MATH 6

60 cm2 – front and back190 cm2 – surface areaIs this the same answer as the one obtained earlier?To solve for the surface area of a rectangular prism, we can use a formula.SA = 2(lw) + 2(wh) + 2(lh) or 2 (lw + wh + lh),where SA = surface area l = length w = width h = heightFor the example we did earlier:l = 10 cm; w = 5 cm; h = 3 cmSA = 2(lw) + 2(wh) + 2(lh) SA = 2(lw + wh + lh) = 2(10)(5) + 2(5)(3) + 2(10)(3) = 2[(10)(5) + (5)(3) + (10)(3)] = 2(50 cm2) + 2(15 cm2) + 2(30 cm2) = 100 cm2 + 30 cm2 + 60 cm2 = 2(50 + 15 + 30) = 190 cm2 = 2(95 cm2) = 190 cm2Which method is easier for you?Do you want to try using any of the formulas in finding the surface area of a rectangularprism? Let’s answer the exercises using any of the methods learned. 4

TRY THESE A. Find the surface area of the following rectangular prisms. Write your answers in your notebook. 1) E 4 cm A B C D 4 cm 1 cm F 1 cm 3 cm 3 cm 2) 2.1 m 5.3 m 8.4 m 3) 5 dm 8 dm 7 dm B. Complete the following charts. Copy and answer in your notebook. Rectangular Prism Area of Each Face (m2) Total Left Right SurfaceLength Width Height Front Back Top Bottom Area1) 9 m 4m 7m2) 3.4 m 1.6 m 7.2 m3) 10 m 4m 8m4) 7 m 8m 12 m5) 5 m 11 m 9m 5

C. Find the surface area of each box. Write the letter of the correct answer corresponding to the surface area in the box.1) 2) 3) l=9m l = 11cm l = 12m w=6m w = 8 cm w = 7m h=8m h = 10 cm h = 4m4) 5) l = 5m w = 3m l = 14cm h = 7m w = 12cm h = 13cm SURFACE BOARD (m2) S ATGER 189142 1,012 348 320 724WRAP UP The surface area of a solid figure is the sum of the areas of all itsfaces. The surface area of a rectangular prism is the sum of the areas of its6 faces. We have two ways of getting the surface area of a rectangular prism. 1. One is by unfolding the figure and finding the area of each face then adding all of these. 2. Another is by using the formula SA = 2(lw) + 2(wh) + 2(lh) or 2(lw + wh + lh). 6

On Your OwnA. Find the surface area of each of the following figures. Write your answers in your notebook. 1. 2. 3. 23 cm 12 dm 20 cm 4m 6m 8m 8 dm 25 cm5 dmB. Read, analyze and solve. Write your answer in your notebook. 1. A wooden cabinet needs repainting. How much surface area needs to be painted if the box has a length of 75 cm, width of 45 cm, and height of 35 cm? 2. What is the surface area of a rectangular solid that is 9.2 cm by 7.5 cm by 2.7 cm? 7

GRADE VI SURFACE AREA OF A TRIANGULAR PRISMObjective: Find the surface area of a triangular prism.REVIEWFind the area of the following plane figures. Write your answers in your notebook.1) 2) 3) 7 cm 4 cm 8 cm 6 cm 10 cm 7 cm Area = ____ Area = ____ Area = ____4) 5) 12 cm 8 cm 2 cm 8 cmArea = ____ Area = ____ STUDY AND LEARNThe surface area of a solid figure is the sum of the areas of all its faces. 1

The surface area of a triangular prism is the sum of the areas of all the faces.height  How many faces does the figure have? 5  What are the shapes of the faces? triangles and rectangles baseTo visualize clearly, let us unfold the triangular prism. Front  The faces may vary depending on how the prism stands. The important thing you must B remember is the shape that each face is made of.SoSiti td o de m e BackHow many faces does the prism have? 5Yes, 5 faces and these faces are triangles and rectangles.How many triangular faces does it have?How many rectangular faces does it have?Are the triangles of the same size? YesWhat kind of triangles are these? equilateral triangleAre the rectangles the same?Since we are after the surface area, we need to recall the formula for finding the areaof a rectangle and a triangle.Do you still remember the formulas?Area of a Rectangle = length x width or A = l x wIf the length of a rectangle is 5 cm and its width is 4 cm, what is the area?20 cm2 because 5 cm x 4 cm = 20 cm2For a triangle, the area is one-half of the product of base and height.In symbol, A = 1 bh or bh . 22If the base of a triangle is 10 cm and its height is 8 cm, what is the area?40 cm2 because 10 cm x 8 cm = 80 cm2  2 = 40 cm2 2

