula and the hexagon shortcut, but if you remember the general formula (½ × perimeter × apothem), you can apply it to any regular figure. That said, knowing the shortcuts saves time.

What if my homework gives me the radius instead of the apothem?

You'll need to convert. In real terms, for a regular polygon, the relationship between the radius (distance from center to vertex) and the apothem involves a right triangle. Use trigonometry: the apothem = radius × cos(π/n), where n is the number of sides.

Why does √3 keep showing up?

√3 appears in the equilateral triangle formulas because of the geometry of 30-60-90 right triangles. Practically speaking, when you split an equilateral triangle in half, you create a 30-60-90 triangle, and the ratio of the sides involves √3. Since regular hexagons are made of equilateral triangles, √3 shows up there too.

My answer doesn't match the answer key. What went wrong?

First, check if you rounded differently than the answer key. Practically speaking, second, make sure you used the right formula for the right shape. Still, third, double-check that you squared the side length if the formula required it. If all that checks out, walk through your work step by step — a calculation error is usually the culprit.

The Bottom Line

Unit 11 Homework 4 isn't about memorizing a dozen different formulas. And it's about understanding one key idea: regular figures have predictable symmetry, and you can use that symmetry to find their area. Whether you're breaking a hexagon into triangles or using the perimeter-apothem method, the logic is the same every time.

Start by identifying your shape, write down what you know, pick the right approach, and check your work. You've got this.