MATH 3 part 2

Solution: Step 1. Draw and label the triangle h = 10 cm 12 cm Step 2. Substitute the data in the formula A = 1 bh 2 = 1 (12) (10) 2 = 60 cm The area is 60 square centimetersExample 5 Find the base of the triangle with an altitude of 12 cm and an area of 66 cm2.Solution Step 1. Draw and label the figure h = 12 cm b=? Step 2. Substitute the data in the formula. A = 1 bh 2 66 = 1 (b) (12) 2 66 = 6 b 6 b = 66 b = 11 The base is 11 centimeters. 16

Try this out 3. 4 mm 4.Set A 13 mmFind the area of each parallelogram.1. 8m 6m 4m 12 mm 6m 15 mm2.6 cm 5.2 cm 8. 20 mFind the area of each triangle 9. 25 m5. 10. 2 6m 46. 11 m 5 37. 5 cm 6 cm 17

Set B.Complete the table below Base Altitude Area of the Parallelogram 1 16 5 2 10 cm 180 cm2 3 14 km 168 km2 4 8m 5 4.5 cm 92 m2. 36 cm2Complete the table below base Altitude Area of the Triangle 6 14 cm 84 cm2 7 16 m 80 m2 8 16 cm 64 cm2 9 7.6 cm 4.5 cm __cm2 10 5.6 m 23.8 m2Set CFind the area of the parallelogram in which1. b = 30, h=62. b = 3 1 , h = 6.5 23. b = 24. h = 12Find the area of each triangle described below.4. b = 40, h = 245. b = 26, h = 146. b = 28, h = 16Solve the following7. Find the area of a parallelogram with base 17 m and height 14 m.8. Find the area of a triangle with base of 16 m and height of 12 m.9. A piece of cardboard is in the form of a parallelogram. Find its area if the base is 17 cm and the altitude is 12 cm.10. A piece of paper is in the form of a triangle. What is its area if the base is 18 cm and its altitude is 8.6 cm? 18

Lesson 3 Area of a Trapezoid Recall that a trapezoid is a quadrilateral with two bases which are parallel. The areaof a trapezoid is one half the product of the length of an altitude and the sum of the lengthsof the two bases. D b2 C A b1 B A = 1 h ( b1 + b2) 2 Where h = altitude b1 = lower base b2 = upper baseExample 1Find the area of trapezoid ABCD. D6 C h =4 A B 8 A = 1 h ( b1 + b2) 2 = 1 ( 4) (8 + 6 ) 2 = 1 ( 4 ) ( 14 ) 2 = 28 The area is 28 square units. 19

Example 2In the following trapezoid, HG = 6 cm and EF = 12 cm. If its area is 36 cm2, find itsaltitude h. H 6 cm G E h=? F 12 cm Solution: A = 1 h ( b1 + b2) 2 36 = 1 h ( 6 + 12) 2 36 = 1 h ( 18) 2 36(2) = 18 h 18h = 72 h = 4 cm The altitude h is 4 cm.Example 3 Find the longer base of a trapezoid with shorter base 5, height 4 and area 24..Step 1 Draw and label the figure. Represent the longer leg by x. 5 A = 24 4 xStep 2. Substitute the data in the formula. A = 1 h( b1 + b2) 2 24 = 1 (4)( x + 5) 2 24 = 2( x + 5)2(x + 5 ) = 242x + 10 = 24 2x = 24 – 10 2x = 14 x=7 The longer base is 7 . 20

Try this out CSet AABCD is a trapezoid. D AB1. If AB = 16, DC = 8, DE = 6, find the area2. If AB = 17, DC = 9 , DE = 8, find the area3. If AB = 20, DC = 18, DE = 10, find the area4. If AB = 30, DC = 20, DE = 10, find the area5. If AB = 40, DC = 30, DE = 20, find the areaEFGH is a trapezoid. HG EF6. If EF = 8, HG = 10, HE = 6, find the area.7. If EF = 6, HG = 8, HE = 4, find the area8. If EF = 12, HG =16, HE = 6, find the area9. If EF = 10, HG =12, HE = 9, find the area10. If EF = 12, HG =14, HE = 7, find the areaSet BSupply the missing information for each trapezoid H b1 b2 Area 14 8 11 28 10 13 140 3 12 10 14 112 4 12 16 5 10 18 21

