Practice Questions

What is the definition of the 'area' of a closed figure?

Explain how to find the perimeter of a triangle with sides of lengths a, b, and c.

State the formula for the area of a rectangle and explain what each variable in the formula represents.

State the formula for calculating the perimeter of a square.

List the formulas for the perimeter of a rectangle, a square, and an equilateral triangle.

The perimeter of a regular hexagon is $108 \text{ cm}$. Calculate the length of one of its sides.

Calculate the perimeter of a square park whose area is $81 \text{ m}^2$.

The area of a rectangular hall is $240 \text{ sq m}$. If its length is $20 \text{ m}$, calculate its breadth.

Justify why simply adding the perimeters of two rectangles that are joined together does not give the correct perimeter of the new composite shape.

Propose a general formula for the perimeter of a regular polygon with 'n' sides, where the length of each side is 's'. Justify your formula.

Formulate a step-by-step method to calculate the area of an L-shaped figure with given outer dimensions. Justify your method by explaining why it works.

Define the term 'perimeter' as it relates to a closed plane figure.

Evaluate the following statement: 'If the area of rectangle A is greater than the area of rectangle B, then the perimeter of rectangle A must also be greater than the perimeter of rectangle B.' Is this always true? Justify your conclusion.

Design two different rectangles that both have an area of $36$ square units. Evaluate which of your designs has the smaller perimeter and justify your choice with calculations.

Describe in detail the procedure for finding the area of a floor that is not covered by a carpet. Assume the floor is rectangular and the carpet is square.

A room is $12 \text{ m}$ long and $9 \text{ m}$ wide. Calculate the number of square tiles of side $30 \text{ cm}$ required to cover the floor of the room.

Explain why the perimeter of a square with side length 's' can be calculated as $4 \times s$.

Explain the general rule to find the perimeter of any regular polygon. Provide examples for a regular pentagon and a regular hexagon.

Summarize the steps to find the perimeter of a rectangular park with a length of 50 m and a breadth of 30 m.

Calculate the area of a square whose perimeter is $60 \text{ m}$.

Ayan runs 5 rounds around a rectangular park of length $80 \text{ m}$ and breadth $45 \text{ m}$. Calculate the total distance he covers.

A wire is in the shape of a rectangle of length $18 \text{ cm}$ and breadth $12 \text{ cm}$. If the same wire is rebent into the shape of a square, calculate the length of the side of the square. Analyze which shape encloses more area.

The cost of fencing a square field at a rate of ₹$22$ per metre is ₹$8,800$. Calculate the length of the side of the field.

By splitting the given T-shaped figure into two rectangles, calculate its total area. All measurements are in centimetres.

The areas of a square and a rectangle are equal. The side of the square is $20 \text{ cm}$. If the length of the rectangle is $25 \text{ cm}$, calculate the breadth and the perimeter of the rectangle.

An architect claims that for any given perimeter, a square will always enclose a smaller area than a non-square rectangle. Critique this statement. Is it correct? Justify your answer with an example.

Identify the standard unit used for measuring area.

Describe the relationship between the area of a rectangle and the area of a triangle formed by cutting the rectangle along one of its diagonals.

A rectangular park is $45$ m long and $30$ m wide. A path $2.5$ m wide is constructed outside the park. Formulate a plan to calculate the cost of paving this path at a rate of ₹120 per square meter. Justify each step of your plan.

Name the type of polygon that has all its sides and all its angles equal.

A rectangular floor is $15 \text{ m}$ long and $10 \text{ m}$ wide. A carpet is laid in the center of the room, leaving a uniform margin of $1.5 \text{ m}$ all around. Calculate: a) The area of the carpet. b) The area of the floor that is not carpeted.

A farmer wants to fence a rectangular field of area $600$ sq m. He wants to use the minimum possible length of fencing. Propose the dimensions of the field that would be most economical. Justify your proposal by comparing it with at least one other possible dimension.

A square garden has a perimeter of $48$ m. A second rectangular garden has the same area as the square garden, but its length is $4$ m longer than its width. Evaluate which garden would require more fencing material. Create a conclusive argument supported by calculations.

Summarize the difference between perimeter and area. Use a square with a side length of 5 cm to illustrate your explanation.

A farmer has a rectangular field measuring $120 \text{ m}$ in length and $80 \text{ m}$ in breadth. He wants to fence it with 3 rounds of barbed wire. The cost of the wire is ₹$15$ per metre. A gate of width $4 \text{ m}$ is to be left for entry. Calculate the total cost of the wire needed.

Create a composite figure using two rectangles, where the total area is $50$ sq m and the total perimeter is exactly $30$ m. Provide a diagram with dimensions and justify your solution with calculations.

From a rectangular piece of cardboard measuring $25 \text{ cm}$ by $20 \text{ cm}$, two square pieces of side $5 \text{ cm}$ are cut from two opposite corners. Calculate the area and perimeter of the remaining piece of cardboard.

Compare two parks: a square park with a side of $35 \text{ m}$ and a rectangular park with dimensions $40 \text{ m} \times 30 \text{ m}$. a) Calculate which park has a larger area and by how much. b) A person walks three rounds of each park. Calculate the total distance walked in each case and analyze in which park the person walks a longer distance.

A rectangular park is $50 \text{ m}$ long and $30 \text{ m}$ wide. A path $2.5 \text{ m}$ wide is constructed outside the park. Calculate the area of the path.

Recall the convention for counting squares to estimate the area of an irregular shape on a grid paper.

A piece of wire is $48$ cm long. Propose two different regular polygons that can be formed using this wire. Evaluate which of the two proposed shapes encloses a greater area. (You may compare a square and an equilateral triangle).

A square of side $4$ cm is cut from a corner of a larger rectangular sheet of paper measuring $10$ cm by $8$ cm. Critique the method of finding the perimeter of the remaining shape by subtracting the perimeter of the small square from the perimeter of the large rectangle. Is this method correct? Justify your reasoning.

Two friends, Rohan and Priya, are given identical rectangular sheets of paper measuring $20$ cm by $12$ cm. Rohan cuts his sheet into four equal smaller rectangles. Priya cuts her sheet along one diagonal to get two triangles. They both rearrange their pieces to form new shapes without overlap. Propose whose set of pieces, when added together, will have a greater total perimeter. Justify your conclusion with detailed calculations and reasoning.

Design a floor plan for a single room that is not a simple rectangle or square, but a composite shape made of two rectangles. The total area of the room must be exactly $120$ square feet. The total length of the wall (perimeter) must be exactly $50$ feet. Create a labeled diagram of your design and provide calculations to justify that it meets both the area and perimeter requirements.

Formulate a general rule to predict the change in the perimeter of a shape made of unit squares on a grid, when a single new unit square is added. Your rule must account for the different possible placements of the new square. Justify your rule with diagrams and examples.