089. Area of Parallelograms

Learning Intentions

• To understand that the area of a parallelogram is related to the area of a rectangle
• Solve 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 and
• Know that the area of a rectangle is found Use
• 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 and width . A parallelogram is made by sliding one side of the rectangle so the base stays and the perpendicular height stays . Find the area of the parallelogram.

Worked Example 2

Find the area of a parallelogram with base and perpendicular height .

Worked Example 3

Find the area of a parallelogram with base and perpendicular height .

Worked Example 4

A parallelogram has base , perpendicular height and a slanted side of . Find the area.

Worked Example 5

A parallelogram has area and base . Find the perpendicular height.

Worked Example 6

A parallelogram-shaped garden bed has base and height . Find its area.

Problems

Problem 1

A rectangle has length and width . A parallelogram is made by sliding one side of the rectangle so the base stays and the perpendicular height stays . Find the area of the parallelogram.

Problem 2

Find the area of a parallelogram with base and perpendicular height .

Problem 3

Find the area of a parallelogram with base and perpendicular height .

Problem 4

A parallelogram has base , perpendicular height and a slanted side of . Find the area.

Problem 5

A parallelogram has area and base . Find the perpendicular height.

Problem 6

A parallelogram-shaped paddock has base and height . Find its area.

Exercises

Understanding and Fluency

Exercise 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 height

c) The slanted side of a parallelogram is always the height

Exercise 2.

Complete the statements.

a) The area of a parallelogram is

b) The height must be measured to the base

c) Area is measured in units

Exercise 3.

Find the area of each parallelogram.

a) Base , height

b) Base , height

c) Base , height

Exercise 4.

Find the area of each parallelogram.

a) Base , height

b) Base , height

c) Base , height

Exercise 5.

A parallelogram has the following measurements. Find the area.

a) Base , height , slanted side

b) Base , height , slanted side

c) Base , height , slanted side

Exercise 6.

Find the missing height.

a) Area , base

b) Area , base

c) Area , base

Exercise 7.

Find the missing base.

a) Area , height

b) Area , height

c) Area , height

Exercise 8.

A rectangle and a parallelogram have the same base and the same perpendicular height. Find the area of each.

a) Base , height

b) Base , height

c) Base , height

Reasoning

Exercise 9.

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

Exercise 10.

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

Exercise 11.

Explain why the formula works for a parallelogram.

Exercise 12.

A parallelogram has base , height and slanted side . One student says the area is , and another says it is . Determine who is correct and explain why.

Problem-solving

Exercise 13.

A parallelogram-shaped banner has base and height . Find its area.

Exercise 14.

A paving stone is shaped like a parallelogram with base and height . Find its area.

Exercise 15.

A parallelogram-shaped garden has base and perpendicular height . How many square metres does it cover?

Exercise 16.

A classroom poster is shaped like a parallelogram. Its area is and its base is . Find its height.

Exercise 17.

A farmer’s field is shaped like a parallelogram with area and height . Find the base.

Exercise 18.

A rectangle and a parallelogram each have base and height . 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