Find the area of each trapezoid 7. 7 cm 8 6. 5 5 cm 11 13 cm8. Given a trapezoid with bases 16 and 20 and height 9, Find the area.9. The bases of a trapezoid are 8 and 20 and the area is 84. Find the height. 8 h=? A = 84 2010. The height of a trapezoid is 12 and the area is 162. If one of the bases is 16, find the other.Set C1. The height of a trapezoid is 4. The bases are 5 and 7. Find the area.2. The height of a trapezoid is 6. The bases are 7 and 9. What is the area?3. The bases of a trapezoid are 4 and 8 and the area is 36. Find the height.4. The bases of a trapezoid are 6 and 8 and the area is 56. Find the altitude.5. The altitude of a trapezoid is 8 and the area is 64. If one base is 5, find the other base.6. The height of a trapezoid is 2 and the area is 16. If the upper base is 9, find the lower base.7. A trapezoid has bases 9 and 10 and the area 38. Find its altitude.8. A trapezoid has bases 8 and 12 and area 80. Find its height.9. Find the longer base of a trapezoid with shorter base 5, height 6, and area 48.10. Find the shorter base of a trapezoid with longer base 12 , altitude 6, and area 54. Lesson 4 Area of a Circle Recall that a circle is a set of points in a plane that have the same distance from agiven point in the plane. The given point is called the center, and the distance from thecenter to any given point on the circle is called the radius. 22

The area of a circle is the measure of the space bounded by it. The ratio of thecircumference to the diameter of any circle is equal to the same number, represented bythe Greek letter π (pi) The approximate value of π is 3.14 or 22/7.The formula for the area of a circle with radius r units is A = πr2 Where A = area r = radiusExample 1The radius of a circle is 2 cm. Find its area.Solution:A = πr2 2 cm ≈ 3.14 (2)2 ≈ 12.56 cm2 ●The area is 12.56 square cm.Example 2The diameter of a circle is 6 cm. Find its area.Solution:Step 1. Find the radius Radius( r ) = diameter (d) divided by 2 d = 6 cm r=6÷2 ● r = 3 cmThe radius is 3 cm.Step 2. Find the area. A = πr2 ≈ 3.14 (3)2 ≈ 3.14 (9) ≈ 28.26 cm2The area is 28.26 square centimeters 23

Example 3 Find the area of the circle in terms of π if its radius is 2 m.SolutionA = πr2 2m ≈ π(2)2 ≈ 4π ●Example 4 Find the area of the shaded region. 22 cm 8 cmSolution: Step 1. Find the area of the rectangle. A = lw = (22) ( 8) = 176 cm2 Step 2. Find the area of the circle The diameter of the circle is the width of the rectangle. r= d 2 r= 8 2 r = 4 cm2 Substitute 4 in the area formula for circles A = πr2 ≈ 3.14 ( 4)2 ≈ 3.14 ( 16) ≈ 50.24 Step 3. Subtract the area of the circle from the area of the rectangle. Area of the shaded region = area of the rectangle – area of the circle = 176 –50.24 = 125.76 cm2 The area of the shaded region is 125.76 square centimeters. 24

Try this outSet A.Find the area of each circle with the given diameter or radius Use 3.14 for π.1. radius = 5 cm2. radius = 1.5 mm3. diameter = 4 cm4. diameter = 12 dm5. radius = 4.6 m6. radius = 2.2 cm7. diameter = 4.8 dm8. diameter = 6.4 cm9. radius = 4.8 m10. radius = 3.4 dmSet BFind the area of each circle described below. Give the answers in terms of π.1. 3. ● d = 10 r=8 ●2. 4. ● d = 11 ● r=9 25 5. The radius is 7 cm 6. The radius is 12 cm 7. The diameter is 14 mm 8. The diameter is 26 mm 9. The radius is 3.4 km 10. The radius is 2.6 km

