How to Find the Perimeter of a Quadrant

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Finding the perimeter of a variety of shapes is an important part of geometry with many practical applications. Quadrants appear in a wide range of places, from a slice of pie to the outer shape of the “diamond” in baseball. Finding the perimeter of a shape like this has two main parts: first you find the length of the curved section, and then you add the lengths of the straight sections to this. Picking up this process will give you a good grounding in finding the perimeters for many shapes, as well as introducing a key strategy to solve problems like this in general.

TL;DR (Too Long; Didn't Read)

Find the perimeter (​p​) of a quadrant with straight sides of length (​r​) using the formula: ​p​ = 0.5π​r​ + 2​r​. The only bit of information you need is the length of the straight side.

The Perimeter of a Circle

Splitting this problem into a curved part and two straight parts is the key to solving it. A quadrant is a pie-slice shaped quarter of a circle, and a perimeter is just the word for the total distance around the outside of something. So to solve the problem, the first thing you need is the distance around a quarter of a circle.

The full perimeter of a circle is called the circumference, and is given by

C = 2πr

where (​C​) means circumference and (​r​) means radius. You need the radius of the quadrant to solve the problem, but this is the only information you need. The first step gives you the circumference of a circle where the radius is the length of one of the straight parts of the quadrant.

The Length of the Quadrant’s 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 to make it clear how the solution works, but you can also calculate 0.5 × π​r​ to do this all in one step. The result of this is the length of the curved section.

The Area of a Quadrant

The method used so far works for the length of a quarter-circle arc, but a small change helps you find the area of a quadrant with a very similar approach. The area of a circle is

A = πr^2

so the area of a quadrant is

A = \frac{πr^2}{ 4}

because it’s a quarter of the area of the circle.

Add the Straight Sections

The final stage in finding the perimeter of a quadrant is to add the missing straight sections to the length of the curved section. There are two straight sections, and they both have length ​r​, so you add 2​r​ to the result for the length of the curve.

Formula for the Perimeter of a Quadrant

Pulling both parts together, the formula for the perimeter (​p​) of a quadrant is:

p = 0.5πr + 2r

This is really easy to use. For example, if you have a quadrant with ​r​ = 10, this is:

\begin{aligned} p &= (0.5×π×10) + (2×10) \\ &= 5π + 20 = 15.7 + 20 \\ &= 35.7 \end{aligned}

Tips

  • If You Don’t Know ​r​: If you aren’t given ​r​ but instead are given the length of the curved section, you can use the result of the first part to find ​r​. Since ​C​ = 2π​r​, this means ​r​ = ​C​÷2π. If you have the measurement for the quarter arc, just multiply that by 4 to find ​C​, and proceed with finding ​r​. Once you’ve found ​r​, add 2​r​ to the length of the curved section to find the total perimeter.

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