Perimeter and area formulas for common geometric figures:


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1 Lesson : Perimeter and Area of Common Geometric Figures Focused Learning Target: I will be able to Solve problems involving perimeter and area of common geometric figures. Compute areas of rectangles, triangles, rhombuses, parallelograms, trapezoids, and kites. Vocabulary: Base of a parallelogram Height of a parallelogram Height of a triangle CA Std 8.0: Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. CA Std 10.0: Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. Altitude of a parallelogram Base of a triangle Height of a trapezoid Base & height (altitude of a parallelogram) Base & height of a triangle Bases & Height of a trapezoid Perimeter and area formulas for common geometric figures: Rectangle P= b + h A= bh Square P = 4s A = s Triangle P = a + b + c Circle Parallelogram Trapezoid P = a + b A= bh Rhombus P = 4s Kite P = a+b 1
2 Example 1: Using the perimeter to find area The perimeter of a rectangle is 48 The perimeter of a rectangle is 34 ft and the base is 8ft. What is the inches and the height is 10 inches. area? Find the area of the rectangle. The perimeter of a rectangle is 36cm and the base is 10cm. Find the area of the rectangle. Example : Using area to find perimeter. A square and a rectangle have A square and a rectangle have equal areas. If the rectangle is 9cm equal areas. If the rectangle is 7 by 4cm. What is the perimeter of inches by 3 inches. What is the the square? perimeter of the square? A square and a rectangle have equal areas. If the rectangle is 16cm by 9cm. What is the perimeter of the square? Example 3: Finding areas parallelograms. Find the area of each parallelogram. Find the area of a parallelogram with the base 1 m and height 9 m. Find the area of each parallelogram. Example 4: Finding a missing dimension. Find the value of h in the Find the value of h in the parallelogram parallelogram Find the value of h in the parallelogram
3 Example 5: Finding the area of the shaded region (triangles): Example 6: Finding the area of trapezoid. Find the area of trapezoid ABCD. What is the area of trapezoid PQRS? (Using a right triangle ) Find the area of trapezoid. Example 7: Finding areas of rhombuses and kites: a) Finding the area of a rhombus: a) Finding the area of a rhombus: a) Finding the area of a rhombus: b) Find the area of a kite: b) Find the area of a kite: b) Find the area of a kite: 3
4 10 3 Areas of Regular Polygons: Focused Learning Target: I will be able to CA Std 10.0: Students compute areas of polygons, Find the area of a regular polygon including rectangles, scalene triangles, equilateral Vocabulary: triangles, rhombi, parallelograms, and trapezoids. Radius of a regular polygon Apothem Radius is the distance from the center to a vertex. Apothem is the perpendicular distance from the center to a side Example 1: Finding angle measures: Find the measure of each numbered angle. A portion of a regular hexagon has apothem and radii drawn. Find the measure of each numbered angle. Find the measure of each numbered angle. Example : Finding the area of a regular polygon: Finding the area of each regular Finding the area of each regular polygon. Round your answer to the polygon. Round your answer to the nearest tenth. nearest tenth. Finding the area of each regular polygon. Round your answer to the nearest tenth. Notice: Regular hexagons have 30 o 60 o 90 o triangles within their 6 equilateral triangles. 4
5 10 4 Perimeters and Areas of Similar Figures: Focused Learning Target: I will be able to find the perimeters and areas of similar figures CA Std 11.0: Students determine how changes in dimensions affect the perimeter and area of common geometric figures. Finding Ratios in Similar Figures: The trapezoids below are similar. The ratio of the lengths of the corresponding sides is 6 or. 9 3 a) Find the ratio (smaller to larger) of the perimeters. b) Find the ratio (smaller to larger) of the areas. The pentagons below are similar. a) Find the ratio of the lengths of the corresponding sides. b) Find the ratio (smaller to larger) of the perimeters. c) Find the ratio (smaller to larger) of the areas. You try: The triangles below are similar. a) Find the ratio of the lengths of the corresponding sides. b) Find the ratio (smaller to larger) of the perimeters. c) Find the ratio (smaller to larger) of the areas. 5