Now, let us try to find the surface area of a triangular prism. D 6 cm4 cm 7 cm A B C 7 cm 6 cm 7 cm 6 cm E 6 cm 6 cm 4 cm What is the area of one rectangle?A =lxw = 7 cm x 6 cm = 42 cm2 How many rectangles does the figure have? 3 Are they all equal? YesSo we will multiply the area of 1 rectangle by 3.42 cm2 x 3 = 126 cm2 What is the area of one triangle?A = bh  6 cm x 4 cm = 24 cm2 = 12 cm2 22 2 How many triangles does the figure have? 2 Are they equal? Yes, because it is an equilateral triangle So, we will multiply the area of one triangle by 2.12 cm2 x 2 = 24 cm2We can also solve the given figure by computing the area of each face thenadding all the areas of the faces.Area of Face A Area of Face B Area of Face CA=lxw A=lxw A=lxw = 7 cm x 6 cm = 7 cm x 6 cm = 7 cm x 6 cm = 42 cm2 = 42 cm2 = 42 cm2 Area of Face D Area of Face E A = bh A = bh 2 2 = 6 cm x 4 cm = 6 cm x 4 cm 3

2 2 = 24 cm2 = 24 cm2 2 2 = 12 cm2 = 12 cm2 Area of Face A - 42 cm2 Area of Face B - 42 cm2 Area of Face C - 42 cm2 Area of Face D - 12 cm2 Area of Face E - + 12 cm2 Total SA - 150 cm2Let us have another one. 12 cm Area of Triangle = bh 5 cm 2 6 cm A = 6 cm x 5 cm  30 cm2  15 cm2 22 2 x 15 cm2 = 30 cm2 Area of 2 triangles Area of Rectangle = l x w A = 6 cm x 12 cm  72 cm2 3 x 72 = 216 cm2 Area of 3 rectangles  So, 30 cm2 + 216 cm2 = 246 cm2 the area of the triangular prism.  It is easy, isn’t it?  But what if the given triangle is not equilateral? What if the given one is an isosceles triangle or a scalene?  Can you still use the formula? No  If the given triangle is not an equilateral triangle, then it’s best to use the flattened method of finding the surface area. It is also important that you can visualize the figure correctly.Let’s have one example. Isosceles Triangle 4 cm 5 cm D8 cm AB C5 cm E 4

 Which faces are the same in size and shape (congruent)? Face A and C and Face D and E  Which face is different from all other faces? Face B 5 cm 5 cm Area of Face A  A = 8 cm x 5 cm Face A Face D 5 cm = 40 cm2 4 cm Face C Area of Face B  A = 8 cm x 4 cm = 32 cm28 cm Face B 8 cm Area of Face C  A = 8 cm x 5 cm 8 cm 8 cm = 40 cm2 5 cm 4 cm 5 cm Area of Face D  A = bh  4 cm x 5 cm 22 Face E = 20 cm2  10 cm2 5 cm 2 5 cm Area of Face E  A = bh  4 cm x 5 cm 22 = 20 cm2  10 cm2 2Face A - 40 cm2  Did you find a shorter way of solving the B - 32 cm2 surface of a triangular prism with an C - 40 cm2 isosceles triangle? D - 10 cm2 E - 10 cm2 Yes, just multiply Face A by two and Face D will be the sum of Face D and ETotal SA = 132 cm2 if you will not divide it by 2.Let’s have a scalene triangle. 5 cm 5 cm 3 cm 5 cm 6 cm D 3 cm 4 cm 6 cm 4 cm 4 cm7 cm A 7 cm B 7 cm C 7 cm 7 cm 5 cm 6 cm 4 cm E 4 cm 3 cmSolution: 5

Area of Face A  A = 7 cm x 5 cm = 35 cm2Area of Face B  A = 7 cm x 6 cm = 42 cm2Area of Face C  A = 7 cm x 4 cm = 28 cm2Area of Face D  A = 6 cm x 3 cm = 18 cm2 = 9 cm2 22Area of Face E  A = 6 cm x 3 cm = 18 cm2 = 9 cm2 22Surface Area = 35 cm2 + 42 cm2 + 28 cm2 + 9cm2 + 9cm2 = 123 cm2So, the surface area of the scalene triangle is 123 cm2.TRY THESEA. Find the surface area of the flattened triangular prism. Copy and answer this in your notebook.1) 8 cm D h = 7 cm A BC 8 cm 10 cm EArea of Face A = _____Area of Face B = _____Area of Face C = _____Area of Face D = _____Area of Face E = _____Total Surface Area = _____ 6

2) I h = 8 cm F GH 10 cm 6 cm 15 cm JArea of Face F = _____Area of Face G = _____Area of Face H = _____Area of Face I = _____Area of Face J = _____Total Surface Area = _____3) 7 cm K 5 cm 9 cm h = 4 cm8 cm OL NArea of Face K = _____ MArea of Face L = _____Area of Face M = _____Area of Face N= _____Area of Face O = _____Total Surface Area = _____ 7

