How to Calculate the Area of a Rectangle in Grade 8

Unlike exercises on calculating the area of a rectangle in grade 3, exercises on calculating the area of a rectangle in grade 8 are quite complex, requiring inference, reasoning, and flexible application of properties related to angles and sides of plane geometry from students.

Solving 8th-grade problems on the area of a rectangle, common types, and solutions

To simplify learning and mastering the concept of rectangular area in 8th grade, Mytour has compiled an article sharing an overview of the concept, basic and advanced exercises in the 8th-grade math textbook, and detailed solutions. Let's delve into it!

How to Calculate Rectangular Area in 8th Grade

1. Key Concepts to Remember about Rectangles in 8th Grade

Mastering the concepts, properties, and characteristics of rectangles, students can apply them to solving problems proving a shape is a rectangle or reasoning through exercises calculating the perimeter, area of rectangles, or other planar shapes.

- Concept: A rectangle is a quadrilateral with four right angles. Additionally, a rectangle can also be special cases of parallelograms or trapezoids.

- Properties of Rectangles:
+ Edge Properties: Opposite edges of a rectangle are parallel and equal in length.
+ Angle Properties: All four angles are equal and measure 90 degrees.
+ Diagonal Properties: The two diagonals are equal in length, intersecting at the midpoint of each diagonal.
- Formula for calculating the area of an 8th-grade rectangle: S = a x b.

Where:
+ S is the area of the rectangle.
+ a, b are the lengths of the two sides (length, width) of the rectangle.

Extended Formula:

From the method of calculating the area of a rectangle in 8th grade above, we can easily derive the following related formulas:

For calculating the length of one side of the rectangle:
- If the width is known: Length = Area : Width.
- If the length is known: Width = Area : Length.

If you want to review concepts, formulas related to rectangles, you can refer to the article on how to calculate the area and perimeter of rectangles, previously shared by Mytour.

2. Exercises for calculating the area of rectangles in 8th grade

Not only does it consolidate the knowledge of calculating the area of rectangles at grade 1, but also the problems of calculating the area of rectangles in grade 8 are upgraded to more difficult forms, requiring deeper thinking from learners. Specifically, common types of problems involving calculating the area of rectangles in grade 8 include:

- Calculating the area of a rectangle when the perimeter is known.
- Calculating the area of a rectangle when the diagonal is known.
- Calculating the area of a rectangle when half the perimeter and length are known.
- Calculating the sides of a rectangle when the area is known.
- Proving two rectangles have equal areas,...

To facilitate the review and solving of 8th-grade rectangle area problems for students, Mytour has compiled exercises on calculating the area of rectangles in textbooks (SGK), exercise books (SBT) along with detailed solutions as follows:

2.1. Exercises for calculating the area of rectangles in SGK 8th-grade math

Eighth-grade math textbook exercises on rectangle area in Volume 1 usually focus on pages 116, 117, 118, 119. However, exercises on pages 116, 117 are quite simple, and students can easily find the answers for themselves. Exercises 6, 7, 9, 10, 11, 12, 13, 14, 15 on pages 118, 119 of 8th-grade math Volume 1 gradually increase in difficulty, requiring high-level thinking from the solver.

Exercise 6 (page 118 of 8th-grade Math Volume 1): How does the area of a rectangle change if:
a) Length doubles, width remains unchanged?
b) Length and width triple?
c) Length quadruples, width decreases four times?

Solution:
Let's assume the original rectangle has length a and width b.
Then, the area of the rectangle is: S = a.b.
a) If the length doubles and the width remains unchanged, the dimensions of the new rectangle are: a' = 2a, b' = b.
⇒ S' = a'.b' = 2a.b = 2ab = 2.S
So, the area of the rectangle doubles.
b) If the length and width triple, the dimensions of the new rectangle are:
a' = 3a; b' = 3b. ⇒ S' = a'.b' = 3a.3b = 9ab = 9S
So, the area of the rectangle triples.
c) If the length quadruples and the width decreases four times:
⇒ a' = 4a; b' = b/4.
⇒ S' = a'.b' = 4a.b/4 = ab = S
So, the area of the rectangle remains unchanged.

