## Chapter 13 Perimeter and Area Answer Key

### Introduction

Perimeter and area are two fundamental concepts in geometry that are used to measure the size and shape of two-dimensional figures. In Chapter 13 of your math textbook, you have encountered various problems related to finding the perimeter and area of different shapes. This article will provide you with the answer key to Chapter 13, allowing you to check your solutions and improve your understanding of these concepts.

### Problem 1: Finding the Perimeter of a Rectangle

In this problem, you are given the length and width of a rectangle and asked to find its perimeter. To do this, you add up the lengths of all four sides of the rectangle. Let's say the length of the rectangle is 10 units and the width is 5 units. The formula to find the perimeter of a rectangle is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width. Plugging in the values, we get P = 2(10) + 2(5) = 20 + 10 = 30 units. Therefore, the perimeter of the rectangle is 30 units.

### Problem 2: Finding the Perimeter of a Square

A square is a special type of rectangle where all four sides are equal in length. To find the perimeter of a square, you simply multiply the length of one side by 4. Let's say the length of a side of the square is 8 units. The formula to find the perimeter of a square is P = 4s, where P is the perimeter and s is the length of one side. Plugging in the value, we get P = 4(8) = 32 units. Therefore, the perimeter of the square is 32 units.

### Problem 3: Finding the Area of a Rectangle

The area of a rectangle is found by multiplying its length and width. Let's say the length of a rectangle is 12 units and the width is 6 units. The formula to find the area of a rectangle is A = l * w, where A is the area, l is the length, and w is the width. Plugging in the values, we get A = 12 * 6 = 72 square units. Therefore, the area of the rectangle is 72 square units.

### Problem 4: Finding the Area of a Square

Just like a square, the area of a square is found by multiplying the length of one side by itself. Let's say the length of a side of the square is 9 units. The formula to find the area of a square is A = s^2, where A is the area and s is the length of one side. Plugging in the value, we get A = 9^2 = 81 square units. Therefore, the area of the square is 81 square units.

### Problem 5: Finding the Perimeter of a Triangle

To find the perimeter of a triangle, you add up the lengths of all three sides. Let's say the lengths of the three sides of a triangle are 5 units, 7 units, and 9 units. The formula to find the perimeter of a triangle is P = a + b + c, where P is the perimeter and a, b, and c are the lengths of the sides. Plugging in the values, we get P = 5 + 7 + 9 = 21 units. Therefore, the perimeter of the triangle is 21 units.

### Problem 6: Finding the Area of a Triangle

The area of a triangle is found by multiplying its base by its height and dividing the result by 2. Let's say the base of a triangle is 8 units and the height is 6 units. The formula to find the area of a triangle is A = (b * h) / 2, where A is the area, b is the base, and h is the height. Plugging in the values, we get A = (8 * 6) / 2 = 48 / 2 = 24 square units. Therefore, the area of the triangle is 24 square units.

### Problem 7: Finding the Perimeter of a Circle

The perimeter of a circle is also known as its circumference. It is found by multiplying the diameter of the circle by π (pi), which is approximately 3.14159. Let's say the diameter of a circle is 10 units. The formula to find the perimeter of a circle is P = πd, where P is the perimeter, π is pi, and d is the diameter. Plugging in the value, we get P = 3.14159 * 10 = 31.4159 units (rounded to four decimal places). Therefore, the perimeter of the circle is approximately 31.4159 units.

### Problem 8: Finding the Area of a Circle

The area of a circle is found by multiplying π (pi) by the square of its radius. The radius is half the length of the diameter. Let's say the radius of a circle is 6 units. The formula to find the area of a circle is A = πr^2, where A is the area, π is pi, and r is the radius. Plugging in the value, we get A = 3.14159 * 6^2 = 3.14159 * 36 = 113.097 square units (rounded to three decimal places). Therefore, the area of the circle is approximately 113.097 square units.

### Problem 9: Finding the Perimeter of a Regular Polygon

A regular polygon is a polygon with all sides and angles equal. To find the perimeter of a regular polygon, you multiply the length of one side by the number of sides. Let's say the length of one side of a regular hexagon is 4 units. The formula to find the perimeter of a regular polygon is P = ns, where P is the perimeter, n is the number of sides, and s is the length of one side. Plugging in the values, we get P = 6 * 4 = 24 units. Therefore, the perimeter of the regular hexagon is 24 units.

### Problem 10: Finding the Area of a Regular Polygon

To find the area of a regular polygon, you need to know the length of one side and the apothem, which is the perpendicular distance from the center of the polygon to one of its sides. Let's say the length of one side of a regular pentagon is 7 units and the apothem is 5 units. The formula to find the area of a regular polygon is A = (ns * a) / 2, where A is the area, n is the number of sides, s is the length of one side, and a is the apothem. Plugging in the values, we get A = (5 * 7 * 5) / 2 = 175 / 2 = 87.5 square units. Therefore, the area of the regular pentagon is 87.5 square units.

### Conclusion

Chapter 13 of your math textbook has provided you with a solid foundation in understanding the concepts of perimeter and area. By solving the problems in this chapter and using the answer key provided in this article, you can check your solutions and further enhance your understanding of these fundamental concepts. Remember to practice regularly and seek additional help if needed to strengthen your skills in geometry.