B. What is the surface area of the following figure? Write your answer in your notebook.1) 3) 13 dm h = 4 cm 8 dm 5 cm h = 4 dm 5 cm 10 cm 10 dm SA = _____2) 12 dm 10 dm 10 dm SA = _____ 15 dm 10 dm h = 9 dm 4 dm SA = _____ WRAP UP  The surface area of a solid figure is the sum of the areas of all its faces.  The surface area of a triangular prism is the sum of the areas of all the faces.  A triangular prism has 5 faces composed of two triangles and 3 rectangles.  The best way to solve for the surface area of a triangular prism is to unfold or flatten the figure so that you can visualize clearly the kind of faces it has and their measurement.  The unit used for surface area is square unit. 8

ON YOUR OWNA. Below are triangular prisms that have a letter on it. To complete the message below, you need to find the surface area of each figure. Then write the code letter corresponding to your answer on the correct lines at the bottom of the page. The first one has been done for you. Copy and answer the message in your notebook.1) 10 cm h = 4 2) 8 dm cm C 20 dm 7 cm 8 dm 5 cm N 12 cm 10 dm h = 6 dm3) 4) h=6 4m 5 cm cm 5 cm 4m X 8m T h=5m 10 cm 4m 5 cm 5) E 6) h = 5 cmh=5m 9m 6m 6 cm 12 m 7m 9 cm L 5 cm 5 cm N!306 112 580 306 174 174 306 304 180 9

B. Find the surface area of the following. Write your answers in your notebook. 1)8 cm h = 5 cm SA = _____ 6 cm 6 cm2) 6 cm 5 cm h = 5 cm SA = _____ 10 cm 6 cm3) 8 dm6 dm h = 7 dm SA = _____ 12 dm SA = _____ SA = _____ 7 dm4) h = 7 cm 8 cm 10 cm 9 cm5) 9m h = 8 cm 9m 12 m 10

GRADE VISURFACE AREA OF A REGULAR PYRAMID WITH A SQUARE BASE Objective: Find the surface area of a regular pyramid with a square base.REVIEWFind the area of the following plane figures. Write your answer in your notebook.1) 2) 3) 8 cm 8 cm 7 cm 7 cmA = _____ A = _____ A = _____4) 5) 6 cm 10 cm 6) 10 cm 12 cm 10 dm STUDY AND LEARNIn the previous modules, you have learned how to find the surface area of a cube,rectangular prism and triangular prism.Now, we will explore pyramids. Did you get a perfect score in Review? Very good! Itmeans that you are ready for this module. 1

Do you know what surface area means? We will now explore the surface area of aregular pyramid. Let’s take a look at a regular square pyramid.face A regular square pyramid is a pyramid with height of the triangle five faces such that the four faces are congruent triangles and the fifth face, which is the base, is a base of the square. pyramidbase of the triangleDo you see the 5 faces of the pyramid in the figure?Look at the figure on the right. 8 cmThis is how the pyramid looks like whenunfolded or flattened. 4 cmWhich face is the base of the pyramid? 4 cm O8Face U 8 4 8Which faces are congruent (the same in L 4U4 Vsize and shape)?Face L, Face O, Face V and Face E 4 E8Since the surface area of a solid figure is the sum of the areas of all its faces, to find thesurface area of a regular square pyramid: Multiply the area of the congruent triangular face by 4 Add the area of the square base to the product. In symbols, SA = 4( 1 bh) + s2 2 In words, Surface Area = 4(Area of the triangle) + (Area of the square) 2

For example: Surface = 4(Area of the triangle) + (Area of the square) 5 cm = 4(5 cm x 8 cm) + (5 cm x 5 cm) 2 = 4(40 cm2) + 25 cm2 h = 8 cm 2 = 4(20 cm2) + 25 cm2 = 80 cm2 + 25 cm2 = 105 cm2Using the formula, = 4( 1 bh) + s2 SA 2 = 4[ 1 (5 x 8)] + 52 2 = 4[ 1 (40)] + 25 2 = 4 (20) + 25 = 80 + 25 = 105 cm2Let us have another example. SA = 4[ 1 (bh)] + s2 height=10 cm 2 6 cm = 4[ 1 (6 x 10)] + 62 26 cm = 4[ 1 (60)] + 36 cm2 2 = 4(30 cm2) + 36 cm2 = 120 cm2 + 36 cm2 = 156 cm2 3

TRY THESEFind the surface area of the regular square pyramids below. Solve them in your notebookand encircle your final answer.1) h = 12 m 2) 3) h = 18 cm h = 13 dm 5m 15 cm 9 dm4) 5) h=9m h = 17 cm 12 cm 4m WRAP UP  A regular square pyramid is a pyramid with five faces such that the four faces are congruent triangles and the fifth face, which is the base, is a square.  To find the surface area of a regular square pyramid: - multiply the area of one of the congruent triangular faces by 4 - add the area of the square base to the product  You can also use the formula: SA = 4( 1 bh) + s2 2 Surface Area = 4(Area of the triangle) + (Area of the square) 4

