MATH 6

The general formula to get the volume of a pyramid is Bh , where B stands for the 3area of the base.Example 1:Look at the figure of the rectangular pyramid on the right.  What is the height of the pyramid? (9 cm)  What is the shape of the base? (rectangle)  How do you know it is a rectangle? (because of its dimensions)  How do you get the area of a rectangle? (length x width or l x w)Now, let us solve for its volume.The formula for finding the volume of a pyramid is:V = Bh , where B = area of the base 3Since the base of the pyramid is a rectangle and the formula for the area of rectangleis length times width, then we can replace B with l x w. So, V = Bh = l x w x h 33So, let us find the volume of the pyramid above.Given: l = 12 cm V = Bh w = 8 cm 3 h = 9 cm =lxwxh 3 = (12 cm x 8 cm) x 9 cm 3 cm = 96 cm2 x 9 cm 3 = 864 cm3 3 V = 288 cm3 2

We can also use cancellation for easier computation.V = Bh = l x w x h = 4 = 32 cm2 x 9 cm = 288 cm3 33 12 cm x 8 cm x 9 cm 31 Did we get the same answer? Which of the two methods do you prefer?Now, do this problem: What is the volume of a pyramid when its rectangular base is 23 m long, 18 mwide and the height is 35 m?Given: l = 23 m  Ask: Volume of the pyramid w = 18 m h = 35 m  Plan: V = Bh or l x w x h 33  Solve: V = 23 m x 18 m x 35 m 3 = 414 m x 35 m 3 = 14 490 m 3 = 4 830 m3Look back:  Did you get the correct data?  Did you follow the steps?  Did you write the correct label?TRY THESEA. Find the volume of the following. 6 m 3. 1. 2. 15 cm 12 cm 6 cm 18 m 9 dm 4 dm 10 cmB. Complete the table. 3

l w h Area of Volume the Base1. 10 cm 7 cm 15 cm2. 6 cm 12 cm 96 cm33. 13 cm 18 cm 260 cm2 1 560 cm3C. Solve the problems. 1. A 20-cm high pyramid has a rectangular base with dimensions 15 cm by 9 cm and a height of 20 cm. What is its volume? 2. A pyramid tent has a capacity of 160 m3 of air. If the base is 10 m by 8 m, find its height.WRAP UPTo get the volume of the pyramid, you multiply the area of the base by itsheight, then divide the product by three. In symbol, V = Bh , where B 3stands for the area of the base.We can also use, V = l x w x h for the volume of a rectangular pyramid. 3 4

On Your OwnA. Find the volume. Write the solutions in your notebook.1. 2. 3. 12 cm 10 cm 17 cm 16 cm 6 cm 15 cm8 cm 16 cm 10 cmB. Solve the problems. 1. A group of Grade VI made rectangular pyramid box for their Math project having a base of 7 cm by 5 cm and a height of 12 cm. What is its volume? 2. A tent is shaped like a pyramid with a base of 5 m by 3 m and a height of 2.5 m. What is the volume of air that it enclosed? 5

GRADE VI VOLUME OF A CYLINDERObjective: Find the volume of a cylinder.REVIEWRecall how to find the area of a circle. Then find the area of the following figures.Write your answers in your notebook.1) 2) 3) r = 6 cmr = 10 cm d =8 cmA = ___ A = ___ A = ___4) 5) r = 8 cm d = 18 cmA = ___ A = ___ 1