Set C. Find the area of each circle in terms of π 1. r = 8 dm 2. r = 18 dm 3. d = 5 cm 4. d = 14 cm 5. r = 4.6 mm 6. r = 4.4 mm 7. The students performed their dance number in a circular platform 20 m in diameter. Find the area of the platform. 8. The radius of a circular garden is 40 m. Half of the garden will be planted with roses. How many square meters will be planted with roses. Find the area of the shaded part of the figure. Give the answers in terms of π 9 r of the bigger circle is 8 cm ● r of the smaller circle is 6 cm.10. ● 14 cm 12 cm 26

Let’s Summarize1. The area of a region is the number of square units contained in the region.2. A square unit is a square with a side I unit in length.3. The area (A) of a rectangle is the product of its length (l) and its width (w). A = lw4. The area (A) of a square is the square of the length of a side (s). A = s25. The area (A) of a parallelogram is equal to the product of the base (b) and the height (h). A = bh6. The diagonal separates the parallelogram into two congruent triangles.7. The area (A) of a triangle equals half the product of the base (b) and the height (h). A = 1 bh. Sometimes altitude is used instead of height. 28. The area (A) of a trapezoid is one half the product of the length of its altitude and the sum of the lengths of the two bases. A = 1 h (b1 + b1). 29. A circle is a set of points in a plane that have the same distance from a given point in the plane.10. In all circles the ratio of the circumference to the diameter is always equal to the same number, represented by the Greek letter π.11. The formula for finding the area of a circle with a radius of r units is: A = πr2 What have you learned1. Find the area of a square whose side is 15 dm.2. Find the area of the triangle below. 9 cm 10 cm3. Find the area of a rectangle whose length and width are 14 dm and 8 dm respectively. 27

4. Find the area of the trapezoid ABCD below B 8 cm C 6 cmA 12 cm D5. Find the area of a circle whose radius is 6 cm.6. Find the base of a triangle if the altitude is 6 cm and the area is 36 cm2.7. Find the area of a parallelogram with base 14 cm and height 6 cm.8. The area of the parallelogram ABCD below is 66 cm2. Find h. BC h=?A 11 cm D9. Find the area of the figure 7 5 3 2310. Gen’s garden is 5 meters wide and 7 meters long. Find the area of the garden. 28

Answer Key 6. 8 cm 7. 96 cm2How much do you know 8. 8 cm 9. 44 cm2 1. 81 cm2 10. 24 m 2. 24 cm2 3. 60 cm2 6. 128 cm2 4. 32 cm 2 7. 304 cm2 5. 153.86 mm2 8. 30.25 m2 9. 140Try this out 10. 49 mm2Lesson 1 6. 180 cm2 7. 10 cm2Set A. 8. 13 cm2 9. 28 sq. units 1. 10 10. 96 sq. units 2. 18 3. 14 6. 9 cm 4. 16 7. 14 cm 5. 25 cm2 8. 15 cm 9. 11 cmSet B. 10. 12 cm 1. 225 cm2 6. 7.5 sq. units 2. 84 m2 7. 15 cm2 3. 72.25 m2 8. 90 mm2 4. 40 cm2 9. 250 m2 5. 169 cm2 10. 33 m2Set C 29 1. 16 m2 2. 20.25 cm2 3. 88 cm2 4. 72 cm2 5. 6 cmLesson 2Set A 1. 24 m2 2. 31.2 cm2 3. 52 mm2 4. 48 m2 5. 4 sq. units