ON YOUR OWNA. Solve for the surface area of each of the following regular square pyramids. Write your solutions in your notebook and encircle your final answers. Don’t forget to label with the correct unit.1) 2) 3) h=8m h = 12 cm h = 12 cm7 cm 5m 3mB. Find the surface area of each of the following regular square pyramids. Write your answer in your notebook.1) h = 10 m 2) h = 15 cm3) h = 8 cm 8m 4 cm 6 cmC. Solve. Write your solution in your notebook and encircle your final answer. 1) Find the surface area of a regular square pyramid with a base of length 12 dm and a height of 9 dm. 2) What is the surface area of a regular square pyramid with a side of 8 dm and an altitude of 12 dm? 5

GRADE VI SURFACE AREA OF A REGULAR PYRAMID WITH A RECTANGULAR BASEObjective: Find the surface area of a regular pyramid with a rectangular base.REVIEWFind the area of each of the following plane figures. Write your answer in yournotebook.1) 2) 3) 7 m 8 cm 8 dm15 m 8 cm 10 dm 15 cm 5)4) 9 cm 12 cm h = 4 cm STUDY AND LEARNDo you recall how the surface area of a regular square pyramid may be obtained?In this lesson, we will take a look at the surface area of other types of pyramids.Today, we will take a look at regular pyramids with rectangular bases. 1

A pyramid is a solid figure with a polygonal base and triangular faces that meet ata common point. One kind of a pyramid is a rectangular pyramid, which has a rectangularbase and four triangular faces.faces height of the triangle face face base face when flattened out base of the triangle face base of the pyramidCan you see the 4 triangular faces and the rectangular base of the pyramid in the figurewhen it is unfolded? A Let us put some measurements so that we can identify the congruent faces. 6 Height is 6 cm. 6 4 6R The sides of the rectangular base are 5 cm and 4 cm. C 5S5 Which 2 faces are congruent? Face C and Face R, 4 Face A and E 6 How many pairs of congruent faces do you find? 2 E pairsHow do you find the area of a triangle? A = 1 bh 2What faces are triangular in shape in our figure? Face C, Face A, Face R and Face EHow do you find the area of a rectangle? A = lwWhat face of the pyramid has the shape of a rectangle? The base 2

To get the surface area of the figure, get the area of each face.Area of Face C Area of Face A Area of Face RA = 1 bh A = 1 bh A = 1 bh 2 2 2 = 1 (6 cm x 4 cm) = 1 (5 cm x 6 cm) = 1 (6 cm x 4 cm) 2 2 2 = 1 (24 cm2) = 1 (30 cm2) = 1 (24 cm2) 2 2 2 = 12 cm2 = 15 cm2 = 12 cm2Area of Face E Area of Face S A=lxwA = 1 bh 2 = 5 cm x 4 cm = 20 cm2 = 1 (5 cm x 6 cm) 2 = 1 (30 cm2) 2 = 15 cm2Then add the areas of the faces.Surface Area (SA) = Face C + Face A + Face R + Face E + Face SSA = 12 cm2 + 15 cm2 + 12 cm2 + 15 cm2 + 20 cm2SA = 74 cm2So the surface area of the regular rectangular pyramid is 74 cm2.Can you now present a shorter way for getting the surface area of a regular pyramid witha rectangular base?1st – Get the area of 1 triangle whose base is the length of the rectangle then multiply it by 2. A = 1 bh 2 = 1 (5 cm x 6 cm) 2 = 1 (30 cm2) 2 = 15 cm2 3

So, 15 cm2 x 2 = 30 cm2 – the sum of Face A and Face E2nd – Find the area of 1 triangle whose base is the width of the rectangle then multiply itby 2. A = 1 bh 2 = 1 (4 cm x 6 cm) 2 = 1 (24 cm2) 2 = 12 cm2So, 12 cm2 x 2 = 24 cm2 – the sum of Face C and Face RLastly, get the area of the base of the pyramid. A=lxw = 5 cm x 4 cm = 20 cm2Then, add the areas you got in 1st step, 2nd step and last step. SA = 30 cm2 + 24 cm2 + 20 cm2 = 74 cm2TRY THESEFind the surface area of the following regular pyramids with a rectangular base. Writeyour solutions in your notebook and box your final answers.1) h = 10 m 2) 3) h = 8 dm h = 12 cm8m 6 dm 10 cm 4m 7 dm 6 cm 4

4) 5) h = 14 m h = 9 dm5 dm 10 m 7 dm 12 m WRAP UP To find the surface area of a regular pyramid with a rectangular base:  First, find the area of each of the faces.  Then, add all the areas of the faces. It is best if you can visualize the solid figure by unfolding it or flattening it. We can also get the surface area by: 1. Finding the area of a triangle whose base is the length of the rectangle, then multiply it by 2. 2. Finding the area of a triangle whose base is the width of the rectangle, then multiply it by 2. 3. Finding the area of rectangular base. 4. Adding all the areas. 5

ON YOUR OWNA. Match the surface area in Column B with the pyramids in Column A. Write only the letters in your notebook. COLUMN A COLUMN B1) h = 12 cm a. 172 m2 5m 8m b. 236 m22) h = 10 m c. 296 m2 7m 6m d. 196 m2 e. 79 m23) h=8m 5m f. 114 m2 3m4) h = 12 m 8m 10 m5) h=9m 4m 6mB. Find the surface area of the following regular pyramids. Write your answer in your notebook. 6

