Area of Polygons

Identify the missing sidelengths.

The figure shows a path (the shaded portion) laid around a rectangular park EFGH.

(i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign possible values of your choice to these measurements and find the area of the path. Give a formula for the area.

An example of a formula — $Area\ of\ a\ rectangle = length \times width$.

[Hint: There is a relation between the areas of EFGH, the path, and ABCD.**]

(ii) If the width of the path along each side is given, can you find its area? If not, what other measurements do you need? Assign values of your choice to these measurements and find the area of the path. Give a formula for the area using these measurements.

[Hint: Break the path into rectangles.**]

(iii) Does the area of the path change when the outer rectangle is moved while keeping the inner rectangular park EFGH inside it, as shown?

The figure shows a plot with sides 14m and 12m, and with a crosspath. What other measurements do you need to find the area of the crosspath? Once you identify the lengths to be measured, assign some possible values of your choice and find the area of the path. Give a formula for the area based on the measurements you choose.

Find the area of the spiral tube shown in the figure. The tube has the same width throughout.

[Hint: There are different ways of finding the area. Here is one method.]

What should be the length of the straight tube if it is to have the same area as the bent tube on the left?

In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons.

Divide a square into 4 parts by drawing two perpendicular lines inside the square as shown in the figure.

Rearrange the pieces to get a larger square, with a hole inside.

You can try this activity by constructing the square using cardboard, thick chart paper, or similar materials.

Find the areas of the following triangles:

Find the length of the altitude BY.

Find the area of $\Delta$SUB, given that it is isosceles, SE is perpendicular to UB, and the area of $\Delta$SEB is 24 sq. units.

[Śulba-Sūtras] Give a method to transform a rectangle into a triangle of equal area.

[Śulba-Sūtras] Give a method to transform a triangle into a rectangle of equal area.

ABCD, BCEF, and BFGH are identical squares.

(i) If the area of the red region is 49 sq. units, then what is the area of the blue region?

(ii) In another version of this figure, if the total area enclosed by the blue and red regions is 180 sq. units, then what is the area of each square?

If M and N are the midpoints of XY and XZ, what fraction of the area of $\Delta$XYZ is the area of $\Delta$XMN? [Hint: Join NY]

Gopal needs to carry water from the river to his water tank. He starts from his house. What is the shortest path he can take from his house to the river and then to the water tank? Roughly recreate the map in your notebook and trace the shortest path.

Find the area of the quadrilateral ABCD given that AC = 22 cm, BM = 3 cm, DN = 3 cm, BM is perpendicular to AC, and DN is perpendicular to AC.

Find the area of the shaded region given that ABCD is a rectangle.

What measurements would you need to find the area of a regular hexagon?

What fraction of the total area of the rectangle is the area of the blue region?

Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral.

Observe the parallelograms in the figure below.

(i) What can we say about the areas of all these parallelograms?

(ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?

Find the areas of the following parallelograms:

Find QN.

Consider a rectangle and a parallelogram of the same sidelengths: 5 cm and 4 cm. Which has the greater area? [Hint: Imagine constructing them on the same base.]

Give a method to obtain a rectangle whose area is twice that of a given triangle. What are the different methods that you can think of?

[Śulba-Sūtras] Give a method to obtain a rectangle of the same area as a given triangle.

[Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it?

[Hint: Show that triangles ΔADB and ΔADC can be made into halves of a rectangle. Figure out how they should be assembled to get a rectangle. Use cut-outs if necessary.]

[Śulba-Sūtras] Give a method to convert a rectangle into an isosceles triangle by dissection.

Which has greater area—an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area—two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.

Find the area of a rhombus whose diagonals are 20 cm and 15 cm.

Give a method to convert a rectangle into a rhombus of equal area using dissection.

Find the areas of the following figures:

[Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.

Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area —

Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH?

[Hint: If ΔAHI ≅ ΔDGI and ΔBEJ ≅ ΔCFJ, then the trapezium and rectangle have equal areas.]

Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area $144\text{ cm}^2$.

A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.

ZYXW is a trapezium with ZY || WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of $\Delta\text{ZWB}$.

