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Lesson 4 of 10

Measuring Space: Perimeter and Area · Lesson 4 of 10

Area of Rectangles and Parallelograms

“Rectangles and parallelograms may lean differently, but their areas know the truth.”

Learning Objectives

• Understand area as measurement by unit squares. • Recall the rectangle area formula. • Derive parallelogram area by rearrangement. • Understand base and perpendicular height. • Explain why side lengths alone do not determine parallelogram area.

Area

Area tells us how much flat surface or two-dimensional space is covered by a shape or region. It helps us compare how large different surfaces are, such as the face of a book, the floor of a room, or a rectangular field.

Area is different from perimeter. Perimeter measures the total length around the boundary of a shape, while area measures the space inside that boundary. Since area covers a surface in two dimensions, it is measured in square units such as square centimetres (cm²), square metres (m²), or square kilometres (km²).

Definition
Square Unit

The area of a 1×1 square. It is the basic unit used to measure two-dimensional area.

Rectangle areaLaTeX

The formula A=ab comes from arranging a by b unit squares in a rectangular array.

Turning a Parallelogram into a Rectangle

A parallelogram can be cut and rearranged into a rectangle with the same base and the same perpendicular height. Since cutting and rearranging does not change total area, both shapes have equal area.

Parallelogram rearranged into a rectangle A triangular piece is cut from the left side of a parallelogram and moved to the right side, forming a rectangle with the same base b and perpendicular height h. Parallelogram rearranged into a rectangle Cut the triangular piece and move it without changing the total area. STEP 1 Cut the triangle h Cut piece triangular section b Move STEP 2 Form a rectangle h b A Area of parallelogram = Area of rectangle = b × h Rearranging the triangular piece does not change the total area.
Parallelogram rearranged into rectangle
Parallelogram areaLaTeX

What Height Really Means

The height is not generally the sloping side. It is the perpendicular distance between the two parallel sides chosen as bases.

Common confusion

Using base × sloping side is usually wrong. Use base × perpendicular height.

Thin Parallelograms

For a very slanted parallelogram, the perpendicular from a vertex may meet the extension of the base instead of the base segment itself. The area formula still works because the same cut-and-rearrange idea can be extended geometrically.

Why Side Lengths Alone Are Not Enough

Two parallelograms can have identical side lengths but different angles between adjacent sides. Their perpendicular heights can therefore differ, producing different areas. So side lengths alone do not determine a parallelogram's area.

Worked Example

Problem
A parallelogram has base 12 cm and perpendicular height 7 cm. Find its area.

  1. 1.Use A=bh.
  2. 2.A=12×7=84 cm².

Practice Problems

Practice Problems
  1. Find the area of a rectangle 16 cm by 9 cm.
  2. Find the area of a parallelogram with base 18 cm and height 11 cm.
  3. A parallelogram has area 96 cm² and base 12 cm. Find the height.
  4. Explain why two parallelograms with sides 8 cm and 5 cm can have different areas.
  5. Draw a very thin parallelogram and mark a valid perpendicular height.

Key Takeaways

Key Takeaways

• Area is measured in square units. • Rectangle area is length × width. • Parallelogram area is base × perpendicular height. • Rearrangement explains why the formula works. • Side lengths alone do not determine parallelogram area.

Coming Next

Next, we derive triangle area and discover a surprising equal-area property of medians.

Previous · Lesson 3

Length of an Arc and Perimeter Puzzles

Next · Lesson 5

Area of a Triangle