Set B 1. 80 sq. units 6. 12 cm 2. 18 cm 7. 10 m 3. 12 km 8. 8 cm 4. 11.5 m 9. 17.1 cm 5. 8 cm 10.8.5 mSet C 6. 224 sq. units 7. 238 m2 1. 180 sq. units 8. 96 m2 2. 22.75 sq. units 9. 204 cm2 3. 288 sq. units 10. 77.4 cm2 4. 480 sq. units 5. 182 sq. units 6. 54 sq. units 7. 28 sq. unitsLesson 3 8. 84 sq. units 9. 99 sq. unitsSet A 10. 91 sq. Units 1. 72 sq. units 6. 47.5 sq. units 2. 104 sq. units 7. 50 cm2 3. 190 sq. units 8. 162 sq. units 4. 250 sq. units 9. 6 5. 700 sq. units 10. 11Set B 6. 7 7. 4 1. 38 sq. units 8. 8 2. 92 sq. units 9. 11 3. 144 sq. units 10. 6 4. 10 5. 8 6. 15.1976cm2 7. 18.0864 dm2Set C. 8. 32.1536 cm2 9. 72.3456 m2 1. 24 sq. units 10. 36.2984 dm2 2. 48 sq. units 3. 6 30 4. 8 5. 11Lesson 4Set A 1. 78.5 cm2 2. 7.065 mm2 3. 12.56 cm2 4. 113.04 dm2 5. 66.4424 m2

Set B 1. 64 π 6. 144 π cm2 2. 81 π 7. 49 π cm2 3. 25 π 8. 169 π mm2 4. 30.25 π 9. 11.56 π km2 5. 49 π cm2 10. 6.76 π km2Set C 6. 19. 36 π mm2 7. 100 π m2 1. 64 π dm2 8. 800 π m2 2. 324 π dm2 9. 64 π cm2– 36 π cm2 = 28 π cm2 3. 6.25 π cm2 10. 196 cm2 – 49 π cm2 4. 49 π cm2 5. 21.16 π mm2 6. 12 cm 7. 84 cm2What have you learned? 8. 6 cm 9. 29 sq. units 1. 225 dm2 10. 35 m2 2. 45 cm2 3. 112 dm2 4. 60 cm2 5. 36 π cm2 31

Module 7 Geometry of Shape and Size What this module is about This module is about surface area of solids. As you go over the exercises,you will develop skills in solving surface area of different solid figures. Treat thelesson with fun and take time to go back if you think you are at a loss. What you are expected to learn This module is designed for you to 1. define surface area of solids. 2. find the surface area of solids such as • cube • prism ( rectangular, triangular ) • pyramid ( square, rectangular, triangular ) • cylinder • cone • sphere 3. solve problems involving surface area of solids. How much do you know Find the surface area of each solid. 1. A cube with side (s) = 2.2 cm. 2. A cylinder with h = 15 cm, r = 3.2 cm. 3. A rectangular prism with l = 12 cm, w = 7 cm, h = 6 cm. 4. A square pyramid with s = 4.2 cm, h = 7 cm (slant height).

5. A cone with r = 5 cm , s = 12 cm (slant height). 6. A triangular prism with height 14 cm, base (a right triangle with sides 6, 8 and 10 cm and the right angle between shorter sides). 7. A ball with radius of 6 cm (Use π = 3.14). 8. A triangular pyramid with b = 4 cm, h = 8.2 cm (altitude of the base), s = 7 cm (slant height). 9. A rectangular pyramid with l = 5 cm, w = 3.3 cm, h = 9 cm (slant height). 10. A cylindrical tank is 2.6 meters high. If the radius of its base is 0.92 meters, what is its surface area? 11. find the surface area of a rectangular prism which is 45 cm long, 36 cm wide and 24 cm high. 12. Find the surface area of a pyramid with a square base if the length of the sides of the base is 1.4 m and the height of the triangular face is 1.9 m. 13. Cube with edge 4 2/3 cm 14. Cylinder with radius of base 6.7 cm and height 14 cm 15. Rectangular prism with base 9m by 10m by 12m What you will do Lesson 1 Finding the Surface Area of a Cube, Prism and Pyramid The surface area of a three-dimensional figure is the total area of itsexterior surface. For three-dimensional figures having bases, the surface area isthe lateral surface area plus the area of the bases. 2

Examples : 1. Find the surface area of a cube with side of 5 cm. Figure :lateral face edge Cube If you open this up, face face face face face face This solid is composed of 6 squares or 6 faces. Since the area of a squareis the square of its side or s2, then the surface area of a cube is 6 s2. SA of a Cube = 6s2 Substituting s by 5 cm: SA = 6(52) = 6(25) SA = 150 cm2 3