STUDY AND LEARNExample 1:Read the problem. A water tank is in the shape of a cylinder with a base radius of 3 meters and aheight of 9 meters. What is its volume?As we have learned before, the volume of any prism is equal to the area of its basemultiplied by the height, that is V=BxhLet us illustrate the problem. Radius (r) = 3 STEP 1 – Find the area of the base. rm  What is the shape of the base of the Height (h) = 9 m cylinder? The base is a circle.  How do you find the area of a circle? A = r2 Solution: A = r2 Use  = 3.14 = 3.14 x 32 = 3.14 x 9 A = 28.26 m2STEP 2 – Multiply the area in Step 1 by the height of the cylinder. A = 28.26 m2 h=9m V = 28.26 m2 (9 m) V = 254.34 m3So, the volume of the cylinder is 254.34 m3.The two steps we used to find the volume of a cylinder can be simplified by replacingB (area of the base) with r2 in the general formula since the area of any circle can bewritten as r2.In symbols, V = b x h = r2 x h V = r2h (volume formula for cylinder)This formula for the volume of a circular cylinder is V =  r2h 2

What is a right circular cylinder? A right circular cylinder is a space figure with a circular base and whose sides are perpendicular to the base.height r circular basesradius h rExample 2: A cylindrical drinking glass has a diameter of 7 cm and a height of 11 cm. How much water can it hold?Given:  = 3.14 h = 11 cm d = 7 cm volume = ?We are given the diameter instead of the radius. How can we get the radius when thediameter is given?Change diameter to radius  radius is 1 of the diameter, so, r = 7  2 = 3.5. 2Can we now solve the problem?Solution: V = r2h = (3.14 x 3.52) x 11 = (3.14 x 12.25) x 11 = 38.465 x 11 = 423.115 cm3So, the glass can hold 423.115 cm3 of water. 3

TRY THESEA. Find the volume of each figure below. Write your answers in your notebook.1. 2. 3. r = 7 cm 24 cm10 cm 9 cm 20 cm 12 cmB. Complete the table. Copy and answer this in your notebook. Find the volume. Use  = 3.14. Cylindrical- Radius Height Volume shaped Objects1. mug 4 cm 12 cm 7 cm 30 cm2. water jug 12 dm 18 dm3. water tankC. Read, analyze and solve. 1. A cylindrical tin can has a 10 cm diameter and a height of 11 cm. Find its volume. 2. Find the volume of a cylinder with a radius of 7 inches and a height of 10 inches. 3. A circular water tank has a radius of 2 m and a height of 5 m. How much water can it hold?WRAP UP To find the volume of a cylinder, use the formula, V = r2h. If the diameter is given, divide it by 2 to get the radius. A right circular cylinder is a space figure with a circular base and whose sides are perpendicular to the base. 4

ON YOUR OWNA. Find the volume of the following cylinders. Write your answers on a piece of paper.1. 2. r = 8 cm 3. r = 5 cm 12 cm r = 3 cm 15 cm 10 cm4. 5. r = 4 cm r = 2.5 cm 6.21 5 cm cm 5

GRADE VI READING AND INTERPRETING A CIRCLE GRAPHObjective: Read and interpret data presented in a circle graph. REVIEWStudy the graph. Then, answer the following questions in your notebook. How Mary Spent Her Day Sleep School 8 hrs. 6 hrs.Eat - 1 hr.Play Help at home3 hrs. Study 2 hrs. 4 hrs.1. For what activity does Mary spend the shortest time?2. For what activity does Mary spend most of her time?3. How many hours does Mary spend studying?4. How many more hours does she spend in playing than helping at home? What kind of a child is she?5. What fraction of the day represents her time in school? 1

STUDY AND LEARNA circle graph also sometimes called a pie chart, is another way of presenting data. Itmakes use of a circle divided into different parts. It shows the relation of the parts to thewhole and to each other.Example - The circle graph below represents the four colors preferred by 40 pupils.Sectors of Yellow Red The entire graph = 100%the circle 10% 50% graph Green 15% Blue 25% Favorite Color of 40 Pupils What are the 4 colors preferred by the pupils? (red, blue, green, yellow) What percent of the students prefer each color? (red – 50%; blue – 25%; green – 15%; yellow – 10%) If you add all the percentages, what is the total? (100%) How many pupils are there in all? (40) Note that the entire graph represents 100%. This represents the total number of pupils which is 40. How do we determine the number of pupils referred to when we say 50% of the total number of pupils? (We can get 50% of the total number of pupils by multiplying 40 by 50%.) Example: 50% of 40 = 50% x 40 = 0.50 x 40 = 20.00 = 20 Therefore, 50% of 40 pupils is 20 pupils. What about 25% of 40? (25% x 40  25% x 40  10.00  10)Now it’s time to find 15% and 10% of 40. (15% of 40 is 6; 10% of 40 is 4) 2