Exercise 7 on page 118 of 8th-grade math Volume 1
- A room has a rectangular floor with dimensions of 4.2m and 5.4m, a window with dimensions of 1m and 1.6m, and a door with dimensions of 1.2m and 2m.
- We consider a room to meet the standard for lighting if the area of the doors is 20% of the floor area. Does the room meet the lighting standard?

Solution:

Exercise 14 on page 119 of 8th-grade Math Volume 1
A rectangular plot of land is 700m long and 400m wide. Calculate the area of this plot of land in square meters, square kilometers, acres, hectares.
Solution Guide:
The area of the rectangular plot of land is: S = 700.400 = 280000 (m²)
We have: 1km² = 1000000 m²
1 acre = 100 m² 
1 hectare = 10000 m²
So, the area of the plot of land in the units above is: S = 280000m² = 0.28 km² = 2800 acres = 28 hectares.

Exercise 15 on page 119 of 8th-grade Math Volume 1
Challenge. Draw rectangle ABCD with AB = 5cm, BC = 3cm. 
a) Draw a rectangle with a smaller area but a larger perimeter than rectangle ABCD. How many such rectangles can you draw?
Solution:

a) Rectangle ABCD given has an area of S_ABCD = 3.5 = 15 (cm²).
Rectangle ABCD given has a perimeter of P_ABCD = (3+5) x 2 = 16 (cm).
A rectangle with dimensions of 1cm x 12cm has an area of 12cm² and a perimeter of:
(1 + 12).2 = 26 (cm) (with 26 cm > 16 cm).
A rectangle with dimensions of 2cm x 7cm has an area of 14cm² and a perimeter of:
(2 + 7).2 = 18 (cm) (with 18 cm > 16 cm).
Thus, you can draw many rectangles with smaller areas but larger perimeters than rectangle ABCD given.

Guidance on solving exercises on calculating the area of rectangles in 8th-grade math textbooks, exercise 15 page 119

Note: Note b exercise 15 page 119 8th-grade Math Volume 1 is an advanced exercise, requiring critical thinking and combining the properties of squares and rectangles. Understanding the methods of calculating the area of squares in the next article by Mytour, students will be able to understand and solve this exercise. Therefore, within the scope of the content of this article guiding the formula for calculating the area of rectangles in 8th grade above, we will not present the solution for this point.

2.2. Solution to 8th-grade Math Exercise Books on Rectangle Area

In addition to exercises in textbooks, students can also review the knowledge of calculating the area of rectangles in 8th-grade math exercise books or advanced books. The difficulty level of exercises in exercise books, advanced books has been increased to help students study and improve better.

Note: Exercises 12, 13, 14 on page 157 calculating the area of rectangles in 8th-grade Math Exercise Books are quite similar to the exercises in the Textbooks that Mytour has shared the solutions for above. Therefore, in this section, we will only provide the answers and instructions for students to solve 8th-grade Math Exercise Books on rectangle area from exercise 15 to exercise 18 on page 157.

Exercise 16 8th-grade Math Exercise Books, page 157
Problem:
Calculate the sides of a rectangle, knowing that the square of one side length is 16cm and the area of the rectangle is 28cm².
Solution:
Let the lengths of the two sides of the rectangle be a and b (a > 0, b > 0)
According to the problem, assuming we have: a2 = 16 and ab = 28.
a2 = 16 ⇒ a = 4 (cm) (since a > 0) ⇒ b = 28 : a = 28 : 4 = 7 (cm).
So, the two dimensions are 4cm and 7cm.

Exercise 17 on page 157 (8th-grade Math Exercise Books)
Problem: Calculate the sides of a rectangle, knowing that the ratio of its sides is 4/9 and its area is 144 cm2.
Solution:

In addition to the solution to 8th-grade Math exercises on calculating the area of rectangles above, students can explore all exercises of 8th-grade Math Volume 1, Volume 2 by clicking the link below.

Above, Mytour has shared with students, parents how to calculate the area of rectangles in 8th grade and detailed solutions to basic math problems. With this information, students can summarize knowledge, familiarize themselves with different types of exercises, and perform well on rectangle area problems encountered in the future. For 3rd-grade students, refer to How to calculate the area of rectangles in 3rd grade here. Wishing you all the best in your studies.