3rd Grade · 3.MD.C.7

Area of rectangles in 3rd grade

To find the area of rectangles, the space inside them, your child learns a fast method in 3rd grade: multiply the two side lengths. Your child sees why this works by filling a rectangle with rows of squares. Later in the year, the same idea gives the area of shapes made of several rectangles.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

3.MD.C.7 Relate area to the operations of multiplication and addition.
3.MD.C.7.a Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.C.7.b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.C.7.c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
3.MD.C.7.d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

What it means, in plain words

A unit square is a square with sides 1 unit long. When unit squares fill a rectangle, they make rows of the same length. A rectangle 3 units wide and 5 units long holds 3 rows of 5 squares. Counting gives 15 squares, and so does $3 \times 5$ . So the area of a rectangle is its length times its width, the two side lengths.

The area of rectangles also gives a useful trick for multiplication. A 4 by 7 rectangle can be cut into a 4 by 5 rectangle and a 4 by 2 rectangle. The two areas add up to the whole: $4 \times 7 = 4 \times 5 + 4 \times 2 = 20 + 8 = 28$ . This lets your child work out a hard fact from two easier ones.

The last step of the year is the area of composite figures. A composite figure is a shape made of rectangles joined together, like an L. Your child cuts it into rectangles, finds the area of each one and adds them. All the side lengths in 3rd grade are whole numbers.

Your child should be able to:

Find the area of a rectangle by tiling it with unit squares
Multiply the side lengths to find the area, and explain why it matches the count
Split a rectangle into two smaller ones to break up a multiplication
Find the area of a shape made of rectangles by adding the parts

Example

A rug is 3 feet wide and 8 feet long. What is the area of the rug?

Squares 1 foot long on each side would fill the rug in 3 rows of 8.
$3 \times 8 = 24$
The area of the rug is 24 square feet.

Example

A floor is shaped like an L. It splits into two rectangles: one is 8 meters by 5 meters, and the other is 3 meters by 4 meters. What is the area of the floor?

Area of the first rectangle: $8 \times 5 = 40$ square meters.
Area of the second rectangle: $3 \times 4 = 12$ square meters.
Add the two parts: $40 + 12 = 52$ .
The area of the floor is 52 square meters.

Where children usually get stuck

For a rectangle 5 inches by 9 inches, your child adds the sides and says the area is 14.
Adding is the older habit, and both numbers sit right there on the picture. The child has not yet linked area to rows of squares.
What helps. Draw the rectangle on grid paper and count the squares together. Then ask which calculation gives that count, $5 + 9$ or $5 \times 9$ .
On an L-shaped figure, your child multiplies the longest side by the tallest side, as if the shape were a full rectangle.
Multiplying two sides worked for rectangles, so the rule gets used on every shape. An L is a rectangle with a corner missing, so this counts squares that are not there.
What helps. Have your child draw a line that cuts the L into two rectangles first. Find each area, then add.
To find $4 \times 7$ by splitting the 7, your child writes $4 \times 5 + 2 = 22$ .
The child splits the 7 into 5 and 2 but multiplies only the first part. The second rectangle is 4 rows of 2, so it holds 8 squares, not 2.
What helps. Draw both rectangles on grid paper and shade each one. Count the squares in the small one together.

