Applying Area and Perimeter Formulas

Grade 4 Measurement and Data: 4.MD.A.3

In this lesson:

• Use A = l x w and P = 2l + 2w for any rectangle
• Find a missing side from the area or the perimeter
• Decide which formula a real problem actually needs

Learning Objectives for This Lesson

By the end of today, you can:

1. Find area with A = l x w
2. Find perimeter with P = 2l + 2w
3. Find a missing side from the area
4. Find a missing side from the perimeter
5. Decide whether a problem needs area or perimeter

Soil or Edging: Same Question?

Mr. Torres is building a rectangular garden bed, 12 feet by 5 feet.

He needs to know how much soil fills the inside, and how much edging goes
around the outside.

Are those the same question?

Count the Squares, Then Check the Formula

A rectangle on a grid is 6 units by 4 units.

Count the unit squares inside. Then compute 6 x 4. Do they match?

Finding a Poster's Rectangular Area

A rectangular poster is 8 inches long and 5 inches wide. What is its area?

Quick Check: The Missing Factor

What number goes in the blank?

Finding a Missing Width From Area

A Longer Banner, Same Idea

A banner has an area of 168 square inches and a width of 8 inches.

Verify:

Is 384 Feet a Reasonable Width?

A room has area 48 square feet and length 8 feet.

A student computes 48 x 8 = 384 feet for the width. Does that make sense?

Your Turn: Try Both Directions

A table is 3 feet wide with an area of 15 square feet. What is its length?

Then solve two more: one direct area computation, and one unknown-dimension
problem with larger numbers.

One Formula, Read Two Ways

• Given two sides: multiply to find the area
• Given the area and one side: divide to find the other side
• Same formula, A = l x w, read in whichever direction the problem needs

From the Inside to the Outside

You've been finding what fills the inside of a rectangle.

Now: what about the distance around the outside?

Tracing the Rectangle's Full Boundary

10 + 4 + 10 + 4 = 28 m, so P = 2(10) + 2(4) = 28 m

Fencing a Rectangular Garden Plot

A garden is 15 feet long and 8 feet wide. How much fencing goes around it?

Finding a Missing Width From Perimeter

A picture frame uses 56 inches of framing. The frame is 16 inches long.

Half-perimeter:

Equation:

Is the Perimeter Really 14 Meters?

A rectangle is 10 meters by 4 meters.

A student adds 10 + 4 = 14 meters for the perimeter. Trace the boundary:
does that cover the whole trip around?

A Longer Playground Perimeter Problem

A playground has a perimeter of 100 meters and a width of 18 meters.

Verify:

Perimeter Works Both Ways, Too

Just like area, the perimeter formula connects three quantities.

Knowing the perimeter and any one side always lets you find the other side.

Your Turn: Unknown Sides and Big Numbers

Solve for the unknown side in two more perimeter problems, including one
with multi-digit numbers.

Use whichever method — half-perimeter or the equation — feels more natural
to you.

Back to the Garden Bed

Words That Signal Which Formula

• Area words: cover, fill, paint, carpet, soil, sod, tile
• Perimeter words: fence, frame, border, edge, trim, ribbon, walk around
• Underline the action word before choosing a formula

The Anchor Phrase to Remember

Area asks: "How much covers the inside?" Answer in square units.
Perimeter asks: "How far around?" Answer in length units.

The unit in your answer tells you which question you actually answered.

Commit First: Wall or Pen?

A wall 14 ft by 9 ft needs paint. A dog pen 20 ft by 12 ft needs fencing.

Name "area" or "perimeter" for each — before you compute anything.

Commit First: Find Unknown Dimensions

A closet floor is 48 square feet with length 8 feet. A pool has perimeter
80 meters and length 25 meters.

Name the formula for each, then find the missing width.

No Obvious Keyword This Time

A rectangular banner needs a ribbon border sewn along its outer stitching.

What does this problem need — area or perimeter? Name it before you
compute.

The Yard: No Help This Time

• Plant grass inside; build a fence around it

Two Questions About One Rectangle

Area asks: how much covers the inside? Answer in square units.
Perimeter asks: how far around? Answer in length units.

Decide which question a problem is asking before you compute anything.

Four Mistakes to Watch Out For

Same two numbers, two questions — check your unit to know which
Area and one side known: divide, don't multiply
Four sides, not two — double the length-plus-width, don't just add once
Square units mean area; plain units mean perimeter — the unit is graded

Key Takeaways From Today's Lesson

✓ A = l x w and P = 2l + 2w work forward and backward
✓ Area and one side, or perimeter and one side, find the third
✓ Area measures inside; perimeter measures around
✓ The unit in your answer always tells you which one you found

Coming Up Next: Measuring Angles

You can now compute, reverse, and choose between area and perimeter for any
rectangle.

Next lesson turns to a different kind of measurement — angles — using the
same habit of matching the right tool to what the problem asks.