Perimeter and Area | A

Question 2

Estimate and Verify

Take a rough sheet of paper or a sheet of newspaper. Make a few random shapes by cutting the paper in different ways. Estimate the total length of the boundaries of each shape then use a scale or measuring tape to measure and verify the perimeter for each shape.

Solution

Understand the Question

Perimeter is the total boundary length around a closed two-dimensional shape.
In this activity, we estimate the boundary length of cut-out shapes by visual inspection, measure the actual length accurately using a flexible thread and a ruler, and compare the two values to verify our estimation.

Step 1 · Prepare the Shapes

Cut out a few different shapes (regular or irregular) from a sheet of newspaper or rough paper. Diagram 1

Step 2 · Estimate the Perimeter

Perimeter is the total distance along the outer boundary of a shape.

Look at each shape and visually divide its boundary into smaller straight or slightly curved parts.
Guess the length of each part and add them up to find the total estimated perimeter.

Step 3 · Measure the Perimeter

To find the exact perimeter:

Place a piece of thread or string along the entire boundary of the shape, following all curves and corners.
Mark the start and end points on the string.
Stretch the string straight along a ruler or measuring tape to measure the distance between the marks.

Step 4 · Compare and Verify

Compare the estimated value with the measured value.

$\text{Verification: } \text{Estimated Perimeter} \approx \text{Measured Perimeter}$

If the estimated perimeter is close to the actual measured perimeter, the estimation is verified.

Answer

Perimeter is estimated by visually summing the guessed lengths of boundary segments, measured accurately using a thread and a ruler, and verified by comparing the two values.

Common Mistakes

Measuring Curved Boundaries with a Rigid Ruler: A straight ruler cannot bend along curved edges. Always use a flexible piece of thread or string to trace the boundary first, then measure the straightened thread against a ruler.
Confusing Perimeter with Area: Perimeter is the one-dimensional length of the boundary line, whereas area is the two-dimensional region covered inside the shape.

More questions in A

Deep Dive: In races, usually there is a common finish line for all the runners. Here are two square running tracks with the inner track of $100\text{ m}$ each side and outer track of $150\text{ m}$ each side. The common finishing line for both runners is shown by the flags in the figure which are in the center of one of the sides of the tracks.

If the total race is of $350\text{ m}$ , then we have to find out where the starting positions of the two runners should be on these two tracks so that they both have a common finishing line after they run for $350\text{ m}$ . Mark the starting points of the runner on the inner track as 'A' and the runner on the outer track as 'B'.

Estimate and Verify

Find various objects from your surroundings that have regular shapes and find their perimeters. Also, generalise your understanding for the perimeter of other regular polygons.

Draw a circle on a graph sheet with diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region.

Try using different shapes (triangle and rectangle) to fill the given space (without overlaps and gaps) and find out the merits associated with using a square shape to find the area rather than another shape. List out the points that make a square the best shape to use to measure area.

Find the area (in square metres) of the floor outside of the corridor.

Find the area (in square metres) occupied by your school playground.

On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units.

a. Which rectangle has the greatest perimeter?

b. Which rectangle has the least perimeter?

c. If you take a rectangle of area 32 sq cm, what will your answers be? Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answer.

Draw a rectangle on a piece of paper and draw one of its diagonals. Cut the rectangle along that diagonal and get two triangles.

Check! whether the two triangles overlap each other exactly. Do they have the same area?

Try this with more rectangles having different dimensions. You can check this for a square as well.

Can you draw any inferences from this exercise? Please write it here.

Q10

Draw suitable triangles on grid paper to verify your inferences and relationships observed in the above exercises.