3.1 Perimeter, Circumference, and Area of Rectangles and Triangles

Transcription

1 3.1 Perimeter, Circumference, and Area of Rectangles and Triangles The Perimeter of a Polygon is the of the lengths of its sides. Perimeters of Common Quadrilaterals Theorem 3.1 Description of Figure Formula Drawing Square with sides of length s P = Rectangle with side lengths of a and b P = Parallelogram with side lengths of a and b P = Rhombus with sides of length s P = Kite with side lengths of a and b P = Find the following perimeters: Square with sides of length 2.2 cm Kite with side lengths of 3.5 cm and 5.7 cm History of π: For any circle the ratio of the circumference (distance around a circle or perimeter) to the diameter,, is constant and is represented by the Greek letter π. Today π (read pi ) has 10 1 been refined from Archimedes approximation of between 3 and 3 to over a trillion 71 7 decimal places and is identified as an IRRATIONAL number. Postulate 3.1 ( Circumference of a Circle ) The circumference of a circle is the product of π and the diameter of the circle. C = dπ = 2πr 1

2 Find the perimeter of the following figures cm Read and discuss Postulate 3.2 on page 109 and Postulate 3.3 given below. - Any two triangles whose corresponding angles and sides are congruent have the same area. Area of Rectangles and Triangles Definition or Theorem Formula Drawing(s) Dfn: The Area of Rectangle is the of its length and width. A = lw Thm 3.2: The Area of a Right Triangle is half the product of the lengths of its. Thm. 3.3: The Area of a Triangle is half the product of the length of one side and the or to that side. A = ½ab A = ½bh Find the area of each polygon. 10cm 6.2 cm cm 13 cm Converting Units of Area Using Dimensional Analysis How many square centimeters (cm 2 ) are in a square meter (m 2 )? Remember the exponent on the unit of measure tells you how many times to use the conversion factor. 1 m 1 m 2

3 Thm 3.7 Thm 3.6 Thm 3.5 Thm 3.4 Mth 97 Fall 2013 Chapter 3 How many square inches are in a rectangle with An Olympic size swimming pool of uniform depth a length of 3 feet and a width of 2 feet? measures 50 m by 21 m. The bottom of the pool is to be treated with a sealer and 1 gallon of sealer 2 ft covers 300 square feet of surface. How many gallons of sealer will be required for the job? 3 ft 6 mi 2 = km 2 Do problem 1 of ICA More Area Formulas Description Area Formula Drawing Area of a Parallelogram is equal to the product of the length of one side and the to that side. A = bh Area of a Trapezoid is equal to half the product of the height and the of the length of its bases. A = ½ h(b 1 + b 2 ) Area of a Regular Polygon is equal to half the product of the perimeter of the polygon and the perpendicular distance from its center to one of its sides, called the. A = ½ Ph where h is the length of the apothum Area of a Circle is equal to the product of π and the square of its. A = πr 2 3

4 12.5 cm Mth 97 Fall 2013 Chapter 3 Find the area of the following figures. a) b) 11 cm 16 cm 10 cm 15 cm 20 cm 20 cm c) d) Do # 16 on page 129 Do the rest of ICA cm 3.3 The Pythagorean Theorem and Right Triangles Thm Pythagorean Theorem The sum of the of the lengths of the legs (a and b) of a right triangle is equal to the square of the length of the hypotenuse (c). a 2 + b 2 = c 2 Find the missing lengths in each right triangle. Give the exact answer and the answer rounded to hundredths, if the exact answer is irrational.. a) b) c) x cm c 8 in y x 10 cm 12 in x + 1 4

5 d) Find the area of a square that has a diagonal 10 inches long. Use the Pythagorean Theorem to test the lengths below to determine if they are the sides of a right triangle. a) 11, 19, 15 b) 7, 16.8, 18.2 Special Right Triangles Theorem triangles In a right triangle, the length of the leg opposite the x 60 o x 3 2 x 30 o angle is half the length of the hypotenuse. The length of the leg opposite the 60 angle is 3 2 times the length of the hypotenuse. Show that the dimensions for the triangle above are the dimensions of a right triangle. Theorem triangles x 45 o x x 2 45 o In a right triangle the length of the hypotenuse is 2 times the length of a leg. Show that the dimensions for the triangle above are the dimensions of a right triangle. 5

6 Find the missing side lengths. a) b) c) y 9 cm 10 ft y y 20 cm 30 o 45 o x x x x = x = x = y = y = y = 3.4 Surface Area The surface area of a right prism is the sum of the area of its lateral faces and bases. Find the surface area of a rectangular prism whose length is 8 cm, width is 3 cm and height is 5 cm. 8cm 5 cm 3 cm A formula you ve probably used before is: SA = 2lw + 2wh + 2lh Let s look at the problem in the previous example by drawing a net (unfold the box) for the prism. Examine your sketch, looking for an easier way to find the prism s surface area. SA = 2(area of the base) + ( )(height) SA = This method works for all prisms. The surface area of a right pyramid is the sum of the area of its base and lateral faces. Find the surface area of a right regular triangular pyramid if the sides of the base are 4 cm and slant height is 5 cm. SA = Base area + lateral area 1 + lateral area 2 + lateral area 3 5 cm 4 cm Draw a net for this pyramid. Is there an easier way to find surface area involving the perimeter of the base? SA = Base area + 1/2 (perimeter of base)(slant height) = 6

7 The surface area of a right circular cylinder is the same as that of a prism except the perimeter becomes the of the circular base. SA = 2(area of base) + C (height) 12 cm Find the surface area of the right cylinder to the left using this formula. 7 cm l h The surface area of a right circular cone is the same as a pyramid except the perimeter becomes the of the circular base. A = area of the base + ½ C (slant height) Find the surface area of a right circular cone whose slant height is 8 cm and whose base has a radius of 6 cm using the above formula. The formula for the surface area of a sphere is 4π times the square of its radius. SA = 4 π r 2 or 4(area of a circle) Find the surface area of a sphere having a diameter of 6.7 m. 6.7 m 3.5 Volume Read and discuss Postulate 3.4, the Volume Postulate, on page 159. Remember: units are used in measuring perimeter and circumference. units are used in measuring area and surface area. units are used in measuring volume. The Volume of a Right Prism is the product of the area of its base and its height of the prism. V = A h 7

8 Name each right (lateral faces must be ) prism and then find the volume of each prism. 8 cm 5 in 4 cm 5 in 15 cm 6 cm 12cm 5 in 6 cm V = A h or V = h V = A h = h V = A h = h The Volume of a Right Cylinder is the product of the area of its base and its height of the cylinder. 6 cm V = A h = 10 cm The Volume of a Right Pyramid is the product of the area of its base and its height. Right Regular Pentagonal Pyramid 10 cm V = ⅓Ah = 5 cm 4 cm The Volume of a Right Cone is one- the product of the area of its base and its height. Right Circular Cone V = ⅓Ah = 6 in r = 3 in The Volume of a Sphere is 4/3 π times the cube of its radius. Find the volume of the sphere below. r = 2 cm 8