Geometry Unit 6 Areas and Perimeters


 Abner Wilson
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1 Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose depends on the size of the figure being measured. Basically it is the one that makes the most sense. Large areas of land are usually measured in square miles or square kilometers. Small surfaces, like a tabletop, might be measured in square inches. If we don t want to be pinned down, we just say square unit. Example: A "square foot" is a unit of area measurement equal to a square measuring one foot on each side. 1 Square Foot = square meters. Abbrev.: ft 2, sq.ft, SqFt. 1 square foot 1 square foot 1 square meter 1 square yard 1 square centimeter Area of a Rectangle A = lw or A = bh The simplest areas to find are those of rectangles. The area of a rectangle is its length multiplied by its breadth. Sometimes the dimensions of a rectangle are referred to as the base and the height, instead of the length and the breadth. The area is then expressed as the base multiplied by the height. To find the height or the base of a rectangle given its area, substitute values into the area formula and solve for the height or base. Area of a Square A = s 2 A square is a special kind of rectangle in which the length is equal to the breadth. Hence its area is the length of one side multiplied by itself, or the length of one side squared. To find the side of a square given its area, substitute values into the area formula and solve for side. Page 1
2 Area of a parallelogram: A = bh In the formula for the area of a parallelogram, the height is the perpendicular distance from the base to the opposite side. In order to avoid ambiguity it is sometimes called the perpendicular height rather than just the height. The height is not the length of the sloping side. h b The area of a parallelogram is the product of h a base and its corresponding height. The area formula for a parallelogram comes from that of a rectangle. Cutting off piece from the corner and moving it, creates a rectangle with the same base and height as the original parallelogram. To find the height or the base of a parallelogram given its area, substitute values into the area formula and solve for the height or base. More about the parallelogram area formula At first sight, the formula for a parallelogram is quite surprising: it is the same formula as that for a rectangle. Imagine the bottom side of the parallelogram is fixed, but the top side slides along a line, as in the diagram below. The top and bottom of the parallelogram remain the same length and the same distance apart, while the other two sides lengthen or shrink. The shape always remains a parallelogram. (Notice that in one position, the parallelogram will become a rectangle its sides will be at right angles to the base.) The area of the parallelogram stays the same as the parallelogram shifts: it is equal to the area of the rectangle (which, of course, is given by base height). This is easy to see by looking at the next diagram. In this, the first figure consists of two identical triangles and a parallelogram. Imagine the left hand triangle slides to the right: it will fit above the other triangle and leave a rectangle to the left. The second figure shows the same two triangles and the rectangle. Therefore the area of the parallelogram must be the same as the area of the rectangle Page 2
3 Assignment #1: Textbook p #110 all, 17, 19, 23, 24, 26. Stay neat and organized! Staple any additional work papers to this packet. Page 3
4 Lesson 8.2: Areas of Triangle (and rhombus and kite) and Area of Trapezoid Vertex of a triangle: The sides of a triangle always connect at a point called the vertex. The plural of vertex is vertices Sides of a triangle: The sides of a triangle are the segments with a vertex as each end. Base of a triangle: Any side can be called the base of a triangle. Height of a triangle: the height of a triangle is always perpendicular to the named base (or extended base) Area of a triangle: A = 1 2 bh The area of a triangle is one half the product of a base and its corresponding height. If a triangle does not have a side that is horizontal, it is not always clear which side is the base. The beauty of the formula for the area is that it works no matter which side is called the base. Thus the area of the following triangle can be evaluated in three ways. TIP: To find the area of a parallelogram or triangle, you can use any side as the base. But be sure you measure the height of an altitude that is perpendicular to the base you have chosen. To find the height or the base of a triangle given its area, substitute values into the area formula and solve for the height or base. More about the triangle area formula Any triangle can be seen as half of a parallelogram. So the area of a triangle is half the area of a parallelogram. A triangle is half the area of a parallelogram with the same base and height. Page 4