2. Find the surface area of a rectangular prism whose length is 7 cm, width is 5 cm and thickness is 4 cm.Figure : Rectangular prism, add the 7 cm areas of its flat surfaces. To find the surface area of a 5 cm base4 cm base Rectangular Prism Area of top and bottom rectangles (bases) plus area of left and rightrectangles and area of back and front rectangles (lateral areas). SA of rectangular prism = 2B + LA Solution: SA = 2(7cm x 5 cm) + 2(4 cm x 5 cm) + 2(7cm x 4 cm) = 70 cm2 + 40 cm2 + 56 cm2 SA = 166 cm23. Find the surface area of a triangular prism.Figure : 5 cm 5 cm 3.9 cm 2.8 cm 4.5 cm Triangular Prism 4

SA of a triangular prism = 2B + LA Solution: SA = 2 ( 1 bh) + LA 2 SA = bh + LA = (4.5cm x 3.9cm) + 2( 5cm x 2.8 cm) + (4.5 cm x 2.8cm) = 17.55 cm2 + 28 cm2 + 12.6 cm2 SA = 58.15 cm24. Find the surface area of a square pyramid with a side of the base as 3 cm and the height of a triangle as 5 cm. Figure: 5 cm To find the surface area of a square pyramid, add the area of the square base and the areas of the four face triangles.3 cm 3 cmSolution:SA of Square Pyramid = B + 4 ( 1 bh) 2 SA = 32 + 4( 1 x 3 x 5) 2 = 9 + 30 SA = 39 cm2 5

5. Find the surface area of the rectangular pyramid with the given dimensions.Figure : 6 cm 5 cm 8 cm Rectangular Pyramid SA of Rectangular Pyramid = B + 2 A1 + 2 A2 Solution: SA = bh + 2 ( 1 b1h1) + 2 ( 1 b2h2) 22 = bh + (b1h1) + ( b2h2) = (8 cm x 5 cm)+ (8 cm x 6 cm) + (5 cm x 6 cm) = 40 cm2 + 48 cm2 + 30 cm2 SA = 118 cm26. Find the surface area of a triangular pyramid with the given dimensions. Figure : 8 cm(slant height)20 cm 20 cm 5 cm 20 cmTriangular Pyramid 6

SA of Triangular Pyramid = Area of the base + Area of the 3 triangular faces Solution: SA = ( 1 x 20 cm x 5 cm) + (10 cm x 8 cm) + ( 1 x 20 cm x 8 cm) 22 = 50 cm2 + 80 cm2 + 80 cm2 SA = 210 cm2Try this outFind the surface area of each solid.1. 1.6 cm 1.6 cm 1.6 cm 52 mm2. 180 mm 204 mm3. 9.4 cm 6.6 cm 6.6 cm 7

4. 7 cm 6 cm 9 cm3 cm 4 cm5. 10 cm 5 cm 7 cm (slant height)6. 10 cm 3 cm 5 cm (altitude of the base)5 cm 8

7. What is the total surface area of a cardboard box that is 1.2 m long, 0.6 m wide, and 0.3 m high? 1.2 cm Figure: 0.3 cm 0.6 cm8. Find the surface area of a cube whose side measures 10 cm.Figure: 10 cm 10 cm 10 cm9. Find the surface area of a triangular chocolate box . Figure: 6 cm 8 cm 6 cm 6 2 cm 9

10. Find the surface area of a camping tent in a square pyramid shape with a side of the base as 5 cm and the height of a triangle as 7 cm. Figure : 7 cm (slant height) 5 cm 5 cm11. A pyramid has a rectangular base whose length and width are 5.5 cm and 3.2 cm respectively. Find its surface area. Figure: 10 cm (slant height) 3.2 cm 5.5 cm12. Find the surface area of a tetra pack juice drink in triangular pyramid shape with the given dimensions.Figure: 10 cm (slant height) 6 cm 6 cm 4.5 cm (altitude of the base) 6 cm 10

