Grade 7 Area of Shapes
Overview
Measurement in Grade 7 sees us move from simply finding the perimeter and area of a common object, such as a rectangle or triangle, and move into complex shapes that may involve a combination of different shapes or circles.
This unit reviews the basics of finding the area of rectangles, triangles, and parallelograms. It concludes with some real-life problems to which these concepts can be applied.
This unit is designed for the Alberta Grade 7 Curriculum.
Review of the Area of Rectangles
This first section is design to give students a review of what area is and how to calculate it mathematically. This is essential as we move into calculating the area of parallelograms and triangles.
Welcome!
Before we jump in and discuss the Grade 7 content of finding the area of parallelograms and triangles, and finding the missing dimensions on those shapes, let us go back and review what we have learned from Grade 6.
Resource link: https://youtu.be/cAI_twrP8QE
We can see that the area of squares and rectangles can be found by multiplying the length and the width together in order to find out how many square units will fit inside the rectangle (its area).
Let's Try!
Let's see if you remember how to find the area of a rectangle. Go to this link at Khan Academy and try to do at least 8 different problems!
Finding the Area of Parallelograms
Parallelograms
Now that we have reviewed the area of rectangles, we will investigate parallelograms, which are four-sided figures whose opposite sides are parallel.
Adapted from Illustrative Mathematics Grade 6, Unit 1 - Families | Kendall Hunt) (CC BY 4.0). This part of the original document was copied and pasted.
A parallelogram is a quadrilateral having two pairs of parallel sides.
Each of the following quadrilaterals is a parallelogram.
This first video explains how we can cut a right triangle portion off the first parallelogram and slide it to its opposite side to form a rectangle.
For the second parallelogram, we need to apply a different strategy. How can you cut the parallelogram and rearrange the pieces to calculate the area?
Draw lines to show it is possible to form two square units for the right parallelogram. This video again uses the same strategy we saw to find the area of the first parallelogram. The video also highlights a second strategy in which we can enclose the parallelogram in a rectangle that is larger than the parallelogram. The strategy can be applied to the second parallelogram.
Adapted from: Developing the Area of parallelograms, triangles, and trapezoids | OER Commons (CC BY-NC-SA 4.0)
Now, watch the following video on how we use the formula for finding the area of a parallelogram.
Resource link: https://youtu.be/uj6k22WubCk
We can find the area of a parallelogram by breaking it apart and rearranging the pieces to form a rectangle. The diagram shows a few ways of rearranging pieces of a parallelogram. In each one, the result is a rectangle 4 units by 3 units, so its area is 12 square units. The area of the original parallelogram is also 12 square units.
Using these strategies allows you to notice pairs of measurements that are helpful for finding the area of any parallelogram: a base and a corresponding height. The length of any side of a parallelogram can be used as a base. The height is the distance from the base to the opposite side, measured at a right angle. In the parallelogram shown here, we can say that the horizontal side that is 4 units long is the base and the vertical segment that is 3 units is the height that corresponds to that base.
The area of any parallelogram is A = bh (Area = Base x Height)
Try It!
Here is a task to try:
Elena and Noah are investigating this parallelogram.
Elena says, “If the side that is 9 units is the base, the height is 7.2 units. If the side that is 7.5 units is the base, the corresponding height is 6 units.”
Noah says, “I think if the base is 9 units, the corresponding height is 6 units. If the base is 7.5 units, the corresponding height is 7.2 units.”
Do you agree with either one of them? Explain your reasoning. When you are finished, check your answer with the solution, below.
Solution:
Agree with Noah. Explanations vary. Sample explanation: A corresponding height must be perpendicular (drawn at a right angle) to the side chosen as the base. The dashed segment which is 6 units is perpendicular to the two parallel sides that are 9 units long. The dashed segment that is 7.2 units long is perpendicular to the two sides that are 7.5 units.
