Pythagorean Theorem. Pythagorean Theorem

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1 MPM 1D Unit 6: Measurement Lesson 1 Date: Learning goal: how to use Pythagorean Theorem to find unknown side length in a right angle triangle. Investigate: 1. What type of triangle is in the centre of the picture? Pythagorean Theorem 2. Complete the table below Side a = Area of A = Side b = Area of B = Side c = Area of C = 3. How is the area of A and the area B related to the area of C? Pythagorean Theorem Example 1 a) Label the hypotenuse a) Label the hypotenuse 10 cm 9 mm 7 mm 11 cm b) Calculate the missing side length b) Calculate the missing side length

2 Example 2: A television is described as a 20 television if the screen has a diagonal length of 20. If the screen of a 20 flat screen television has a height of 12, what is the width? Example 3: A ladder that is 8.5 m long leans against a wall. The foot of the ladder is 2.3 m from the base of the wall. How far up the wall does the ladder reach? Example 4: The ranger has to search the area of the outlined in the triangle. Determine the perimeter and area that the ranger must cover. Cabin Beach 3 km 2.4 km Ranger Station Example 5: Determine the height of the cone.

3 Assignment 6.1: Pythagorean Theorem **Round all answers to two decimal places** 1. Solve for the missing side in the following triangle: 40 cm 30 cm 2. A wall is supported by a brace 10 feet long, as shown in the diagram below. If one end of the brace is placed 6 feet from the base of the wall, how many feet up the wall does the brace reach? 3. When Arnold swims laps in his rectangular swimming pool, he swims along the diagonal so he doesn t have to turn around so often. Find the distance Arnold travels by swimming once along the diagonal. 4. The three sides of a triangle are 18 cm, 24 cm and 30 cm. Determine whether the triangle is a right triangle. Justify your answers using calculations. 5. Determine the value of x and y. 9.8 m 3.2m B 6. Towns A, B, C, and d are situated as shown on the diagram. a. How far is it from town B to town D? b. How far is it from town B to town C? 10Km A C 6.1 Answers ft m 7K D 6Km 4. It is a right angle triangle. The sum of the smallest two sides squared equals the largest side squared. 5. x=25, y=20 6. a) 7.14km b) 9.33km

4 MPM 1D Unit 6: Measurement Lesson 2 Learning goal: how to calculate the area and perimeter for composite shapes. Date: Composite Shapes Example 1: Calculate the perimeter and area of the following figure. Example 2: The figure to the right is made up of a triangle and a rectangle. Determine the area of the figure. Example 3: Calculate the area of the shaded region.

5 Example 4: Determine a simplified expression for the perimeter and area of the following figures. a) b) Example 5: The circumference of a circle is feet. What is the diameter of the circle? Example 6: The perimeter of a rectangle is 34 cm. If the length of the rectangle is 11 cm then what is the width of the rectangle? Example 7: The area of a square equals the area of a circle. If the square has a side length of 8cm determine the radius of the circle.

6 1. Use the figure on the right to answer the questions. a) What is the area of the square? b) What is the area of the triangle on the left? c) What is the area of the composite figure? Assignment 6.2: Composite Shapes **Round all answers to the nearest hundredth** 2. Find the area of the following shaded regions. a) b) 3. Determine a simplified expression for the perimeter and area of the figure to the right. 4. A figure is a triangle composed of and a square, as shown below. The area of the square is 144 m2. What is the height of the triangle? Show your work. 5. The figure below is made two semicircles. What is the area of the shaded region? Show your work. 6. A circle and a triangle of an equal area. If the circle has a radius of 4cm and the triangle has a height of 15cm, determine the base of the triangle. 7. The area of a triangle is 32 cm 2. If the height of the triangle is 7 inches then what is the base of the triangle? 8. The base of a parallelogram is 17mm. If the area of the parallelogram is 82mm 2, what is the height of the parallelogram? 9. The area of a rectangle is 42cm 2. If the length of the rectangle is 5cm, what is the perimeter of the rectangle? 10. A right angle triangle has a height of 6 inches and hypotenuse of 9 inches. What is the area of the triangle? 6.2 Answers 1. a) 25 units 2 b) 15 units 2 c) 55 units 2 2. a) 7.73 m 2 b) units 2 3. perimeter=18x, area=16 units m cm cm cm mm cm inches 2