Lesson 2 Finding the Surface Area of Cylinder, Cone and SphereSurface Area of a CylinderFigure: base basealtitude (h) h radius (r) base circumference of thebase base = 2πr r Area of the 2 bases = 2πr r2 To find the surface are of a cylinder, add the areas of the circular basesand the area of the rectangular region which is the body of the cylinder.SA = Area of 2 Circular Bases + Area of a rectangleSA = 2πr2 + 2πrhExample: Find the surface area of a cylinder which has a radius of 5 cm and a bodylength of 20 cm. ( Use 3.14 for π)Solution:SA = 2πr2 + 2πrh = 2(3.14)(5) 2 + 2(3.14)(5)(20) = (6.28)(25) + (6.28)(100) = 157 + 628SA = 785 cm2 11

Surface Area of a Cone Consider an ice-cream cone with itsFigure: curved surface opened out to resemble a fan. Slant height radiuss h=s The fan-shaped surface can 2πr r b = πr r be cut into smaller pieces and rearranged to resemble a parallelogram with base b = πr and height h equal to side s.SA = Area of the circular base + Area of the region which resembles a parallelogramSA = πr2 + πrsExample: Find the surface area of a cone if the radius of its base is 3.5 cm and itsslant height is 7.25 cm. (Use π = 3.14)Solution :SA = πr2 + πrs s = 7.25 cm = (3.14)(3.5)2 + (3.14)(3.5)(7.25) r = 3.5 cm = (3.14)(12.25) + (10.99)(7.25) = 38.465 + 79.6775SA = 118.1425 cm2 12

Surface Area of a Sphere A sphere is a solid whereFigure: every point is equally distant from its center. This distance radius is the length of the radius ofSA = area of 4 circlesSA = 4πr2Example:What is the surface area of a ball with radius equal to 7 cm? (Use π = 3.14)Solution : SA = 4πr2 r=7 = 4(3.14)(7)2 = (12.56)(49) SA = 615.44 cm2Try this outFind the surface area of each solid. 1.8 m1. height = 2.4 m 13

2. 2.6 m3. s = 12.3 m r =7.8 cm4. 7.6 cm 8.2 cm 14

5. s = 20 mm r = 8 mm6. r = 6 cm7. A cylindrical water tank is 2.2 meters high. If the radius of its base is 0.8 meter, what is its surface area?Figure: r = 0.8 m h = 2.2 m 15

8. The radius of a ball is 19 cm. What is its surface area? Figure: r = 19 cm9. Find the surface area of a conic solid whose radius is 2.5cm and its height is 3.3 cm.Figure: r = 2.5 cm s = 3.3 cm10. Find the surface area of a spherical tank whose radius is 0.7meter.Figure: r = 0.7 m 16

11. A cone with a diameter of 10 cm and height of 8 cm. Find its surface area. Figure : s = 8 cm r = 10 cm12. A can whose height is 20 cm and 12 cm is the diameter. Find its surface area. Figure: r = 12 cm h = 20 cm 17

Let’s summarize The surface area of a three-dimensional figure is the total area of itsexterior surface. For three-dimensional figures having bases, the surface area isthe lateral surface area plus the area of the bases. • Surface Area of a Cube = 6s2 • Surface Area of Rectangular Prism = 2B + LA • Surface Area of Rectangular Prism = 2B + LA • Surface Area of Square Pyramid = B + 4 ( 1 bh) 2 • Surface Area of Rectangular Pyramid = B + 2 A1 + 2 A2 • Surface Area of Triangular Pyramid = Area of the base + Area of the 3 triangular faces • Surface Area of a Cylinder = Area of 2 Circular Bases + Area of a rectangle SA = 2πr2 + 2πrh • Surface Area of a Cone = Area of the circular base + Area of the region which resembles a parallelogram SA = πr2 + πrs • Surface Area of a Sphere = area of 4 circles SA = 4πr2 18

