Questions: Area of Rectangles

Question 1

A rectangle is 7 centimeters long and 4 centimeters wide. What is its area?

• A. 11 square centimeters — add the length and width
• B. 22 centimeters — add all four sides
• C. 28 square centimeters — multiply length times width
• D. 28 centimeters — multiply length times width

Answer: (C)

Area = length × width = 7 × 4 = 28 square centimeters. Option A confuses area with addition. Option B gives the perimeter (2 × 7 + 2 × 4 = 22 cm). Option D has the right number but wrong units — area is always in square units because it measures a two-dimensional surface, not a one-dimensional length.

Question 2

Why does the formula A = l × w work for finding the area of a rectangle?

• A. Because length and width are the only two measurements of a rectangle, so they must be multiplied
• B. Because multiplying is faster than adding, and area problems need quick answers
• C. Because the rectangle can be filled with rows and columns of unit squares, and multiplication counts that array
• D. Because area formulas are rules mathematicians agreed on, not explanations of why they work

Answer: (C)

The formula comes from the array structure. A 5-by-3 rectangle can be filled with 3 rows of 5 unit squares each — exactly the equal-groups model of multiplication. Multiplying counts all the squares without listing them one by one. The formula is not arbitrary — it is a shortcut for counting a rectangular arrangement of unit squares.

Question 3

A rectangle with a perimeter of 20 cm has an area of 20 square centimeters.

• T. True
• F. False

Answer: False

Perimeter and area are different measurements. A rectangle could have a perimeter of 20 cm with many different areas depending on its dimensions. For example, an 8×2 rectangle has perimeter 20 cm but area 16 sq cm; a 6×4 rectangle has perimeter 20 cm but area 24 sq cm. The two measures are independent.

Question 4

If you know that 6 × 8 = 48, you already know the area of a 6-by-8 rectangle in square units — no separate formula is needed.

• T. True
• F. False

Answer: True

The area formula A = l × w is simply multiplication applied to rectangle dimensions. Knowing 6 × 8 = 48 directly gives the area as 48 square units. The formula is not a separate procedure — it is the same multiplication you already know, applied to the specific case of a rectangular array of unit squares.

Question 5

What is the difference between area and perimeter, and why must area always be expressed in square units rather than regular units?

Model answer: Perimeter measures the distance around the outside of a shape — it is a length, measured in regular units (cm, inches). Area measures the surface covered — how many unit squares fill the shape — measured in square units (sq cm, sq in). Area requires square units because you are counting two-dimensional squares, not one-dimensional lengths.

The distinction comes down to what you are measuring: a line (1D, regular units) versus a surface (2D, square units). Forgetting square units is a persistent error because students write the number correctly but omit the 'square' qualifier. The unit matters: 28 cm describes a length; 28 sq cm describes a surface covering 28 unit squares.