5 Area of Rhombus AND Area of Kite A = dd 2 which is also A = 1 2 dd or A = d 1d 2 which is also A = d d 1 2 The area of a rhombus and of a kite will be equal to the is area of the triangles created by drawing the diagonals. Find the total area of both triangles by multiplying the sum of their heights (the short diagonal d), the sum of their bases (the long diagonal D) and then dividing by 2. To find the length of either diagonal of a rhombus or kite given its area, substitute values into the area formula and solve for the diagonal that you need. Area of Trapezoids Formula #1: A = 1 2 h b 1 + b 2 ( ) or A = h ( b + b 1 2 ) 2 or A = 1 2 h( a + b) Formula #2: A = Mh To find the height or either base of a trapezoid given its area, substitute values into the area formula and solve for the height or missing base. Where do the trapezoid area formulas come from? Formula #1: Start with a trapezoid and made a copy of it rotated it 180 degrees: Next, translate it up to form a parallelogram (the combination of both trapezoids). Using the parallelogram area formula, we find a formula for the area of the trapezoid:. It is HALF the parallelogram we create. Formula #2: Since length median = 1 ( 2 b + b 1 2 ), substitute into formula #1: A = 1 2 h ( b 1 + b 2 ) = 1 ( 2 b 1 + b 2 ) i h = Mh Page 5
6 Assignment #2: Textbook p. 418 #112 all, 17, 19, 20, 25, 26 Stay neat and organized! Staple any additional work papers to this packet. Page 6
7 Lesson 8.3: Areas Problems of more complex shape: You can often use what you know about the areas of rectangles and triangles to find the areas of more complex shapes. 1) Suppose a friend of yours decides to lay crazy paving in his garden which measures 7 m by 5 m, but he wants to leave two rectangular areas, each 2 m by 1 m, for flowerbeds. What area of crazy paving will be needed? The first thing to do when tackling a problem like this is to draw a diagram, and to include on it all the information that has been given. 2) A rug measures 3 m by 2 m. It is to be laid on a wooden floor that is 5 m long and 4 m wide. The floorboards not covered by the rug are to be varnished. (a) What area of floor will need to be varnished? (b) A tin of varnish covers 2.5 m 2. How many tins will be required? a) b) 3) This diagram represents the end wall of a bungalow; the wall contains two windows. The wall is to be treated with a special protective paint. In order to decide how much paint is required, the owner wants to know the area of the wall. Divide the wall up into simple shapes and then find the total area to be painted. Follow the steps below: a) The area of the triangle, large left rectangle (ignore window), right trapezoid (ignore window) b) The window area. c) The total area to be painted. Page 7
8 Assignment #3: Textbook p #1, 2, 49 all Stay neat and organized! Staple any additional work to this packet Page 8
9 8.4 Areas of Regular Polygon A = 1 2 asn or A = 1 2 ap Regular polygons have congruent side lengths and angles. Radius of a regular polygon: the congruent segments drawn from the center of the polygon to each vertex. Apothem of a regular polygon, a: the perpendicular segment from the center of the polygon to a side of a polygon (to the midpoint of that side). Where does the area formula come from? All regular polygons can be divided up into isosceles triangles by drawing segments from the center of the polygon to each vertex. The area of one of the triangles will be A = 1 2 bh = 1 as Where a is the length of the apothem and s is the 2 length of one side of the regular polygon with n sides. The total area of the polygon will be the sum of the triangle areas. A = 1 asn. Since the perimeter is the 2 sum of the sides p = ns so an alternative formula can be found by substituting p in for ns: A = 1 2 ap Page 9
10 Assignment #4: Textbook p #18 all, 12, 13 Stay neat and organized! Staple any additional work to this packet Page 10
11 Lesson 8.5 Areas of Circles There are two very famous formulas for circles: C = πd and A = πr 2 is the Greek letter for p and it has the name pi. Its value is approximately Most calculators have a key for which you can use when carrying out calculations. Finding the Circumference of a circle (length around the circle) Try measuring the circumference and diameter of some circular objects such as tins, bottles or bowls. For each object, divide the circumference by the diameter. You should find that your answer is always just over 3. In fact the ratio is the constant. Therefore: C = πd Since the diameter is twice the radius, this formula can be also be written as circumference = 2 radius = 2 radius = 2πr Finding the area of a circle A = πr 2 The formula for the area of a circle can be explained, as outlined below. The circle here has been divided into equal slices or sectors. The eight sectors can then be cut out and rearranged into the shape shown: this shape has the same area as the circle. You can see that the total distance from A to B along the bumps is the same as half the circumference of the circle that is: 1 i 2π i r = πr. 2 Also the length OA is the same as the radius of the circle. Imagine dividing the circle into more and more sectors and rearranging them as described above. For example, dividing the circle into 16 equal sectors gives the following shape, whose area is still the same as that of the circle. Again the total distance from A to B along the bumps is πr, and the length of OA is the same as the radius. Notice how the rearranged shape is beginning to look more like a rectangle. The more sectors, the straighter AB will become and the more perpendicular OA will be. Eventually it will not be possible to distinguish the rearranged shape from a rectangle. The area of this rectangle will be the same as that of the circle, and its sides will have the lengths radius (for AB) and radius (for OA). So the following formula can be deduced So Area of Circle formula is A = πr 2 Page 11
