Area and Perimeter

Area and perimeter

Area

The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square centimeters, square inches, or square kilometers.

Area of a Square

If l is the side-length of a square, the area of the square is l² or l × l.

Example:

What is the area of a square having side-length 3.4?
The area is the square of the side-length, which is 3.4 × 3.4 = 11.56.

Area of a Rectangle

The area of a rectangle is the product of its width and length.

Example:

What is the area of a rectangle having a length of 6 and a width of 2.2?
The area is the product of these two side-lengths, which is 6 × 2.2 = 13.2.

Area of a Parallelogram

The area of a parallelogram is b × h, where b is the length of the base of the parallelogram, and h is the corresponding height. To picture this, consider the parallelogram below:

We can picture "cutting off" a triangle from one side and "pasting" it onto the other side to form a rectangle with side-lengths b and h. This rectangle has area b × h.

Example:

What is the area of a parallelogram having a base of 20 and a corresponding height of 7?
The area is the product of a base and its corresponding height, which is 20 × 7 = 140.

Area of a Trapezoid

If a and b are the lengths of the two parallel bases of a trapezoid, and h is its height, the area of the trapezoid is 
1/2 × h × (a + b) .

To picture this, consider two identical trapezoids, and "turn" one around and "paste" it to the other along one side as pictured below:

The figure formed is a parallelogram having an area of h × (a + b), which is twice the area of one of the trapezoids.

Example:

What is the area of a trapezoid having bases 12 and 8 and a height of 5?
Using the formula for the area of a trapezoid, we see that the area is 
1/2 × 5 × (12 + 8) = 1/2 × 5 × 20 = 1/2 × 100 = 50.

Area of a Triangle

or

Consider a triangle with base length b and height h. 
The area of the triangle is 1/2 × b × h.

To picture this, we could take a second triangle identical to the first, then rotate it and "paste" it to the first triangle as pictured below:

or

The figure formed is a parallelogram with base length b and height h, and has area b × ×h. 
This area is twice that of the triangle, so the triangle has area 1/2 × b × h.

Example:

What is the area of the triangle below having a base of length 5.2 and a height of 4.2?
The area of a triangle is half the product of its base and height, which is 1/2 ×5.2 × 4.2 = 2.6 × 4.2 = 10.92..

Area of a Circle

The area of a circle is Pi  × r² or Pi × r × r, where r is the length of its radius. Pi is a number that is approximately 3.14159.

Example:

What is the area of a circle having a radius of 4.2 cm, to the nearest tenth of a square cm? Using an approximation of 3.14159 for Pi, and the fact that the area of a circle is Pi × r², the area of this circle is Pi × 4.2² 3.14159 × 4.2² =55.41…square cm, which is 55.4 square cm when rounded to the nearest tenth.

Perimeter

The perimeter of a polygon is the sum of the lengths of all its sides.

Example:

What is the perimeter of a rectangle having side-lengths of 3.4 cm and 8.2 cm? Since a rectangle has 4 sides, and the opposite sides of a rectangle have the same length, a rectangle has 2 sides of length 3.4 cm, and 2 sides of length 8.2 cm. The sum of the lengths of all the sides of the rectangle is 3.4 + 3.4 + 8.2 + 8.2 = 23.2 cm.

Example:

What is the perimeter of a square having side-length 74 cm? Since a square has 4 sides of equal length, the perimeter of the square is 74 + 74 + 74 + 74 = 4 × 74 = 296.

Example:

What is the perimeter of a regular hexagon having side-length 2.5 m? A hexagon is a figure having 6 sides, and since this is a regular hexagon, each side has the same length, so the perimeter of the hexagon is 2.5 + 2.5 + 2.5 + 2.5 + 2.5 + 2.5 = 6 × 2.5 = 15m.

Example:

What is the perimeter of a trapezoid having side-lengths 10 cm, 7 cm, 6 cm, and 7 cm? The perimeter is the sum 10 + 7 + 6 + 7 = 30cm.

Circumference of a Circle

The distance around a circle. It is equal to Pi () times the diameter of the circle. Pi or is a number that is approximately 3.14159.

Example:

What is the circumference of a circle having a diameter of 7.9 cm, to the nearest tenth of a cm? Using an approximation of 3.14159 for , and the fact that the circumference of a circle is times the diameter of the circle, the circumference of the circle is Pi × 7.9 3.14159 × 7.9 = 24.81…cm, which equals 24.8 cm when rounded to the nearest tenth of a cm.