Measurement: Perimeter and Area We have all seen a squares and rectangles, and remember some basic facts from previous studies. A rectangle is a flat, four-sided geometric figure in which both pairs of opposite sides are parallel and equal, and all four angles are right angles. A square is a rectangle in which all four sides are of equal length. Simply put, the perimeter of any flat geometric figure is defined as the distance around the figure. Although we may be presented with different formulas which get used for different figures, the perimeter of a given figure will still always be the distance around it. The area of a flat geometric figure is defined as the amount of surface the figure covers. For rectangle and squares, this can be viewed as the number of square units which can be enclosed by the figure. In fact, for that very reason, area is always given in square units. For example, imagine a square that measures 1 inch on each side. How many of those 1-in squares – also called “square inches” - can fit inside a rectangle that measures 2 inches wide by 5 inches long? If you're not sure, draw a picture.

IMPORTANT: When we are computing areas and perimeters we need to pay attention to the units of measure. If no unit is stated, then only numeric answers should be given. If, however, a unit is stated – such as cm, ft, or in – we must include the appropriate unit with our answers. Linear units should accompany the linear measures of perimeter. Areas should be stated in square units.

Example 1: A rectangular room measures 3 yards by 4 yards. How many square yards of carpet are needed to cover the floor? Also, how many feet of baseboard trim will be needed to go around the base of the room? Assume the door to the room is 1 yd wide. For the carpet, we need the area of the floor. 3 yd × 4 yd = 12 yd2 We need 12 square yards of carpet. For the baseboard trim, we need to find the perimeter of the room, less the width of the door. 3 yd + 4 yd + 3 yd + 4 yd – 1 yd (for the doorway) = 13 yd We need 13 yards of baseboard trim. Notice how we were able to perform the previous example without the use of formulas. In the case of a perimeter, whether we have a rectangle or not, all we need to do is add up the lengths of all the sides of the figure. When dealing with areas, we have already seen the area of a rectangle is the product of the length and width. In formula form, if we use l for the length and w for the width, the formula is A = lw. Keep in mind, if we understand the origin of the formula, there is no need to memorize it.

Parallelograms A parallelogram is a four sided-figure in which both pairs of opposite sides are equal in length and parallel, but the angles are not necessarily right angles.

Just like any flat geometric figure, the perimeter of a parallelogram is the distance around it. To find the area, we first need to identify the lengths of the base and height of the parallelogram. The base, b, of the parallelogram is the length of the bottom side. Actually, we can use any side, but, for simplicity, let’s stick with the bottom. The height, h, of a parallelogram is the perpendicular distance from the base to the opposite side. This is usually represented by a dashed line drawn perpendicular to the base. Do not confuse the height with the length of a side. That is why we use a dashed line instead of a solid line.

To determine the area of a parallelogram, imagine cutting off the triangular region on the right side and moving it to the left side as follows.

The resultant figure would be a rectangle of dimensions b and h. Thus, the area of the parallelogram would be the product of the base and the height, or A = bh.

Example 2: Find the perimeter and area of the following parallelogram.

The perimeter, P, is the sum of the four side lengths. Remember, the 3 in measure is the height, not a side length. P = 4 in + 7 in + 4 in + 7 in = 22 in The area is the product of the base and the height. In this case the base is 7 inches and the height is 3 inches. A = (7 in)(3 in) = 21 in2

Composite Figures A composite figure is made up of two or more smaller figures. Like always, the perimeter of such a figure is still the distance around the figure. If the composite contains only right angles, we cut the figure into squares and rectangles, find the areas of the smaller pieces, and add them together to get the total area, which we will write as ATotal. Do remember, though, composite figures can also be made of other shapes, as well.

Example 3: Find the area of the following composite figure.

The figure can be viewed as a 2 cm by 8 cm rectangle on top of a 2 cm by 2 cm square, or as a 2 cm by 2 cm square next to two rectangles.

Below is the total for the area of the figures split on the left, and you can verify the area of the split on the right. The area of the composite figure is the sum of the rectangle, Region I, and the square, Region II. ATotal = ARectangle + ASquare = 16 cm2 + 4 cm2 = 20 cm2

It is important to note that, for the perimeter, we sum up only the sides that form the border of the composite figure. We should not try to break apart a figure to find the perimeter, but we may need to use some critical thinking skills to determine the lengths of unidentified sides.