Try to think of different creative ways to divide a square into 4 parts of equal area.

Why Can't Perimeter be a Measure of Area?

Why do we count the number of unit squares to assign measures for area? Couldn't we have just used the perimeter of a region, i.e., the length of its boundary as a measure of its area?

Context: Consider two regions, Region 1 and Region 2, such that Perimeter of Region 1 > Perimeter of Region 2, but Area of Region 1 < Area of Region 2.

Q. Find two rectangles that are examples of such regions. If needed, use a grid paper (given at the end of the book) for this.

Also give an example of two regions of other shapes, where the region with the larger perimeter has the smaller area! This property should be visually clear in your example.

In the given figure, which triangle has a greater area: $\Delta$XDC or $\Delta$YDC, if both the rectangles are identical?

In the given figure, which triangle has a greater area: $\Delta$XDC or $\Delta$YBC, if both the rectangles are identical?

Find the area of $\Delta$XDC.

To find the area of a triangle, what measurements do we need?

How do we get the outer rectangle from the given triangle?

Will this formula hold for the kind of triangle, around which we cannot draw a rectangle with BC as the base?

Line $l \parallel \text{BC}$. Consider the different triangles that have BC as their base, and with their third vertex lying anywhere on $l$.

(i) Which of these triangles has the maximum area, and which has the minimum area?

(ii) Which of these triangles has the maximum perimeter, and which has the minimum perimeter?

Analyse whether A lies on the perpendicular bisector of BC.

Area of any Polygon

How do we find the area of this quadrilateral? What measurements do we need for this?

How do we find the area of this pentagon?

Can any polygon be divided into triangles?

Give a method to convert a parallelogram into a rectangle of equal area.

You can try this using a cut-out of a parallelogram.

Can $\Delta\text{AXD}$ and $\text{ABCX}$ fit together, as shown in the figure, to get a rectangle?

Try working this out!

What are the sidelengths of the rectangle WXYZ?

Area of rhombus ABCD can also be determined by finding the areas of $\Delta\text{ADB}$ and $\Delta\text{CDB}$. What formula does this give us?

Context: The area of rhombus ABCD can be written as: $\text{Area of rhombus ABCD} = \text{Area}(\Delta\text{ADB}) + \text{Area}(\Delta\text{CDB}) = \frac{1}{2} \times \text{AO} \times \text{BD} + \frac{1}{2} \times \text{CO} \times \text{BD}$

Q. Simplify the expression to show that we get the same formula for the area of a rhombus in terms of its diagonals.

Find the areas of the following trapeziums by breaking them into figures whose areas can be computed.

Will this formula hold for a trapezium that looks like this?

Will Approach 2 work for any type of trapezium?

What figure will we get when the two trapeziums are joined along BC?

What type of a quadrilateral is this?

What do you think is the area of an A4 sheet? Its sidelengths are 21 cm and 29.7 cm. Now find its area.

What do you think is the area of the tabletop that you use at school or at home? You could perhaps try to visualise how many A4 sheets can fit on your table.

Express the following lengths in centimeters:

(i) 5 in

(ii) 7.4 in

Express the following lengths in inches:

(i) 5.08 cm

(ii) 11.43 cm

How many $\text{cm}^2$ is $1\text{ in}^2$?

Context: Convert $161.29\text{ cm}^2$ to $\text{in}^2$. Every $6.4516\text{ cm}^2$ gives an $\text{in}^2$. Hence, $161.29\text{ cm}^2 = \frac{161.29}{6.4516}\text{ in}^2$.

Q. Evaluate the quotient.

What do you think is the area of your classroom?

How many $\text{in}^2$ is $1\text{ ft}^2$?

What do you think is the area of your school? Make an estimate and compare it with the actual data.

Find out the local unit of area measurement in your region.

What do you think is the area of your village/town/city? Make an estimate and compare it with the actual data.

How many $\text{m}^2$ is a $\text{km}^2$?

How many times is your village/town/city bigger than your school?

Find the city with the largest area in:

(i) India

(ii) the world

Find the city with the smallest area in:

(i) India

(ii) the world