Correct! 15% of 40 is 6 and 10% of 40 is 4.Try adding the number of pupils who prefer each color. What is the sum?50% Red – 2025% Blue – 1015% Green – 610% Yellow – 4100% ? (40) this represents 100% of the entire graph, correctWhat if the given data on the graph is not in percent but in fractions? Can we still getthe quantity in each part? (Yes!)Let us have another example to illustrate this.Example 2: Twenty-four Grade VI pupils were asked what subject they prefered. The circlegraph below tells us about the fractional part of the pupils who preferred Science,Math, Filipino and English. Number of Students Who Like Each Subject Area Science Filipino 1 1 Math 8 2 1 8 English 1 4 How many pupils are there in all? (24)Can we determine the number of pupils who prefer each subject?To get the number of pupils who prefer each subject area when the data given is infraction, we need to multiply the fraction by the total number of pupils. 3

Example:1 of 24 = 1 x 24 = 1 x 24 = 24 = 24  2 = 12 pupils2 2 21 2What about 1 of 24? 4Correct! 1 of 24 is 3. 8If you add the fractions, 1 , 1 , 1 and 1 , will it be equal to 1? (Yes, it will be equal 248 8to 1 because the whole circle represents 100% or 1 whole part.) If you total the number of pupils who prefer each subject will the sum be equal to24? (Yes!) 1 or ( 4 ) Filipino – 12 28 1 or ( 2 ) English – 6 48 1 Math – 3 8 Science – + 3 1+ 8 8 =1 24 8 4

TRY THESEA. Study the graph. Then, answer the questions about the graph. Write your answers in your notebook. Sleep Office 1 1 3 3 Others Recreation 1 Meals 1 12 8 1 8 Ellen’s Daily Activities1. On which activity does Ellen spend the shortest time?2. What are the activities with the same time spent?3. How many hours does Ellen spend for recreation?4. What is the total number of hours spent working in the office and sleeping?5. How many hours does she spend for other activities? 5

B. Use the circle graph to answer the questions that follow. Write your answer in your notebook. Shelter Food 20% 30%Savings 5% Education 25%Miscellaneous 10% Health 10%Monthly Family Budget of Php20,000.001. What amount is spent for shelter?2. Which item is given the least budget?3. How much is spent for food?4. Which item in the graph gets the biggest share?5. How much is the budget for health? WRAP UP A circle graph or a pie graph is used to show the relation of the parts to the whole and to each other. The entire graph always represents 100%. Always multiply the given percent or fraction by the total amount or item to get the number of items or amount for each part.6

ON YOUR OWNStudy the circle graph. Then answer the questions that follow using the graph. Writeyour answer in your notebook.Potato Cabbage 50% 25% Ampalaya 5% Carrot 20% Vegetable Choices of 40 Students1. What vegetable is least liked by the students?2. Which vegetable is most liked by the students?3. How many students like carrots?4. How many students like cabbage?5. How many more students like cabbage than carrots? 7

GRADE VI CONSTRUCTION OF A CIRCLE GRAPHObjective: Construct a circle graph.REVIEWA. Round the following to the nearest whole number. Write your answers in your notebook. 1. 12.5o 2. 72.4oB. Express each as a decimal and as percent. Write your answers in your notebook. Decimal Percent1. 1 _____________ _____________42. 3 _____________ ______________ 8C. Express each as a fraction in simplest form and as a decimal. Write your answers in your notebook. Fraction Decimal1. 37.5% ____________ ____________2. 12 1 % ____________ ____________ 23. 15% ____________ ____________ 1