1) h = 10 cm 2) h=9m 3) h = 22 mm 8 cm 4 cm 20mm6 cm 5 cm 18 mm4) 5) h = 12 cm h = 10 dm 5m 8 dm8m 9 dm 7

GRADE VI SURFACE AREA OF A CYLINDERObjective: Find the surface area of a cylinder.REVIEWFind the area enclosed by the following figures. Write the answer in your notebook.1) 2) 3) r= 14 dm r = 7 cm 6 cm 8 cm4) 5) r= 10 dm 9m 7m 1

STUDY AND LEARN A cylinder is a solid shape that contains a lateral surface and two circular baseswhich are parallel and congruent. base (the surface is a circle) lateral surface (the surface is a curve) base (the surface is a circle) What will the figure look like when it is opened? base Lateral Surface Lateral Surface baseTherefore, a cylinder is made up of 2 circles and a rectangle.Let us recall the dimensions of a rectangle and circle. width radius length diameter How do we get the area of a rectangle? A = length x width How do we get the area of a circle? A = r2 where  = 3.14 or 22 and r = radius 7 How do we get the circumference of a circle? C = 2r or d, where r = radius, d = diameter and  = 3.14 or 22 7 How do you think can we find the total surface area of a cylinder? Do you still recall the meaning of “surface area?” 2

Surface area is the sum of all the areas of the faces of a solid figure.To find the surface area of a cylinder, we get the sum of the areas of the lateral surfaceand twice the area of one base.In symbols, SA = 2rh + 2(r2) lateral surface area area of a base (circle)Let us try to discover how the formula was derived. radius  top base bottom base height length height  width circumference of the cylinder  length radius opened body of cylinder Think FindArea of a circle: r2  Area of 2 circles: 2r2Circumference: 2r  Area of rectangle: 2rhNote that the circumference of the circular base is the same as the length of arectangle and the height of the cylinder is the same as the width of the rectangle.The total surface area is the sum of the areas of the rectangle and the area coveredby the two circles.Therefore, SA = 2rh + 2r2Example 1:Let’s find the total surface area of this cylinder. Use  = 3.14. 3

r = 5 cm STEP 1 – Find the area of the circular bases. Area of 2 Bases = 2(r2) h = 12cm = (2)(3.14)(5 cm)(5 cm) = (2)(3.14)(25 cm2) = (2)(78.5 cm2) = 157 cm2 STEP 2 – Find the area of the lateral surface. Lateral Area = 2rh = 2(3.14)(5 cm)(12 cm) = 6.28(5 cm)(12 cm) = (31.40 cm)(12 cm) = 376.8 cm2 STEP 3 – Find the sum of the areas. 157 cm2 + 376.8 cm2 = 533.8 cm2So, the surface area of the cylinder is 533.8 cm2.Example 2:Here is another example. The top and bottom of cylindrical milk can have a radius of 40 mm. Its height is80 mm. What is the surface area of the can?Given: r = 40 mm SA = 2rh + 2(r2) h = 80 mm = 2(3.14)(40 mm)(80 mm) + 2(3.14)(40 mm)2  = 3.14 = (6.28)(3 200mm2) + (6.28)(1 600 mm2) = 20 096 mm2 + 10 048 mm2 SA = 30 144 mm2Now, let’s find the surface of a cylinder with a height of 28 mm and a radius of 8 mm.Use  = 3.14. STEP 1 Area of 2 bases = 2(r2) STEP 2 = 2(3.14)(8 mm)2 = (6.28)(64 mm2) = 401.92 mm2 Area of lateral surface = 2rh = 2(3.14)(8 mm)(28 mm) = (6.28)(224 mm2) = 1 406.72 mm2 4

STEP 3 Find the sum. 401.92 mm2 + 1 406.72 mm2 = 1 808.64 mm2Or SA = 2rh + 2r2 = 2(3.14)(8 mm)(28 mm) + 2(3.14)(8 mm)2 = (6.28)(224 mm2) + (6.28)(64 mm2) = 1 406.72 mm2 + 401.92 mm2 SA = 1 808.64 mm2Let’s explore another way of finding the answer.Try this formula: SA = 2r (r + h)Given: r = 8 mm SA = 2r(r + h) h = 28 mm  = 3.14 = 2(3.14)(8 mm) (8 mm + 28 mm) = (6.28)(8 mm) (36 mm) = 50.24 mm2 (36 mm) SA = 1 808.64 mm2 TRY THESEA. Find the total surface area of each cylinder below. Write your solutions in your notebook and box your final answers. 1) 9 cm 2) 4m   21 cm 8m 5 cm 4) 3)  6 cm 7 dm h = 15 dm 5

