How to Calculate the Perimeter of a Quadrant with Examples
Geometry includes a crucial concept called perimeter finding that has many real-world applications. Quadrants can be seen in a variety of contexts, such as the outer shape of a baseball's "diamond" and a slice of cake.
Finding the perimeter of a form like this involves adding the lengths of the straight sections to the length of the curved section, which is the first step in determining the perimeter. Learning this method will provide you with a solid foundation for calculating the perimeters of several forms and expose you to an important tactic for handling issues similar to this in general.
The Quadrant
Quarter refers to a $\dfrac{1}{4}$ part of a thing or shape. A quarter circle is created by cutting a circle into four equal parts, or a semicircle into two equal sections.
The Perimeter of a Circle
Perimeter of circle is nothing but the length of boundary of the circle which is also known as circumference of the circle. The entire perimeter of a circle is called the circumference and is given by $c=2 \pi r$, where $(c)$ means circumference and $(r)$ means radius.
The Length of the Quadrant Curve
Since a quadrant is a quarter of a circle, to find the length of the curved part, take the circumference from the last step and divide it by 4. This helps clarify how the solution works, but you can also compute $0.5 \times \pi r$ by doing all of this in one step. The length of the curved portion is the outcome of this.
The Quadrant Circle with Radius
The Perimeter of a Quadrant
To find the perimeter of a quadrant we need length of curved part which can be found using the formula $0.5 \times \pi r$ and length of two straight lines which is equal to the radius as quadrant is combination of two radii and a curved part.
Therefore the formula can be written as:
The Perimeter of a Quadrant = $2r + 0.5 \times \pi r$
where r is radius of quadrant circle.
The Area of a Quadrant
The method used so far works for the length of an arc of a quarter circle, but a small change helps you find the area of a quadrant with a very similar approach. the area of a circle is $\mathbf{a}=\pi \mathbf{r}^{2}$, so the area of a quadrant is $\mathbf{a}=\dfrac{\pi \mathbf{r}^{2}}{4}$, because it is one fourth the area of the circle.
Solved Examples
Q 1 If you have a quadrant with r = 10, what will be its perimeter?
Sol: Put r = 10 in the perimeter of a quadrant formula
$p=0.5 \pi r+2 r$
After substituting the values, we get
$\Rightarrow p=0.5 \pi(10)+2(10)$
Now on solving it further, we get
$\Rightarrow p=5 \pi+20$
$\Rightarrow p=15.7+20$
$\Rightarrow p = 35.7$
Therefore, the perimeter will become 35.7 unit.
Q 2 What will be the perimeter of a quadrant if the diametre of the circle is 10 m.
Sol: The radius of the circle will be half of its diameter. Therefore,
$d=10 \mathrm{~m}$
$r=\dfrac{d}{2}$
$r=\dfrac{10}{2}$
$r=5 \mathrm{~m}$
Now the perimeter of the quadrant will be
$p=0.5 \pi r+2 r$
Now, on substituting the values, we get
$\Rightarrow p=0.5 \pi(5)+2(5)$
Solving further the above equation, we get
$\Rightarrow p=2.5 \pi+10$
$\Rightarrow p=7.85+10$
$\Rightarrow p=17.85 m$
Therefore, the perimeter of a quadrant if the diameter of the circle is 10 m will be 17.85m.
Practice Questions:
Q 1 What will be the perimeter of a quadrant circle with radius r = 15m.
Ans: 53.55m
Q 2 If the diametre of a circle is 12m, find the perimeter of the quadrant of that circle.
Ans: 21.42m
Q 3 What will be the perimeter of a quadrant circle with radius r = 75m .
Ans: 267.75m
Summary
The locus of all points that are equally distant from the centre is referred to as a circle. A circle is evenly divided into four pieces, or more specifically, four quadrants, by a set of two perpendicular lines. A quadrant is one-fourth of a circle's segment. The coordinate plane system's four quarters are referred to as quadrants. A quadrant is the name for one of the four divisions. A quadrant, or sector of 90 degrees, is what is referred to as a quarter of a circle in the context of circles. We have also discussed the perimeter of the quadrant.
FAQs on Perimeter of a Quadrant Explained
1. What exactly is a quadrant, and how is its perimeter defined?
A quadrant is a sector of a circle that covers one-fourth of its total area, formed by two radii meeting at a 90-degree angle. The perimeter of a quadrant is the total length of its boundary. This boundary consists of two straight sides (the radii) and one curved side (the arc). It is crucial to understand that the perimeter is the entire enclosing path, not just the curved part.
2. What is the standard formula to calculate the perimeter of a quadrant?
The formula to calculate the perimeter of a quadrant with a given radius 'r' is: Perimeter = 2r + (πr/2). This formula is derived by summing the lengths of the two radii (r + r = 2r) and the length of the arc, which is exactly one-fourth of the circle's total circumference (2πr ÷ 4 = πr/2).
3. How can I calculate the perimeter of a quadrant if its radius is 14 cm?
To find the perimeter with a radius of 14 cm, apply the formula P = 2r + (πr/2). You can follow these steps:
- First, calculate the combined length of the two straight sides (radii): 2 × 14 cm = 28 cm.
- Next, calculate the length of the curved arc: (π × 14) / 2 = 7π cm. Using the approximation π ≈ 22/7, the arc length is 7 × (22/7) = 22 cm.
- Finally, add the lengths of the straight sides and the arc: 28 cm + 22 cm = 50 cm. The total perimeter is 50 cm.
4. Is the 'perimeter' of a quadrant the same as its 'circumference'?
No, these terms are not interchangeable, and this is a common point of confusion. The term 'circumference' specifically refers to the perimeter of a full circle. For a quadrant, the correct term is 'perimeter', as it refers to the total length of the boundary of any two-dimensional shape. The perimeter of a quadrant includes the curved arc *plus* the two straight radii that enclose the shape. Calculating just one-fourth of the circle's circumference only gives you the arc's length, not the complete perimeter.
5. What is the key difference between calculating the perimeter and the area of a quadrant?
The fundamental difference lies in what is being measured:
- Perimeter measures the total length of the boundary around the shape. It is a one-dimensional measurement (e.g., in cm or m) and answers the question, "What is the length of the border?" The formula is P = 2r + (πr/2).
- Area measures the total surface space enclosed within that boundary. It is a two-dimensional measurement (e.g., in cm² or m²) and answers, "How much space does the shape cover?" The formula is A = (1/4)πr².
6. How is the perimeter of a quadrant calculated if the radius is unknown but the arc length is given?
If the arc length of a quadrant is known, you can work backwards to find the radius and then the total perimeter. Here is the method:
- The formula for a quadrant's arc is: Arc Length = (πr)/2.
- Rearrange this formula to solve for the radius 'r': r = (2 × Arc Length) / π.
- Once you have calculated the radius 'r', you can find the total perimeter by adding the length of the two radii (2r) to the given arc length.
7. Where might the concept of a quadrant's perimeter be used in a real-world scenario?
The concept of a quadrant's perimeter has many practical applications. For instance, in landscaping, if you are designing a corner garden plot shaped like a quadrant, you would need to calculate its perimeter to determine the exact amount of fencing or decorative edging required. In architecture and construction, calculating the perimeter of a quarter-circle window or a structural arch is essential for sourcing framing materials and ensuring a perfect fit.