What have you learnedFind the surface area of each solid. 1. A cube with side (s) = 4.3 cm 2. A cylinder with h = 13 cm, r = 4.1 cm 3. A rectangular prism with l = 15 cm, w = 8 cm, h = 7 cm 4. A square pyramid with s = 5.4 cm, h = 9 cm (slant height). 5. A cone with r = 4cm , s = 9 cm (slant height). 6. A triangular prism with height 16 cm, base (a right triangle with sides 7 cm, 8 cm and 10 cm and the right angle between shorter sides). 7. A ball with radius of 12 cm. (Use π = 3.14) 8. A triangular pyramid with b = 5 cm, h = 7.2 cm (altitude of the base), s = 8 cm (slant height) 9. A rectangular pyramid with l = 6 cm, w = 6.3 cm, h = 8 cm (slant height) 10. A cylindrical tank is 3.6 meters high. If the radius of its base is 1.9 meters, what is its surface area? 11. Find the surface area of a rectangular prism which is 40 cm long, 35 cm wide and 23 cm high. 12. Find the surface area of a pyramid with a square base if the length of the sides of the base is 1.6 m and the height of the triangular face is 2.8 m. 13. Cube with edge 5 1 cm. 2 14. Cylinder with radius of base 8.7 cm and height 13 cm. 15. Rectangular prism with base 10m by 12m by 14m. 19

Answer KeyHow much do you know 1. 29.04 cm2 2. 365.74742 cm2 3. 396 cm2 4. 135.24 cm2 5. 266.9 cm2 6. 384 cm2 7. 452.16 cm2 8. 100.4 cm2 9. 91.2 cm2 10. 20.337152 cm2 11. 7, 128 cm2 12. 12.6 m2 13. 130 2 cm2 3 14. 870.9732 cm2 15. 636 m2Try this outLesson 1 1. 15.36 cm2 2. 113, 376 mm2 3. 167.64 cm2 20

4. 159 cm2 5. 132 cm2 6. 85 cm2 7. 2.52 m2 8. 600 cm2 9. 181.88 cm2 10. 95 cm2 11. 104.6 cm2 12. 78.3 cm2Lesson 2 1. 18.6516 m2 2. 21.2264 m2 3. 492.2892 cm2 4. 301.3576 cm2 5. 703.36 mm2 6. 452.16 cm2 7. 15.072 m2 8. 4,534.16 cm2 9. 45.53 cm2 10. 6.1544 m2 11. 204.1 cm2 12. 979.68 cm2 21

What have you learned 1. 110.94 cm2 2. 131.3148 cm2 3. 562 cm2 4. 101.16 cm2 5. 163.24 cm2 6. 228 cm2 7. 1,808.64 cm2 8. 138 cm2 9. 136.2 cm2 10. 65.626 m2 11. 6,250 cm2 12. 20.48 m2 13. 181.5 cm2 14. 1,185.6012 cm2 15. 856 m2 22

Module 8 Geometry of Shape and Size What this module is about This module is about volume of solids. The volume of a solid is thenumber of cubic units contained in the solid. If measures are given in centimeter,the volume is stated in cubic cm, written as cm3. What you are expected to learn This module is designed for you to: 1. define volume of solids. 2. find the volume of solids such as: • cube • prism (rectangular, triangular) • pyramid (square, rectangular, triangular) • cylinder • cone • sphere 3. solve problems involving volume of solids. How much do you know Find the volume of each solid: 1. a cube with side (s) = 2.4 m 2. a cylinder with h = 20 cm, r = 22 cm. 3. a rectangular prism with l = 25 cm, w = 17 cm, h = 30 cm

4. a square pyramid with s = 5 m, h = 6 m. 5. a cone with r = 2 cm, h = 6 cm. 6. a triangular prism with height 10 cm, base (a right triangle with sides 3, 4 and 5 cm and the right angle between shorter sides). 7. a ball with radius of 17 cm (use = 3.14) 8. a triangular pyramid with b = 4 cm, h = 8.2 cm (altitude of the base), h = 7 cm (height of the pyramid). 9. a rectangular pyramid with l = 6 cm, w = 4.3 cm, h = 8 cm (height of the pyramid) 10. a cylindrical tank is 5.3 meters high. If the radius of its base is 2.8 meters, what is its volume? 11. Find the volume of a rectangular prism which is 46 cm long, 37 cm wide and 25 cm high. 12. Find the volume of a pyramid with a square base if the length of the sides of the base is 2.4 m and the height of the triangular face is 3.5 m. 13. cube with edge of 6 2 cm. 3 14. cylinder with radius of base 8.7 cm and height 12 cm. 15. rectangular prism with base 8 m by 10 m by 15 m. What you will do Lesson 1 Finding the Volume of a Cube, Prism and Pyramid One problem with rooms that have high ceilings is that they are hard toheat and cool. The amount of air in a room determines the heating or coolingpower needed. To find the amount of air in a room, you need to find the volumeof the room. In finding volume of solids, you have to consider the area of a face andheight of the solid. If the base is triangular, you have to make use of the area ofa triangle, if rectangular, make use of the area of a rectangle and so on. 2