6 If you know the area of one of two similar figures, you can find the other by using proportions. Finding areas using similar figures: The area of the smaller regular pentagon is about 7.5cm. Approximate the area of the larger regular pentagon. We ll do one: The two triangles below are similar. The area of the larger triangle is about the smaller triangle. 65m. Approximate the area of You Try: The two trapezoids below are similar. The area of the larger trapezoid is about 10in. To the nearest tenth, find the area of the smaller trapezoid. Application Problems: For some medical imaging, the scale is 3:1. That means that if an image is 3cm long, then the corresponding length on the person s body is 1cm. Find the actual area of a lesion if its image has area of.7cm We ll do one: Two similar rectangles have areas 7in and 48in. The length of one side of the larger rectangle is 16 in, what are the dimensions of both rectangles? You Try: 6
7 Two rectangles are similar. The smaller has an area of 60in and the shorter side is 6in long. The larger has a shorter side that is 8 in long. Find the area of the larger rectangle. Challenge problem: Draw a square with an area 8in. Draw a nd square with an area that is four times as large. What is the ratio of their perimeters? 10 6 Circles and Arcs Focused Learning Target: I will be able to Find the measures of central angles and arcs Find circumference and arc length CA STD 7.0 Students prove and use theorems involving the properties of circles. CA STD 8.0 Students know, derive, and solve problems involving circumference. Vocabulary: Circle Center Radius Congruent circles Diameter Central Angle Semicircle Minor Arc Major Arc Circumference Pi ( ) Concentric Circles 7
8 Example Identifying Arcs You try: From the picture to the left, identify a minor arc: From the picture to the left, identify a minor arc: From the picture to the left, identify a minor arc: A semicircle: A major arc: A semicircle: A major arc: A semicircle: A major arc: You try: From the picture to the left, find the measure of each arc: From the picture to the left, find the measure of each arc: From the picture to the left, find the measure of each arc: a. a. a. b. b. b. 8
9 Find the circumference of the circle. Leave your answer in terms of The diameter of a bicycle wheel is in. To the nearest whole number, how many revolutions does the wheel make when the bicycle travels 100 ft? The radius of a tire is 1 in. To the nearest whole number, how many revolutions does the tire make when the car travels 580 ft? As we ve seen so far, the measure of an arc is in degrees out of 360 o. Arc length is different because it refers to the distance around a circle; a fraction of a circle s circumference. 9
10 Find the length of the minor arc XY: Find the length of the major arc XPY: Find the length of a semicircle with radius 1.3 m. Leave your answer in terms of. 10
11 10 7 Areas of Circles and Sectors Focused Learning Target: I will be able to Find the areas of circles and sectors Find segments of circles Vocabulary: Sector of a circle Segment of a circle CA STD 8.0 Students know, derive, and solve problems involving circumference and area of common geometric figures Sector of a circle is a region bounded by an arc of the circle and the two radii to the arc s endpoints. Segment of a circle is a part of a circle bounded by an arc and the segment joining its endpoints How much more pizza is in a 1 in pizza compared to a 10 in. pizza? We ll do one: Two sprinklers spray water in a circular path. One sprinkler sprays in a circle with 6 ft. diameter, the larger sprays with an 8 ft diameter. How much more ground is covered by the larger sprinkler? You try: How much more pizza does a 14 in. pizza have compared to a 1 in. pizza 11
12 Find the area of sector ZOM. Leave your answer in terms of. We ll do one: Find the area of the shaded sector of a circle. Leave your answer in terms of. You Try: Find the area of the shaded sector of a circle. Leave your answer in terms of. A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle. To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed. 1
13 Find the area of the shaded segment. Leave your answer in terms of. Finding the area of shaded segment. Round your answer to the nearest tenth. You Try: Finding the area of shaded segment. Round your answer to the nearest tenth. Find the shaded region. Leave your answer in terms of. 13
14 We ll do one: Find the shaded region. Leave your answer in terms of. Challenge A: Find the area of the shaded region. Leave your answer in terms of. Challenge B: Find the total area of the shaded segments. Round your answer to the nearest tenth. 14
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