089. Area of Parallelograms

Learning Intentions

• To understand that the area of a parallelogram is related to the area of a rectangle
• find the area of a parallelogram given its base and height

Pre-requisite Summary

• Know that area measures the amount of surface inside a two-dimensional shape
• Know that area is measured in square units such as ${\text{cm}}^2$ and ${\text{m}}^2$
• Know that the area of a rectangle is found using $A=l\times w$
• Know that a parallelogram has two pairs of parallel sides
• Know that the perpendicular height is the shortest distance between the base and the opposite side
• Know that the slanted side length is not usually the height unless it is perpendicular to the base

Worked Examples

Worked Example 1

A rectangle has length $8\text{ cm}$ and width $5\text{ cm}$. A parallelogram is made by sliding one side of the rectangle so the base stays $8\text{ cm}$ and the perpendicular height stays $5\text{ cm}$. Find the area of the parallelogram.

Worked Example 2

Find the area of a parallelogram with base $12\text{ cm}$ and perpendicular height $7\text{ cm}$.

Worked Example 3

Find the area of a parallelogram with base $15\text{ m}$ and perpendicular height $4\text{ m}$.

Worked Example 4

A parallelogram has base $9\text{ cm}$, perpendicular height $6\text{ cm}$ and a slanted side of $8\text{ cm}$. Find the area.

Worked Example 5

A parallelogram has area $48{\text{ cm}}^2$ and base $8\text{ cm}$. Find the perpendicular height.

Worked Example 6

A parallelogram-shaped garden bed has base $11\text{ m}$ and height $3\text{ m}$. Find its area.

Problems

Problem 1

A rectangle has length $10\text{ cm}$ and width $4\text{ cm}$. A parallelogram is made by sliding one side of the rectangle so the base stays $10\text{ cm}$ and the perpendicular height stays $4\text{ cm}$. Find the area of the parallelogram.

Problem 2

Find the area of a parallelogram with base $14\text{ cm}$ and perpendicular height $6\text{ cm}$.

Problem 3

Find the area of a parallelogram with base $18\text{ m}$ and perpendicular height $5\text{ m}$.

Problem 4

A parallelogram has base $13\text{ cm}$, perpendicular height $7\text{ cm}$ and a slanted side of $9\text{ cm}$. Find the area.

Problem 5

A parallelogram has area $63{\text{ cm}}^2$ and base $9\text{ cm}$. Find the perpendicular height.

Problem 6

A parallelogram-shaped paddock has base $16\text{ m}$ and height $4\text{ m}$. Find its area.

Exercises

Understanding and Fluency

1. Decide whether each statement is true or false.
   a) Area measures the space inside a shape
   b) The area of a parallelogram can be found using base $\times$ height
   c) The slanted side of a parallelogram is always the height

2. Complete the statements.
   a) The area of a parallelogram is $A=\mathrm{\_}\mathrm{\_}\mathrm{\_}$
   b) The height must be measured $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ to the base
   c) Area is measured in $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ units

3. Find the area of each parallelogram.
   a) Base $6\text{ cm}$, height $4\text{ cm}$
   b) Base $9\text{ cm}$, height $3\text{ cm}$
   c) Base $20\text{ m}$, height $7\text{ m}$

4. Find the area of each parallelogram.
   a) Base $11\text{ cm}$, height $8\text{ cm}$
   b) Base $13\text{ m}$, height $5\text{ m}$
   c) Base $4.5\text{ cm}$, height $2\text{ cm}$

5. A parallelogram has the following measurements. Find the area.
   a) Base $10\text{ cm}$, height $6\text{ cm}$, slanted side $7\text{ cm}$
   b) Base $15\text{ cm}$, height $9\text{ cm}$, slanted side $11\text{ cm}$
   c) Base $8\text{ m}$, height $4\text{ m}$, slanted side $5\text{ m}$

6. Find the missing height.
   a) Area $24{\text{ cm}}^2$, base $6\text{ cm}$
   b) Area $45{\text{ m}}^2$, base $9\text{ m}$
   c) Area $56{\text{ cm}}^2$, base $8\text{ cm}$

7. Find the missing base.
   a) Area $35{\text{ cm}}^2$, height $5\text{ cm}$
   b) Area $72{\text{ m}}^2$, height $8\text{ m}$
   c) Area $27{\text{ cm}}^2$, height $3\text{ cm}$

8. A rectangle and a parallelogram have the same base and the same perpendicular height. Find the area of each.
   a) Base $7\text{ cm}$, height $4\text{ cm}$
   b) Base $12\text{ m}$, height $5\text{ m}$
   c) Base $9\text{ cm}$, height $2\text{ cm}$

Reasoning

9. Explain why a parallelogram and a rectangle with the same base and height have the same area.

10. A student uses the slanted side instead of the perpendicular height when finding the area of a parallelogram. Explain why this is incorrect.

11. Explain why the formula $A=b\times h$ works for a parallelogram.

12. A parallelogram has base $10\text{ cm}$, height $4\text{ cm}$ and slanted side $6\text{ cm}$. One student says the area is $10\times 4$, and another says it is $10\times 6$. Determine who is correct and explain why.

Problem-solving

13. A parallelogram-shaped banner has base $18\text{ cm}$ and height $9\text{ cm}$. Find its area.

14. A paving stone is shaped like a parallelogram with base $25\text{ cm}$ and height $12\text{ cm}$. Find its area.

15. A parallelogram-shaped garden has base $14\text{ m}$ and perpendicular height $6\text{ m}$. How many square metres does it cover?

16. A classroom poster is shaped like a parallelogram. Its area is $84{\text{ cm}}^2$ and its base is $12\text{ cm}$. Find its height.

17. A farmer’s field is shaped like a parallelogram with area $96{\text{ m}}^2$ and height $8\text{ m}$. Find the base.

18. A rectangle and a parallelogram each have base $15\text{ m}$ and height $4\text{ m}$. Compare their areas.

Potential Misunderstandings

• A student may think the slanted side length is always the height
• A student may forget that the height must be perpendicular to the base
• A student may confuse area with perimeter
• A student may think a parallelogram has a different area formula from a rectangle even when the base and perpendicular height match
• A student may use mismatched units and not write the answer in square units
• A student may multiply the wrong pair of measurements
• A student may not understand that cutting and rearranging a parallelogram can form a rectangle with the same area

Next: 090. Area of Triangles