Area · Lesson 7 of 12
Area of a Parallelogram
“Derive the parallelogram area formula by dissection, distinguish side length from height, and solve equal-area and missing-dimension problems.”
• Transform a parallelogram into an equal-area rectangle by dissection. • Derive A = base × height from the rearranged rectangle. • Distinguish a sloping side from the perpendicular height. • Use any side as base with its corresponding height. • Compare parallelograms with equal bases and heights and solve missing-dimension problems.
A parallelogram looks like a slanted rectangle, but multiplying two adjacent side lengths usually gives the wrong area. The reason is that a sloping side is not the perpendicular distance between the parallel sides. To find the correct formula, we can literally convert the parallelogram into a rectangle without changing its area.
From Parallelogram to Rectangle
Drop a perpendicular from a top vertex to the opposite base line. This creates a right triangle at one end. Cut that triangle and slide it to the other end. The remaining piece and moved triangle fit together as a rectangle.
The new rectangle has the same base length as the parallelogram and the same perpendicular height. Since the pieces are unchanged, the rectangle and parallelogram have equal area.
Problem
A parallelogram has base 13 cm and perpendicular height 7 cm. Find its area.
- 1.A = bh.
- 2.A = 13 × 7.
- 3.The area is 91 cm².
Why the Sloping Side Is Not the Height
Suppose a parallelogram has base 10 cm and sloping side 6 cm, but the perpendicular height is only 4.8 cm. Multiplying 10 × 6 would treat the sloping side as though it were perpendicular, overestimating the area. The correct value is 10 × 4.8.
Problem
A parallelogram has base 10 cm, adjacent side 6 cm, and perpendicular height 4.8 cm. Find the area.
- 1.The base is 10 cm.
- 2.Use the perpendicular height 4.8 cm, not the sloping side 6 cm.
- 3.Area = 10 × 4.8 = 48 cm².
The height is not simply the side drawn upward on the page. It must meet the chosen base line at 90°. In a slanted parallelogram, the side and height are usually different.
Changing the Base
Any side can be chosen as the base, but each choice has its own corresponding perpendicular height. If base b₁ with height h₁ and base b₂ with height h₂ describe the same parallelogram, their products are equal because both equal the same area.
Problem
A parallelogram has area 72 cm². One side is 9 cm. Find the perpendicular height corresponding to that side.
- 1.Use A = bh.
- 2.72 = 9h.
- 3.h = 8 cm.
Parallelograms Between the Same Parallels
If several parallelograms share the same base and lie between the same pair of parallel lines, they all have the same base length and the same perpendicular height. Therefore they all have the same area, no matter how much their top sides are shifted.
Their perimeters can still differ because the sloping side lengths can change. This repeats a theme from triangles: fixed area does not imply fixed perimeter.
Problem
Three parallelograms each have base 12 cm and lie between parallel lines 5 cm apart. Compare their areas.
- 1.Every parallelogram has height 5 cm.
- 2.Every area is 12 × 5 = 60 cm².
- 3.Their shapes and perimeters may differ, but their areas are equal.
Rectangle and Parallelogram with the Same Side Lengths
A rectangle with side lengths 5 cm and 4 cm has area 20 cm² because its height relative to the 5 cm base is exactly 4 cm. A genuinely slanted parallelogram with side lengths 5 cm and 4 cm has height less than 4 cm relative to the 5 cm base, so its area is less than 20 cm². Equality occurs only when the angle becomes 90° and the parallelogram is a rectangle.
Problem
Compare a 5 cm by 4 cm rectangle with a parallelogram having side lengths 5 cm and 4 cm but height 3.2 cm to the 5 cm base.
- 1.Rectangle area = 5 × 4 = 20 cm².
- 2.Parallelogram area = 5 × 3.2 = 16 cm².
- 3.Equal side lengths do not guarantee equal areas because the perpendicular heights differ.
Parallelogram and Triangle Connections
A triangle with the same base and height as a parallelogram has exactly half its area. Therefore a rectangle or parallelogram of base b and height h has twice the area of a triangle with the same base-height pair.
Quiz
How can a parallelogram be converted into a rectangle of equal area?
Which measurement is required with a chosen base?
A parallelogram has base 11 cm and height 6 cm. Its area is
Parallelograms on the same base between the same parallels have
A triangle and parallelogram have the same base and height. The triangle area is
Practice Problems
- Find the area of a parallelogram with base 17 cm and height 8 cm.
- A parallelogram has area 108 cm² and base 12 cm. Find its height.
- A parallelogram has area 84 m² and height 7 m. Find its base.
- A parallelogram has side 10 cm, sloping adjacent side 8 cm, and perpendicular height 6 cm to the 10 cm base. Find its area and explain which measurement is not used.
- Several parallelograms share a 9 cm base and lie between lines 4 cm apart. Find the area of each.
- Compare a 6 cm by 5 cm rectangle with a slanted parallelogram having the same side lengths but height 4 cm to the 6 cm base.
- Give dimensions for a rectangle whose area is twice that of a triangle with base 10 cm and height 7 cm.
Key Takeaways
• A parallelogram can be dissected and rearranged into an equal-area rectangle. • Its area is base × perpendicular height. • A sloping side is not the height unless it is perpendicular to the base. • Any side can be the base if the matching perpendicular height is used. • Parallelograms on the same base between the same parallels have equal areas. • A triangle with the same base and height has half the area of the parallelogram.