7 MPM 1D Unit 6: Measurement Lesson 3 Learning goal: how to determine the volume of prisms & cylinders. Date: Volume of Prisms & Cylinders is the amount of space a 3-dimensional object occupies. 3-dimensional objects are called if the figure has the same shape on both the top and the bottom. RECTANGULAR PRISM It is called a rectangluar prism because the base is a Example: height length width A prism is a 3- dimensional object because Volume To find the volume of a prism, find the base area and multiply by the height V = A H Because we are multiplying 3 dimensions, volume is expressed in cubic units. For example: cm 3, or m 3. Example 1: Calculate the volume of the rectangular prism to the right. 12 cm 5cm 7cm The base of the prism is the side that is sitting on the ground. This shape is a rectangle with length 5 cm and width 7 cm.

8 Example 2: Calculate the area of the base, and then calculate the volume. Example 3: Calculate the volume of the following composite shape. Example 4: Stacey is filling her fish tank with her cylinderical cup. Her cup has a radius of 3.5cm and a height of 10cm. Her fish tank is 30cm wide, 50 cm long, and 40cm tall. How many cups will it take to fill her fish tank?

9 Assignment 6.3: Volume of Prisms & Cylinders **Round all answers to the nearest hundreth** For questions 1-4, find the volume of the shape drawn. 5. Dairy Fresh packages two cartons of milk. The large carton is 8cm by 8cm by 18cm. The small carton is 6cm by 6cm by 8cm. How many small cartons of milk would fill one large one? 6. A gold bar is 15 cm by 8cm by 3cm and is worth $ What is the volume of the bar in cubic centimetres? How much is one cubic centimetre work? 7. A cylinder just fits inside a 10cm by 10cm by 10cm box. a) Determine the volume of the box. b) Determine the volume of the cylinder. c) Determine the volume of the empty space in the box when the cylinder is placed inside. 8. Tracy plans to create an ice sculpture for a table centrepiece at her friend s housewarming. The ice she will start with is shown. Calculate the volume of ice Tracy will need to make the sculpture. 6.3 Answers m cartons ft 3 6. $ mm 3 7a) 1000 cm 3 b) cm 3 c) cm in cm 3

10 MPM 1D Unit 6: Measurement Lesson 4 Learning goal: how to calculate the surface area of prisms & cylinders. Date: Surface Area of Prisms & Cylinders Remember that when a 2D net is folded together it turns into a 3D shape. Example 1: Find the surface area of the rectangular prism è Suraface area is. Example 2: Find the surface area of the following prism. 12 cm 20 cm 10 cm

11 Example 3: Find the surface area of the following cylinder. 2 cm 8 cm Example 4: Find the surface area of the figure below. The upper cylinder is centered on the lower one.

12 Assignment 6.4: Surface Area of Prisms & Cylinders **Round all answers to the nearest hundreth** 1. Calculate the surface area of the following shapes. (b) 2. A can of soup is 10.3cm high and its diameter is 6.7cm. How much paper is needed to make the label? 3. A wedge of cheese is in the shape of a triangular prism. It is 6cm high and each triangular face has a side length of 10cm and a base of 2.9cm. (a) How much plastic wrap is needed to cover the cheese? (b) The plastic wrap costs $0.01/cm 2. What is the cost to package the cheese? 4. Find the surface area of the following composite shape. 5. A box with a square base must have a volume of 200cm 3. Find the dimensions of the box that minimizes the amount of material used. 6.4 Answers 1 a) 216 cm 3 b) mm cm 3 3 a) cm 3 b) $ ft cm x 5.85cm x 5.85cm