B. Complete the table. Copy and answer this in your notebook. Radius Height Area of Area of Area of the Surface (dm) (dm) Lateral Surface One Base 2 Bases Area1. 6 10 (dm2) (dm2) (dm2) (dm2)2. 8 123. 15 9C. Solve the following problems. Write your solution in your notebook and encircle your final answers. 1. A cylindrical tank open at the top is to be painted to prevent it from rusting. The radius is 12 dm and the height is 18 dm. What is the total area to be painted? 2. A closed circular water tank in a subdivision is made of concrete. Its outer radius is 3.2 m and its height is 8 m. If it is painted on the exterior, how much total surface is to be painted? WRAP UP A cylinder is a solid shape that contains a lateral surface and two circular bases which are parallel and congruent. A diameter is twice the length of the radius. (d = 2r) To find the surface area of a cylinder, find the sum of the areas of the lateral surface and twice the area of one base. In symbol, SA = 2rh + 2(r2) You may also use the factored form of the formula, SA = 2r (r + h). 6

ON YOUR OWNA. Find the total surface area of each cylinder described below. Write your solutions in your notebook and box your final answers.1. r = 5 dm 2. r = 10 cm 3. r = 3 cm h = 10 dm h = 30 cm h = 9 cm SA = ____ SA = ____ SA = ____B. Find the surface area of the following cylinders. Write your answers in your notebook. Use  = 3.14.1)  7 cm 2) 3) height = 10 cm 9 cm 7 cm   4 cm radius = 4 cmC. Read, analyze and solve. Write your answers in your notebook. 1. A cylinder has a diameter of 13 cm and a height of 9 cm. What is its surface area? 2. A cylindrical milk can has a diameter and a height of 22 cm. What is the area of the material of which it is made? 7

GRADE VI VOLUME OF A CUBE AND RECTANGULAR PRISMObjective: Find the volume of a cube or rectangular prism.REVIEWFind the area of the following figures. Write the answers in your notebook.1) 2) 3) 10 m 6m 6.2 cm 9m 6.2 cm 4m4) 5) 3m 4m4m 7m STUDY AND LEARNBefore we proceed to explore the volume of a rectangular prism, let us see what a prismis. A prism is a solid with two bases that are parallel and identical polygons. 1

These are the examples of a prism. bases bases basesrectangular prism square prism or cube triangular prism Are the two bases in each solid parallel? Are the bases identical polygons?Now that you know what a prism is, we will proceed to our lesson.Read the following problem.Example 1:A box is 10 cm long, 5 cm wide and 3 cm tall. What is its volume?Let us visualize the problem.3 cm - cubic A cube has 3 dimensions:height centimeters length, width and height. 5 cm 1 cm3 Volume is expressed in width cubic units. We write 10 cm cubic units: unit3. lengthThe figure above is a box whose dimensions are 10 cm long, 5 cm wide and 3 cm high.To find the volume, we need to know the number of cubes in the whole figure.We know that the bottom of the prism is 10 centimeters long and 5 centimeters wide andthat it is also 3 cubes high.To find the total number of centimeter cubes,  We multiply to find the number of cubes on the bottom layer.  Then multiply by the number of layers 2

We multiply 10 by 5 by 3.10 x 5 x 3 = 150 centimeter cubeslength width heightnumber of cubes numberon bottom layer of layersThe volume of the box is 150 cm3.So, the formula in finding the volume of a rectangular prism is: Volume = length x width x heightIn symbols: V = l x w x h, we may also use the general formula V = B x h, where B stands for the area of the baseTherefore, we can use the formula, V = l x w x h or V = B x h in finding the volume of arectangular prism.Let us take a look at some more examples.Example 2:a) b) c) 4m 2 cm 5 cm 6m 3 cm 3 cm 10 m 3 cm V=? V=? 3 cm V=?V=lxwxh V=Bxh V=lxwxh = 5 cm x 3 cm x 2 cm = (10 m x 6 m) x 4m = 3 cm x 3 cm x 3 cm = 30 cm3 = 60 m2 x 4 m = 27 cm3 = 240 m3The solid figure in numbers 1 and 2 are both rectangular prisms.The third figure is also a prism but since the length, width and height are all the same, it has a special name. It is called a cube.If the length, width and height of a cube are all equal, we can substitute s for length,width and height and form a specific formula for finding the volume. 3

V=l x w x h sss V = s x s x s  V = s3 (the formula for the volume of a cube)Example:Find the volume of a cube whose side is 4 dm.Solution: V = s3 = (4 dm)3 = (4 dm) (4 dm) (4 dm) = (16 dm2)(4 dm) V = 64 dm3Reminder: Some use e for edge instead of s for side. So you may also encounter the formula V = e3 for cube. It is important to remember to express volume in cubic units. TRY THESEA. Find the volume of each of the following prisms. Write the answers in your notebook.1) 2) 3) 7 cm 12 cm 3 cm 7 cm 4 cm 8 cm 5 cm 6 cm 20.5 cm4) 5) 8 cm 8 cm 14 cm 14 cm 8 cm 14 cmB. Solve each problem. Write your solutions in your notebook and encircle your finalanswers. 4