12 Assignment #5: Textbook p #110 all, 13, 17, 18, 20 Stay neat and organized! Staple any additional work to this packet Page 12
13 Lesson 8.6 Areas of Sectors, Segments, and Annulus Sector of a circle: the region between two radii of a circle and the included arc Segment of a circle: the region between a chord of a circle and the included arc Annulus: the region between two concentric circles Sector of a circle Segment of a circle Annulus A different technique is used to find the area of each of these Area of a Sector : Use the proportion: A arc measure = 2 πr 360 or use formula: A = arc measure 360 πr 2 Example: Area of a Segment: A segment = A sector A triangle Example 1: Example 2: Find x given that the shaded area is 14π cm 2 Area of a Annulus: A annulus = A big circle A small circle Page 13
14 Assignment #6: Textbook p #114 all, all Stay neat and organized! Staple any additional work to this packet Page 14
15 Lesson 8.7 Surface Area of Polyhedra Lesson 8.7 Part 1: Prisms and Cylinders: Prisms have TWO congruent polygonal bases and the lateral faces that are rectangles (right prisms) or parallelograms (oblique prisms). Cylinders have two congruent circular bases and a lateral face that folds out to 1 rectangle. Faces: the flat surfaces Bases: the 2 congruent faces on the prism. Lateral faces: the remaining faces other than the bases (always are rectangles and parallelograms) Edges: a line segment where two faces intersect Vertex: a point of intersection of 3 or more edges. Height of the prism or cylinder: the perpendicular distance between the bases Base Area: B This is the area of the base polygon and is found by using the formulas in lessons #14 Surface area formula for Prisms and Cylinders: Prisms: S.A. = 2B + L.A. Cylinders: S.A. = 2πr 2 + 2πrh Where do Surface Area formulas come from? We are adding the areas of the bases and the lateral faces of polygons. Surface Area = 2B + LA B= Base area LA= Total Lateral Area Page 15
16 Assignment #7: Textbook p. 450 #1, 2, 3, 6, 7, 9, 10, 12 Stay neat and organized! Staple any additional work to this packet Page 16
17 Lesson 8.6 Part 2: Pyramids and Cones Pyramids have ONE polygonal base and lateral faces that are triangles. Cones have ONE circular base and a lateral face that folds out to 1 rectangle Apex: the vertex not on the base Base edge length: b Base Area: B Height of Pyramid (altitude): h the perpendicular distance from the apex to the base. Height of Base polygon (apothem): The perpendicular distance from center of base to the side of the base. Slant height: l or s: The perpendicular distance on a lateral face from the apex to the base edge. Surface area formula for Pyramids and Cones: S.A. = B + L.A. Pyramids: L.A. = 1 2 bl i number of sides of base = 1 2 pl Cones: B = πr 2 L.A. = πrl = πrs S.A. = B + L.A. = πr 2 + πrl Where does surface area formula for a pyramid come from? The surface area of a pyramid will be the sum of the areas of all the triangles L.A. = 1 2 bl i n added to the base area B. The base area formula will be the formula for the area of a regular polygon. Page 17
18 Deriving the Surface Area of a Cone The base The base is a simple circle, so we know from Area of a Circle that its area is given by base of the cone. Where r is the radius of the The top If we were to cut the cone up one side along the red line and roll it out flat, it would look something like the shaded pieshaped section below. 1. This shaded section is actually part of a larger circle (NOT the base of the cone) that has a radius of s, the slant height of the cone. (To flatten it, the cone was cut along the red lines, the length of this cut is the slant height of the cone.) The area of the larger circle is therefore the area of a circle radius s, or 2. The circumference of the larger circle, radius s is 3. The arc AB originally wrapped around the base of the cone, and so its length is the circumference of the circle that is the base of the cone. Recall that circumference of a circle is given by Where r is the radius of the base of the cone. 4. The ratio of area x of the shaded sector to the area of the whole circle, is the same as the ratio of the arc AB to circumference of the whole circle*. Put as an equation 5. Substituting the values from above: Canceling the 2π on the right and solving for x we get 6. Finally, adding the areas of the base and the top part produces the final formula: * For example, if the arc AB is one third the circumference of the large circle, then the area of the sector AB is one third the area of the large circle Page 18
19 Assignment #8: Textbook p #4, 5, 8, Stay neat and organized! Staple any additional work to this packet Page 19
20 Assignment #9: Review p #110, 1731, 34, 35, 45, 46 Stay neat and organized! Staple any additional work to this packet Page 20
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