Neft Teacher · Learn It

Unit 5 · Standard 6.GR.1

Area of Regular Polygons

Lesson 5-9 · Learn It — how the math works

Why we're learning this

Your architecture firm is designing a hexagonal skylight for the lobby of a new museum. The glass fabricator needs to know the total area so they can cut the correct amount of glass. Each side of the hexagon is 6 feet, and the height of each triangle formed from the center is 5.2 feet. The height is rounded to the nearest tenth of a foot, so the area is approximate.

✱ Math Workbench

Learn It

How do we find the area of a regular polygon?

A regular polygon is a closed shape with straight sides where every side is the same length and every angle is the same size. We can decompose it (break it apart) into equal triangles from the center, then add up their areas.

Picture itSee what the math looks like before you read about it

The firm's blueprint shows a regular pentagon-shaped plaza. Dashed lines run from the center point to each of the 5 corners, dividing the plaza into 5 triangular sections.
Where you see this in real lifeThe firm's blueprint shows a regular pentagon-shaped plaza. Dashed lines run from the center point to each of the 5 corners, dividing the plaza into 5 triangular sections.

Watch me solve itI do — read one step at a time

The problem looks like this
Regular hexagon with 6 sides of 6 feet, split into 6 equal triangles with height 5.2 feet.5.2 ft6 ft
  1. My hexagon has 6 equal sides. Each side is 6 feet, and each center has a height of 5.2 feet. Use the rounded height to estimate the area.
  2. First I find one : A = ½ × base × height = ½ × 6 × 5.2.
  3. I multiply: 6 × 5.2 = 31.2, then ½ × 31.2 = 15.6 square feet.6 × 5.2 = 31.2
  4. A hexagon has 6 sides, so it has 6 equal .
  5. I multiply: 6 × 15.6 = 93.6. So the total area is 93.6 square feet.6 × 15.6 = 93.6

Watch out — mistakes students make here

  • 11.2 comes from adding 6 + 5.2. Area means multiply base × height, then take half.
  • The number is right, but ft measures length. Area needs square units (sq ft).
  • 31.2 is 6 × 5.2 — the full rectangle. One triangle is half of that: multiply by ½.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Now a smaller regular pentagon: each of its 5 has base 4 cm and height 3 cm. Use the rounded height to estimate the area.
  2. First find one : ½ × 4 × 3. What is 4 × 3?
    Check this step
    Yes, 12, and half of 12 is 6 sq cm.
  3. A pentagon has 5 sides, so how many ?
    Check this step
    Yes, 5.
  4. Multiply: 5 × 6 = 30, so the total area is 30 square centimeters.5 × 6 = 30

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Remember: find one 's area, then multiply by the number of sides.

Solve: A regular hexagon is divided into 6 equal from the center. Each triangle has a base of 6 ft and a height of 5.2 ft. What is the area of one triangle? Use the given heights; nonsquare heights are rounded, so their areas are estimates.

My answer

Stuck? Open one hint at a time.

Hint 1

Focus on just ONE of the 6 triangles: its base is 6 ft and its height is 5.2 ft.

Hint 2

One triangle uses A = ½ × b × h — don't multiply by 6 yet; that step is for the whole hexagon.

Hint 3

Compute ½ × 6 × 5.2: take half of 6 first, then multiply by 5.2. Label it in sq ft.

Check my answer
✅ Answer: 15.6 sq ft
A = ½ × b × h = ½ × 6 × 5.2 = 15.6 square feet.

Turn & TalkSay it out loud with a partner

The hexagonal skylight has sides of 6 feet, and each from the center has a height of 5.2 feet. Why is it helpful to break a hexagon into triangles to find its area?

Try starting with:

  • We decompose the hexagon into ___ because ___.Descomponemos el hexágono en ___ porque ___.
  • Once it is split, I can find each area using ___.Una vez dividido, puedo hallar cada área usando ___.
Use these wordsregular polygondecomposetrianglecompositeformula