1. A storage box is 50 cm by 30 cm by 40 cm. What is the volume of the box?2. Find the volume of a cube whose side is 4.3 dm. WRAP UP  Prism – a solid with two bases that are parallel and identical polygons. Ex. Rectangular prism, square prism, triangular prism.  Cubic unit – the unit used to find the volume of a solid  Volume – the number of cubic units needed to fill a solid figure.  The formula for finding the volume of: Rectangular prism: V = l x w x h Square prism or cube: V = s3 or V = e3  The general formula for any prism is: V = B x h, where B stands for the area of the baseON YOUR OWNA. Find the volume of each of the following prisms. Write your answers in your notebook.1) 2) 3) 5 cm 2 cm 2 cm 10.5 cm 9 cm 15 cm 10.5 cm 10.5 cm3 cm4) 5) 11 cm 8 cm 15 cm 31 cm 15 cm15 cm 5

B. Find the volume of each prism. Write your answers in your notebook.1) 2) 3) 12 cm 8 cm 5 cm 5 cm 2 cm 5 cm 12 cm 5 cm 5 cm4) 5)7 cm 7 cm 15 cm 7 cm 4 cm 12 cm 6

GRADE VI VOLUME OF A TRIANGULAR PRISMObjective: Find the volume of a triangular prism.REVIEWFind the area of the following figures. Write your answerS on your notebook.1) 2) 3) 4) h = 10 cm 8m 6 cm h = 9 cm 12 cm 5.4 9 m 10 cm cm5) 12 m h=4m STUDY AND LEARNIn the previous module, you learned how to get the volume of cube and a rectangularprism.Let us recall the meaning of prism.A prism is a solid figure with two bases that are parallel and congruent.The figure on the right is an example of a triangular prism. 1

 What is the shape of the figure that is congruent and parallel? triangle  What is the shape of the other faces? rectangleHow do we get the volume of a triangular prism?  The volume of a space figure is the number of cubic units contained in the solid figure.We have learned from the previous module that the general formula for volume is V =Bx h, where B stands for the area of the base.Example 1:5 cm 12 cm 6 cm 12 cm 5 cm 6 cm Is the figure a triangular prism? YesLooking at the two positions of the triangular prism, can you locate where the bases are.To get the volume of a triangular prism, we will use the general formula for finding thevolume of any prism which is:V = B x h, where B stands for the area of the base What is the shape of the base of a triangular prism? triangle How do you get the area of a triangle? A = 1 bh or bh 22Let us solve for the volume of the figure.V = Bh = ( 1 bh)(h) 2= 1 (6 cm x 5 cm) x 12 cm 2= 1 (30 cm2 x 12 cm) 2 2

= 15 cm2 x 12 cm V = 180 cm3 the volume of the given figure Example 2: A prism has a height of 8 cm and a base that is an isosceles triangle. If the isosceles triangle has a base of 6 cm and a height of 4 cm, what is the volume of the prism? Let us illustrate the problem. 8 cm Solution: V = Bh or V = (bh)(h) 6 cm 4 cm V = ( 1 bh)(h) 2 2 = (6x4)(8) = 1 (6 cm x 4 cm) x 8 cm 2 2 = (24)(8) = 1 (24 cm2) x 8 cm 2 2 = 192cm 3 = 12 cm2 x 8 cm = 96 cm3 2 = 96 cm3 What do you notice? Is the answer the same? Which of the two procedures do you prefer? = (bh)(h) 2 Let us have another example. = (9 x 8) x 20 Find the volume of the prism below. 29 cm 8 cm V = Bh = ( 1 bh)(h) = 72 x 20 2 2 = 1 (9 x 8) x 20 = 1440 2 20 cm 2 = 1 (72) x 20 V = 720 cm3 2 = 36 x 20 V = 720 cm3 So the volume of the triangular prism is 720 cm3. 3

Were you able to follow the steps in finding the volume of a triangular prism? If you are,then you are ready to answer the next exercises. TRY THESEA. Find the volume of each of the following prisms. Write your answers in your notebook. 1) 2) 3) 4) 16 cm 7 cm10 cm 18 cm 5) 5 cm 9 cm 8 cm 14 cm 12 cm 22 cm h = 8 cm h = 4 cm 14 cmB. Find the volume of each prism. Write your answers in your notebook. 1) 2) 3) 4) 15 cm 12 cm 10 cm 20 cm 8 cm 4 cm 12 cm1`0 cm 6 cm 7 cm 8 cm5) 10 cm 5 cm 10 cm 6) 6 cm 11 cm6 cm 12 cm 4

WRAP UP The volume of a prism is the number of cubic units contained in the solid figure. The volume of any prism is equal to the product of the area of its base, B and its height h. In symbols, V = B x h, where B stands for the area of the base Since, the base of a triangular prism is a triangle, we can also use, V = ( 1 bh)(h) or V = (bh)(h) . 22 ON YOUR OWNFind the volume of each prism. Write your answers in your notebook.1) 2) 10 cm 3) 12 cm 6 cm 10 cm 14 cm 8 cm 6 cm 8 cm 10 cm4) 5) 20 cm 10 cm 18 cm15 cm 4 cm 12 cm 5