Try four practice questions

Real 3rd grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Maya found the area of the figure below in square inches.
Maya's equation:
$\displaystyle 5 \times 4 = 20$
Select the statement that best describes Maya's equation.
A
The equation finds the perimeter, not the area.
B
The equation uses the wrong width.
C
The equation is correct.
D
The equation uses addition where multiplication is needed.
Show the answer and how to solve it
The area is $5 \times 4 = 20$ square inches. Maya's equation uses multiplication and the right numbers, so the equation is correct.
1. 1Look at the rectangle. It is 5 inches tall and 4 inches wide.
2. 2Think of 5 rows of 4 squares. Each square covers 1 square inch. Multiplication counts all 20 squares.
3. 3Maya's equation, $5 \times 4 = 20$ , multiplies the correct numbers shown on the figure. The equation is correct.
Common mistake. A common mistake is to think every equation must have an error. Always check carefully — an equation can be right.
The idea underneath. To find the area of a rectangle, multiply length × width. Check that the equation uses the right operation and the numbers shown on the figure.
2On level
The rectangle below is split into two smaller parts. Which equation can be used to find the total area of the whole rectangle?
A
$\displaystyle 6 \times 2 \times 7$
B
$\displaystyle 6 + {}$
$\displaystyle (2 \times 7)$
C
$\displaystyle 6 \times (2 + 7)$
D
$\displaystyle (6 + 2) \times 7$
Show the answer and how to solve it
The equation that finds the total area is $6 \times (2 + 7)$ , which equals $54$ square units. The height $6$ multiplies the total width $2 + 7$ .
1. 1Look at the rectangle below. It is one rectangle split into two smaller parts. The whole rectangle is $6$ units tall, and the top is split into two widths of $2$ and $7$ .
2. 2First, find the total width of the whole rectangle by adding the two widths.
3. 3Now find the area of the whole rectangle by multiplying its height ( $6$ ) by its total width ( $2 + 7$ ).
4. 4The equation that finds the total area is $6 \times (2 + 7)$ .
Common mistake. A common mistake is multiplying all three numbers ( $6 \times 2 \times 7 = 84$ ) when seeing three numbers in the figure. The $2$ and $7$ are two parts of the same side, so they should add to make the total width of $9$ before multiplying by the height of $6$ .
The idea underneath. When a rectangle is split into two parts along one side, the total area is the OTHER side times the SUM of the parts — not the product of all three numbers.
3On level
The rectangle below is split into two pieces.
Drag tiles to show how to find the rectangle's area.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
×
(
)
3
2
6
+
-
×
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
Use $3 \times (2 + 6)$ to find the area. The height is $3$ and the total width is $2 + 6 = 8$ , giving an area of $3 \times 8 = 24$ square units.
1. 1The rectangle has a height of 3. It is split by a vertical line into a piece with width 2 and a piece with width 6.
2. 2To find the total width of the whole rectangle, add the two piece widths together.
3. 3The area of a rectangle is its height times its width. Here that means the height ( $3$ ) times the total width ( $2 + 6$ ).
4. 4Drag $3$ into the slot before the multiplication sign. Drag $2$ and $6$ into the two number slots inside the parentheses, and drag + into the slot between them.
Common mistake. A common mistake is using $\times$ inside the parentheses to write $3 \times (2 \times 6)$ . The two pieces of the rectangle are added to make the total width, so the inside operator must be $+$ , not $\times$ .
The idea underneath. When a rectangle is split into pieces side by side, the total area equals the height multiplied by the SUM of the piece widths.
4Stretch
A student tried to find the area of each figure. Read both calculations carefully.
Figure A
$\displaystyle 7 \times 6 = 42$
Figure B
$\displaystyle 5 \times (4 + 3) = 35$
Which statement best describes the work?
A
Both calculations are wrong.
B
Both calculations are correct.
C
Figure A's calculation is wrong.
D
Figure B's calculation is wrong.
Show the answer and how to solve it
Figure A's calculation $7 \times 6 = 42$ is correct. Figure B's calculation $5 \times (4 + 3) = 35$ is also correct. The answer is: Both calculations are correct.
1. 1Look at Figure A. It is a rectangle that is $7$ wide and $6$ tall. To find its area, multiply the two side lengths: $7 \times 6 = 42$ . The student wrote $7 \times 6 = 42$ — that is correct.
2. 2Look at Figure B. It is a rectangle that is $5$ tall and $7$ wide total, split into a $4$ -wide piece and a $3$ -wide piece. To find its area, multiply $5$ by the total width: $5 \times (4 + 3) = 5 \times 7 = 35$ . The student wrote $5 \times (4 + 3) = 35$ — that is correct.
3. 3Figure A's calculation is correct. Figure B's calculation is correct. The statement that best describes the work is: Both calculations are correct.
Common mistake. A common mistake is to check one figure carefully and then assume the other must have an error too. Both $7 \times 6 = 42$ and $5 \times (4 + 3) = 35$ are correct. Always verify each calculation against its own figure.
The idea underneath. When a rectangle is split into two pieces side by side, the area is the height times the total width of both pieces together.