Question

Which of the following figure will have maximum area if the. perimeter of all figures is same? (a) Square (b) Octagon (c) Circle (d) Hexagon

Short answer

Answer: (c) Circle

Step-by-step solution

Square - Area Calculation

Firstly, let's consider the square. If we denote the length of one side of the square as 'a', then the perimeter P is given by: P = 4a Now, to find the area of the square, we use the following formula: Area = a^2 We can solve for 'a' in terms of 'P' and substitute it into the area formula: a = P/4 Area = (P/4)^2 = P^2/16

Octagon - Area Calculation

Moving on to the octagon, let's denote the length of one side as 'b'. Since there are eight sides, the perimeter P is given by: P = 8b The area of a regular octagon can be calculated using the formula: Area = 2(1+√2) * b^2 Now, we solve for 'b' in terms of 'P' and substitute it into the area formula: b = P/8 Area = 2(1+√2) * (P/8)^2 = (1+√2) * P^2/32

Circle - Area Calculation

For the circle, let's denote the radius as 'r'. Using this, we can express the perimeter (circumference) and area as: P = 2πr Area = π * r^2 We can solve for 'r' in terms of 'P' and substitute it into the area formula: r = P/(2π) Area = π * (P/(2π))^2 = P^2/(4π)

Hexagon - Area Calculation

Lastly, let's consider the hexagon with one side length denoted by 'c'. The perimeter P is given by: P = 6c The area of a regular hexagon can be calculated using the formula: Area = (3√3/2) * c^2 Now, we can solve for 'c' in terms of 'P' and substitute it into the area formula: c = P/6 Area = (3√3/2) * (P/6)^2 = √3 * P^2/24

Comparing Areas

Considering the common perimeter 'P', the areas of the respective figures are: Square: P^2/16 Octagon: (1+√2) * P^2/32 Circle: P^2/(4π) Hexagon: √3 * P^2/24 Since we want to find the figure which has the maximum area given the same perimeter, we will not derive exact values for 'P'. Instead, we will compare the coefficients in each area expression. Upon comparison, we can see that the circle has the smallest coefficient in the denominator, thus implying the maximum area. So, the answer is (c) Circle.

Key concepts

Area Calculation

Calculating the area of different shapes can help us understand which shapes make optimal use of space when perimeter remains constant. The area gives us a numerical representation of how much space is enclosed within a shape.

For instance, if you know the perimeter of a square, you can determine its side length and subsequently, calculate its area with the formula:

• Square: If one side is 'a', the area is: \( Area = a^2 \).
• By using the perimeter length \( P \), find \( a \) through the equation: \( a = \frac{P}{4} \).
• This results in an area of: \( Area = \frac{P^2}{16} \).

With these formulas, you're better equipped to determine how the area changes relative to the perimeter, helping you compare different geometric shapes.

Perimeter

The perimeter is the total length of the outer edge of a shape. It's essentially the distance you would walk if you traversed around the figure's boundary.

• For a square, the perimeter \( P \) is \( 4a \) where 'a' is the length of one side.
• An octagon's perimeter is given by \( 8b \), where 'b' is the side length.
• The perimeter of a circle, also known as its circumference, is \( 2\pi r \).
• For a hexagon, the perimeter is \( 6c \) with 'c' being the side length.

By understanding perimeter, one can set boundaries for how much material is needed to cover or enclose a space. It's a foundational concept in determining how shapes compare when fixed "boundary distances" are required.

Circle

The circle stands out among other geometric shapes for having continuous symmetry. Circles are defined by their radius \( r \), which is the constant distance from the center to any point on the circumference.

The key calculations related to circles include:

• The circumference (perimeter) is calculated with: \( P = 2\pi r \).
• Area is given by the equation: \( Area = \pi r^2 \).
• Rewriting in terms of perimeter provides: \( r = \frac{P}{2\pi} \), leading to: \( Area = \frac{P^2}{4\pi} \).

Due to its ratio of area to perimeter, the circle maximizes the area when compared to other geometric shapes sharing the same perimeter. This is an essential property that showcases the efficiency of a circular shape in enclosing maximum space.

Geometric Shapes

Geometric shapes are fundamental structures in mathematics, each having unique properties and formulas for their perimeter and area. By mastering them, you can solve complex problems involving space and design.

Common shapes in geometry include:

• **Square:** Equal-length sides and right angles.
• **Circle:** Endless curves without edges.
• **Hexagon:** Six-sided with equal angles in a regular hexagon.
• **Octagon:** Eight-sided figure often seen in stop signs.

Each shape follows distinctive rules in geometry, contributing different applications and usability. Through understanding their spatial properties, learners can better grasp how these figures apply to real-world scenarios, from architecture to everyday objects.