GRADE VI VOLUME OF A PYRAMID WITH A SQUARE BASEObjective: Find the volume of a pyramid with a square base.REVIEWA. Find the area of the following figures. Write the answerS in your notebook.1) 2) s = 1.4 m 3) s = 9 cm s = 27 cmArea = ____ Area = ____ Area = ____B. Find the volume of the following figures. 3) r = 4 cm 1) 2) h = 10 cm h = 6 cm l = 10 cm w = 7 cm V = _____ e = 5 cm V = _____V = _____ STUDY AND LEARN A pyramid is a three-dimensionalfigure with only one base. The base is apolygon and the other faces are triangles. Apyramid is classified by the shape of itsbase. 1

 The formula for the volume of a rectangular prism can be used to find the formula for the volume of a pyramid.Let us have a simple experiment. l w h1. Make a model of a rectangular prism out of cardboard with these measures. h2. Make a model of a pyramid out of l cardboard. Use the same measures for w base and height.3. Fill the pyramid with sand.4. Pour the contents of the pyramid into the rectangular prism.5. Approximately how full is the rectangular prism?6. The volume of the pyramid is what part of the volume of the rectangular prism? V= 1Bxh So, the volume of a pyramid is 1 B x h or Bh 3 where B stands for the area of the 33 base.Explore first the pyramid with a square base.Example 1: height = 9 cm  Try to solve for the volume of the pyramid on the left.  What is the shape of the base of the pyramid?  Why did you say that the base is a square? How do you get the area of square? 8 cm 8 cm Using the formula, V = 1 Bh or Bh , where B stands for the area of the base. 33 You will get  V = base area x height  V = area of square x h h=9 cm 3 3 V = side x side x height 3 8 cm V = 8 cm x 8 cm x 9 cm 38 cm V = 64 cm2 x 9 cm 2

3 V = 576 cm3 V = 192 cm3 3Example 2: h= 6 m V = Bh 8m 38m =8mx8mx6m 3 = 64 m2 x 6 m 3 = 384 m3 3 V = 128 m3Now, will you get the volume of the figure below? h = 9 cm 20 cm 20 cmWhat is your answer?If your answer is 1200 cm3, then you got the correct answer. Congratulations!A pyramid has a height of 12 meters and a square base of side 6 meters. Find itsvolume.Understand Base  square  side = 6 meters height = 12 metersPlan V  Bh , where B = s x s because the base is a square 3Solve s = 6 meters h = 12 meters V = 6 m x 6 m x 12 m = 36 m2 x 12 m = 432 m3 3 33 V = 144 m3Check Did you follow correctly the formula? 3

Did you use the correct operation? Did you write the correct unit? Now, it’s your turn to solve a problem. Find the volume of a pyramid when the side of its square base is 40 m and the height is 27 m.Understand Plan Solve Check If your answer is 14,400 m3, then you have done right. TRY THESEA. Find the volume of the following pyramids with square base. Write the answers in your notebook.1) 10 cm 2) 12 cm 3) 6 m 6m 15 cm 12 cm 15 cm 5m 12 cm4) s = 9 cm 5) s = 10.2 m 6) s = 14 dm h = 8 cm V = ____ h=9m h = 12 dm V = ____ V = ____B. Read, analyze then solve.1. Find the volume of a pyramid with a square base if the side is 24 cm and the height is 18 cm. 4

2. What is the volume of a square pyramid 10.3 cm on each side and 12 cm high? WRAP UP  A pyramid is a three-dimensional figure with only one base. The base is a polygon and the other faces are triangles. It is classified by the shape of its base.  To find the volume of a pyramid, you can use the formula V  Bh 3 where b stands for the area of the square base or V = s x s x h 3ON YOUR OWNA. Find the volume of each of the following. Write the solutions in your notebook.1) 2) 3) 12 cm12 cm 15 cm 9 dm10 cm 12 cm 12 cm 9 dm 10 cmB. Read, analyze and solve. 4. What is the volume of a pyramid whose base is a square with a side 13 m long and whose height is 20 m? 5. How much air could a square pyramid hold if its base area is 125 m2 and 15 m high? 5

GRADE VIVOLUME OF A PYRAMID WITH A RECTANGULAR BASEObjective: Find the volume of a pyramid with a rectangular base.REVIEWA. Find the area of the following figures. Write your answers in your notebook.1. 2. 3. w = 17 cm l = 19 m l = 25 cm w = 12 m l = 20 cm w = 9 cmB. Find the volume of the following square pyramids.1. 2. 3. 5m h = 12 cm h=9m h = 12 cm 10 cm 15 cm 10 cm 5m 15 cm STUDY AND LEARNYou have learned from your previous lessons how to get the volume of a squarepyramid. This time, you will study how to get the volume of a rectangular pyramid. 1