D. Change the following to similar fractions. Write your answers in your notebook.1. 1 , 2 , 1 2. 3 , 4 , 1 254 10 5 2E. Find the fractional part or percentage of the following. Write your answers in your notebook.1. 3 of 40o is _____ 3. 30% of 70o is _____ 82. 2 of 50o is _____ 5 STUDY AND LEARNYou will now study how to construct a circle graph.A circle graph is used to show the relation of the parts to the whole and to each other.The circle represents the entire quantity, the whole unit or 100%. A fractional part of thewhole number is represented by an equivalent fractional part of the circle, and is usuallynamed as percent.Study the circle graph below. 2

25 % 37.5 % 25% is 1 of 100, so 25% is represented School Sleep 4 25 % by 1 of the circle. Others 4 12.5 % So, how are the sectors of the graph Play determined? There are 2 ways to determine the sectors of the graph.sectors First, draw the central angle way. A central angle is an angle whose vertex is at the center of a circle. Draw a circle on a bond paper using your compass. One revolution in degree measure is 360o for this is the sum of the measures of all the central angles in a circle. Then, we compute each given percent to find the size of the sector that will represent the item. Item Percent Decimal Angle Measure ofSchool 25% 0.25 x 360o SectorSleep 37.5% 0.375 x 360o 90o 12.5% 0.125 x 360 135o Play 25% 0.25 x 360o 45oOthers 90o 100% 1 Total 360o 3

Therefore, the central angle that will be labeled School (25%) will measure 90o, Sleep(37.5%) will measure 135o and so forth. A protractor is used to draw the angles on thecircle. 25 % 37.5 % School Sleep 90o 135o 90o 45o 25 % 12.5 % Other PlayAnother way of doing this is by partitioning the circle into equal parts, so that we candetermine the number of sectors needed.First, we need to change the given percent to fraction in its simplest form.School – 25% → 0.25 → 25 = 1 100 4Sleep – 37.5% → 0.375 → 375 = 3 1000 8Play – 12.5 % → 0.125 → 125 = 1 1000 8Other – 25% → 0.25 → 25 = 1 100 4Second, change the fractions that you got to similar fractions.1, 3, 1, 1 2484884 x 224 x 1 x 2 = 8 → LCD Get the LCD Rename the fractions with the LCD as their denominator 4

1= 2 1= 1 - Play 8 - School 833481= 3 1 = 2 - Others 4 - Sleep 83388 Draw a circle in your notebook. Divide the circle that you have drawn into 8 equal parts.Now, color with green the parts that represent school, blue for sleep, red for play andyellow for others.Which of the two ways do you prefer to use?  How many eighths represent School? (2)  How many eighths represent Sleep? (3)  How many eighths represent Play? (1)  How many eighths represent Others? (2) TRY THESEA. Draw the circle graph for the data presented. Write your answer on a bond paper.Favorite Sports of 60 Pupils Basketball – 50% Volleyball – 25% Chess – 15% Badminton – 10%Direction: Draw a circle graph showing the following data. Write your answer 5

in your notebook. The Gonzales family budget is planned as follows: food – 35% rent – 20% savings – 3% clothing – 10% car – 10% education – 7% miscellaneous expenses – 15% WRAP UP  A circle graph is used to show the relation of the parts to the whole and to each other.  The interior of the circle represents an entire quantity, a whole unit or 100% while a part is called a sector.  A central angle is an angle whose vertex is at the center of a circle. The sum of all the central angles in a circle is 360o.  There are 2 ways to make the graph. One is by central angle method and the other is by partitioning the circle into equal parts according to the given in the problem. ON YOUR OWNDraw a circle graph showing the following data. Write your answer on a bond paper.You may color your graph. Sources of Air Pollution Industries – 60% Vehicles – 30% Others – 10% 6