13 MPM 1D Unit 6: Measurement Lesson 5 Learning goal: how to determine the volume of cones, spheres, and pyramids. Date: Volume of Cones, Spheres, and Pyramids SPHERES Spheres are very unique shapes and as such, it is very difficult to figure out exactly how to find the volume of a sphere. Fortunately for us, Archimedes, the Greek mathematician already figured it out a long time ago. Volume of Spheres V = 4 3 πr! Example 1: Find the volume of the given figure to the right. RECTANGULAR PYRAMIDS The pyramid on the right has similar dimensions with respect to its length, width and height, but its volume is clearly much less than that of the rectangular prism. In fact, the volume of the pyramid is exactly the volume of the similar rectangular prism. Volume of Rectangular Pyramids V = l w h 3

14 IMPORTANT TERMS Height: (altitude) Slant height: How does pythagorus help us here? Example 6: Find the volume of the pyramid to the right. 11 m 16 m 16 m CONES The relationship between cylinders and cones is similar to the relatioship between prisms of pyramids. V = πr! h V = πr! h 3 Example 2: Find the volume of the cone below.

15 Example 3: A cone has volume cm 3 and base radius 6 cm. What is the height of the cone? Example 4: Determine the amount of empty space in the square box.

16 Assignment 6.5: Volume of Cones, Spheres, and Pyramids **Round all answers to the nearest hundreth** 1. Calculate the volume of each solid. a) b) 23 m 18 in. 15 in. 25 m 12.5 m c) d) 13 m 24 m 24 m 2. Calculate the volume of a sphere with a diameter of 78 cm. 3. Calculate the volume of the figure shown in the diagram to the right 4. A sphere with a radius of 46 cm is centered inside a sphere with a radius of 76 cm. What is the volume of the space between the two spheres? 5. A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the base of the cone and cylinder is 2 feet. What is the volume of the stone that the sculptor must remove? 6.5 Answers

17 1 a) m 3 b) in 3 c) mm 3 d) 960 m cm m cm 3 5. π ft 3 MPM 1D Unit 6: Measurement Lesson 7 Learning goal: I can apply agebra to surface area and volume. Date: Surface Area and Volume with Algebra Example 1: The Earth has a surface area of 509, 904, 262.8km 2. a) What is the earth s diameter? b) What is the surface area of each hemisphere? Example 2: The area of a square equals the area of a circle. If the square has a side length of 8cm determine the radius of the circle.

18 Example 3: a cone-shaped paper cup has a volume of 67 cm 3 and a diameter of 6cm. What is the height of the paper cup, to the nearest tenth of a centimeter? Example 4: Patrice is in charge of the excavation for the foundation of a building. If a hole must be dug that is 35 m by 25 m by 12 m, how many trips will be required to remove the dirt if a trailer can carry only 15 cubic meters of dirt? Example 5: A cylindrical juice container has a diameter of 8cm. If the surface area of the juice container is 754cm 2, what is the heighto f the container?

19 Assignment 6.7: Surface Area and Volume with Algebra **Round all answers to the nearest hundreth** 1. Determe the height of a cone if the volume of the cone is 56cm 3 and the diameter is 5cm. 2. If a can of soup can hold 64cm 3 of soup, and the height of the container is 6cm, what is the radius of the filled can of soup? 3. A circle and a triangle of an equal area. If the circle has a radius of 4cm and the triangle has a height of 15cm, determine the base of the triangle. 4. Find the height of a can with a surface area of cm 2, if the radius is 4cm. 5. If the volume of a square-based pyramid is 680 cm 3 and the side lengths of the square are 16 cm, what is the height of the pyramid? 6. A gift is put into a rectangular box measuring 6 x 6 x 15 inches. If each wrapping paper sheet covers 60 in 2, how many sheets must be purchased to completely wrap the gift? 6.7 Answers cm cm cm 4. 9 cm cm 6. 8 sheets