Resource adapted from Illustrative Mathematics Grade 6, Unit 1 - Families | Kendall Hunt. (CC BY 4.0)
Ready to Practice (Resource: Khan Academy)
Follow this link to Khan Academy and practice the 4 questions. When you are done, close the window and return here. The link is at Plane figures | 6th grade | Math | Khan Academy
You may also find extra worksheet practice at Math Worksheets for Kids
Finding the Missing Dimension
Sometimes you are given the area of the parallelogram and one of the dimensions. Watch the following video to see how:
Resource link: https://youtu.be/SKe3ItfdzTY
Let's Practice!
Now that you know how to find the missing dimension of a parallelogram when given the area, try these practice questions over at Khan Academy. Follow this link.
Area of Triangles
Triangles
You will now use your knowledge of the area of parallelograms to find the area of triangles. For example, to find the area of the blue triangle on the left, we can make a copy of it, rotate the copy, and use the two triangles to make a parallelogram.
This parallelogram has a base of 6 units, a height of 3 units, and an area of 18 square units. So the area of each triangle is half of 18 square units, which is 9 square units.
A triangle also has bases and corresponding heights. Any side of a triangle can be a base. Its corresponding height is the distance from the side chosen as the base to the opposite corner, measured at a right angle. In this example, the side that is 6 units long is the base and the height is 3 units.
Resource adapted from Illustrative Mathematics Grade 6, Unit 1 - Families | Kendall Hunt. (CC BY 4.0)
Because two copies of a triangle can always be arranged to make a parallelogram, the area of a triangle is always half of the area of a parallelogram with the same pair of base and height. We can use this formula to find the area of any triangle. The formula for the area of a triangle is A=bh/2 (Area = base x height. Divide that answer by 2).
Watch the video below:
Resource link: https://youtu.be/pvMuDPVOm7Y
You can also find the base or the height of a triangle when you are given an area. Watch this video to see how:
Resource link: https://youtu.be/42cZch-B9kw
Try It!
Now, try to solve some of these equations on your own:
Find the area of each triangle. Show your reasoning. When you are finished, check your answer by reading the solution, below.
Solution:
- 12 square feet. Sample reasoning: The triangle is half of a rectangle that is 3 feet by 8 feet, and has an area of 24 square feet.
- square units. Sample reasoning: The triangle is half of a parallelogram with a base of 5 units and a height of 3 units.
Resource adapted from Illustrative Mathematics Grade 6, Unit 1 - Families | Kendall Hunt. (CC BY 4.0)
Let's Practice!
These practice activities are available at Khan Academy. You will be asked a series of questions and can put your answer into the program. This will give you immediate feedback on your learning. If you need more practice, you can come back to this page and watch the videos again or watch the videos on the Khan Academy page to get a different perspective.
Practice 1
Plane figures | 6th grade | Math | Khan Academy
Practice 2
Plane figures | 6th grade | Math | Khan Academy
Problem Solving
Let's Apply What We Have Learned!
It's time to put what you have learned about finding the area of a parallelogram and triangle into practice. Complete the questions, below. When you are finished, you will find the solutions to these questions on the attached sheet.
1. A farmer has four pieces of unfenced land as shown to the right in the scale drawing where the dimensions of one side are given. The farmer trades all of the land and $10,000 for 8 acres of similar land that is fenced. If one acre is equal to 43,560 ft2, how much per square foot for the extra land did the farmer pay rounded to the nearest cent?
2. An ordinance was passed that required farmers to put a fence around their property. The least expensive fences cost $10 for each foot. Did the farmer save money by moving the farm?
3. The Smith family is renovating a few aspects of their home. The following diagram is of a new kitchen countertop. Approximately how many square feet of counter space is there?
4. In addition to the kitchen renovation, the Smiths are laying down new carpet. Everything but closets, bathrooms, and the kitchen will have new carpet. How much carpeting must be purchased for the home?
Resource: math-g7-m6-topic-d-lesson-20-student.docx (sharepoint.com) from Engage NY Resources: EngageNY Resources | New York State Education Department (nysed.gov) Creative Commons Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) Some questions were eliminated from the original file.