Geometry Notes Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: • Calculate the area of given geometric figures. • Calculate the perimeter of given geometric figures. • Use the Pythagorean Theorem to find the lengths of a side of a right triangle. • Solve word problems involving perimeter, area, and/or right triangles. Vocabulary: As you read, you should be looking for the following vocabulary words and their definitions: • polygon • perimeter • area • trapezoid • parallelogram • triangle • rectangle • circle • circumference • radius • diameter • legs (of a right triangle) • hypotenuse Formulas: You should be looking for the following formulas as you read: • area of a rectangle • area of a parallelogram • area of a trapezoid • area of a triangle • Heron’s Formula (for area of a triangle) • circumference of a circle • area of a circle • Pythagorean TheoremGeometry Notes Perimeter and Area Page 2 of 57 We are going to start our study of geometry with two-dimensional figures. We will look at the one-dimensional distance around the figure and the two-dimensional space covered by the figure. The perimeter of a shape is defined as the distance around the shape. Since we usually discuss the perimeter of polygons (closed plane figures whose sides are straight line segment), we are able to calculate perimeter by just adding up the lengths of each of the sides. When we talk about the perimeter of a circle, we call it by the special name of circumference. Since we don’t have straight sides to add up for the circumference (perimeter) of a circle, we have a formula for calculating this. Example 1: Find the perimeter of the figure below 8 11 14 4 Solution: It is tempting to just start adding of the numbers given together, but that will not give us the perimeter. The reason that it will not is that this figure has SIX sides and we are only given four numbers. We must first determine the lengths of the two sides that are not labeled before we can find the perimeter. Let’s look at the figure again to find the lengths of the other sides. Circumference (Perimeter) of a Circle rCπ2= r = radius of the circle π = the number that is approximated by 3.141593 perimeter circumferenceGeometry Notes Perimeter and Area Page 3 of 57 Since our figure has all right angles, we are able to determine the length of the sides whose length is not currently printed. Let’s start with the vertical sides. Looking at the image below, we can see that the length indicated by the red bracket is the same as the length of the vertical side whose length is 4 units. This means that we can calculate the length of the green segment by subtracting 4 from 11. This means that the green segment is 7 units. 8 11 14 4 4 11 ― 4 = 7 In a similar manner, we can calculate the length of the other missing side using 6814=− . This gives us the lengths of all the sides as shown in the figure below. 8 11 14 4 7 6 Now that we have all the lengths of the sides, we can simply calculate the perimeter by adding the lengths together to get .5067811144=+++++ Since perimeters are just the lengths of lines, the perimeter would be 50 units. areaGeometry Notes Perimeter and Area Page 4 of 57 The area of a shape is defined as the number of square units that cover a closed figure. For most of the shape that we will be dealing with there is a formula for calculating the area. In some cases, our shapes will be made up of more than a single shape. In calculating the area of such shapes, we can just add the area of each of the single shapes together. We will start with the formula for the area of a rectangle. Recall that a rectangle is a quadrilateral with opposite sides parallel and right interior angles. Example 2: Find the area of the figure below 8 11 14 4 Solution: This figure is not a single rectangle. It can, however, be broken up into two rectangles. We then will need to find the area of each of the rectangles and add them together to calculate the area of the whole figure. There is more than one way to break this figure into rectangles. We will only illustrate one below. Area of a Rectangle bhA= b = the base of the rectangle h = the height of the rectangle rectangleGeometry Notes Perimeter and Area Page 5 of 57 8 11 14 4 8 11 14 4 8 11 14 4 We have shown above that we can break the shape up into a red rectangle (figure on left) and a green rectangle (figure on right). We have the lengths of both sides of the red rectangle. It does not matter which one we call the base and which we call the height. The area of the red rectangle is 56144=×==bhA We have to do a little more work to find the area of the green rectangle. We know that the length of one of the sides is 8 units. We had to find the length of the other side of the green rectangle when we calculated the perimeter in Example 1 above. Its length was 7 units. 8 11 14 4 4 11 ― 4 = 7 Thus the area of the green rectangle is 5678=×==bhA. Thus the area of the whole figure is 1125656rectanglegreenofarearectangleredofarea=+=+ . InGeometry Notes Perimeter and Area Page 6 of 57 the process of calculating the area, we multiplied units times units. This will produce a final reading of square units (or units squared). Thus the area of the figure is 112 square units. This fits well with the definition of area which is the number of square units that will cover a closed figure. Our next formula will be for the area of a parallelogram. A parallelogram is a quadrilateral with opposite sides parallel. You will notice that this is the same as the formula for the area of a rectangle. A rectangle is just a special type of parallelogram. The height of a parallelogram is a segment that connects the top of the parallelogram and the base of the parallelogram and is perpendicular to both the top and the base. In the case of a rectangle, this is the same as one of the sides of the rectangle that is perpendicular to the base. Example 3: Find the area of the figure below 15 6 15 Solution: In this figure, the base of the parallelogram is 15 units and the height is 6 units. This mean that we only need to multiply to find the area of 90615=×==bhA square units.
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