Example 4: Find the area and perimeter of the following figure.

Before we can accurately compute the area and perimeter, we must first determine the lengths of the two unlabeled sides. For the missing horizontal measure, we need to notice the length of the bottom is 13 ft, and the length of the top-most side is 5 ft. That means the missing horizontal measure must be 8 ft. Likewise, the missing vertical measure can be found to be 3 ft. Next, for the area, we can imagine the composite as 5 ft by 3 ft rectangle on top of a 13 ft by 4 ft rectangle. Thus, the total area, A = (5 ft )(3 ft) + (13 ft)(4 ft) = 15 ft2 + 52 ft2 = 67 ft2. For the perimeter, we add together the lengths of all six sides. Be careful not to forget the lengths of the unlabeled sides. Starting at the top and going clockwise, we find the perimeter, P = 5 ft + 3 ft + 8 ft + 4 ft + 13 ft + 7 ft = 40 ft.

Example 5: Find the area and perimeter of the following figure.

For the area, we should imagine the figure as a 4 cm by 9 cm rectangle with a triangular piece next to it. BE CAREFUL! The base length of that triangle part is not 12 cm; it is only 8 cm. Also, how do we find the height of the triangular piece?

As we have seen, since the length of the base for the whole composite figure is 12 cm, and 4 cm of that is composed of the rectangle, the triangle-shaped region has a base of 8 cm. For the height of the triangular region, we see the entire height of the composite figure is 9 cm, and the triangle height is 3 cm lass than that. So, the height of the triangle is 6 cm. Now, taking the entire composite figure in mind, ATotal = ARectangle + ATriangle ATotal = (4 cm)(9 cm) + (1/2)(8 cm)(6 cm) = 36 cm2 + 24 cm2 = 60 cm2. For the perimeter, start at the top and go clockwise around the figure. P = 4 cm + 3 cm + 10 cm + 12 cm + 9 cm = 38 cm.

Circles All of us can identify a circle by sight, so we will focus on a few certain properties of circles. The diameter of a circle is the distance across a circle as measured through the circle's center, and is indicated with the letter d. Instead of the word perimeter, the distance around a circle is called its circumference, and is indicated with the letter C.

Experiment Select three circular objects, and use a piece of string and a metric ruler to measure the circumference and diameter of each. Either print this sheet or make a list that looks like the one below. In each case, compute C/d (rounded to the nearest hundredth). Object

Circumference, C

Diameter, d

C/d

_________________________

_____ cm

_____ cm

_____

_________________________

_____ cm

_____ cm

_____

_________________________

_____ cm

_____ cm

_____

Pi and r For each of the above circular objects, you should have found C/d to be close to 3.14. This value is represented by the lower-case Greek letter pi, π. If you did everything accurately in the above experiment, you should have found the value of C/d to be really close to pi for each one. Also, the radius, r, is defined as half of the circle's diameter. Example 6: Find the radius of a circle with diameter 12 m. The radius is half the diameter, so r = 6 cm.

Example 7: Find the diameter of a circle with a radius of 9 ft. The diameter is twice the radius, so d = 18 ft.

Area and Circumference Formulas Using r for the radius, and d for the diameter, the area and circumference of a circle are defined as: Area, A = πr2 Circumference, C = πd. Or, since d = 2r, we often see this as C = 2πr.

Also, to approximate the calculations, it is customary to use π = 3.14. Use these formulas to find the area of the three circles in your experiment.

Example 8: Find the area and circumference for a circle of radius 3 inches. Be sure to label your answers. Use π = 3.14, and round your answers to the nearest hundredth. Area = AC = πr2 = (3.14)(3 in.)2 = 28.26 in2 Circumference = C = 2πr = 2(3.14)(3 in) = 18.84 in

Example 9: Find the area and circumference of the following circle. Be sure to label your answers. Use π = 3.14, and round your answers to the nearest tenth.

15 cm

Area = (3.14)(15 cm)2 = 706.5 cm2 Circumference = 2(3.14)(15 cm) = 94.2 cm