The next examples will help you to understand more about volume or theamount of space in three – dimensional figures.Volume of a cube The volume V of a cube with edge e is the cube of e. That is, V = e3.Example:Find the volume of a cube with edge ( e ) of 3 cm.Figure: 3 cm 3 cm 3 cm CubeSolution: V = e3Substituting e by 3 cm: V = 33 V = 27 cm3 3

Volume of Prism The volume V of a rectangular prism is the product of its altitude h, the length l and the width w of the base. That is, V = lwh.Example: Find the volume of a rectangular prism whose length is 7.5 cm, width is4.3 cm and thickness is 5.1 cm.Figure: 7.5 cm 5.1 cm 4.3 cmSolution: V = lwh V = (7.5 cm)(4.3 cm)(5.1 cm) V = 164.475 cm3The volume of a prism can also be expressed in terms of area of the base, B. The volume V of a prism is the product of itsaltitude h and area B of the base. That is, V = Bh. 4

Example: Find the volume of a triangular prism whose dimensions is given in the figure below.4.5 cm 3.9 cm 2.8 cmSolution: Let B = area of the triangular base B = 1 bh 2 = 1 (4.5 cm)(3.9 cm) 2 B = 8.775 cm2Finding the volume of the prism: V = Bh = 8.775 cm2 (2.8 cm) = 24.57 cm3 5

Volume of Pyramids Consider a pyramid and a prism having equal altitudes and bases withequal areas. If the pyramid is filled with water or sand and its contents pouredinto a prism, only a third of the prism will be filled. Thus the volume of a pyramidis 1 the volume of the prism. 3 The volume V of a pyramid is one third the product of its altitude h and the area B of its base. That is, V = 1 Bh. 3Example: 1. Find the volume of the rectangular pyramid with the given dimensions. Figure: h = 6 cm w = 4 cm base l = 9 cmSolution: Let B = the area of the rectangular base B = lw = (9 cm)(4 cm) = 36 cm2Finding the volume V: V = 1 Bh 3 = 1 (36 cm2) (6 cm) 3 = 72 cm3 6

2. Find the volume of a square pyramid with a side of the base as 4 cm and the height of a pyramid as 6 cm. Figure: h = 6 cm s = 4 cmSolution: Let B = area of the square base = s2 = (4 cm)2 B = 16 cm2Finding the volume of the pyramid: V = 1 Bh 3 = 1 (16 cm2 )(6 cm) 3 = 32 cm3 7

3. Find the volume of a triangular pyramid with the given dimensions. Figure: h = 8 cm (height of the pyramid)h = 5 cm b = 20 cm(height ofthe base)Solution: Let B = area of the triangular base B = 1 bh 2 = 1 (20 cm)(5 cm) 2 B = 50 cm2Finding the volume of the pyramid: V = Bh = (50 cm2)(8cm) V = 400 cm3 8

Try this out 6 cm 6 cmFind the volume of each solid:1. 6 cm 12 mm 6 mm2. 9 mm3. h = 3 cm 1.5 cm 15 cm 9

Find the volume of each pyramid:4. h = 15 cm w = 10 cml = 25 cm5. h=7m 3.5 m3.5 m6. h = 10.4 cm (pyramid) h = 4.3 cm (base)b = 16.2 cm 10

7. What is the volume of a cardboard box that is 9 m long, 6 m wide, and 3 m high? 9m 3m 6m8. Find the volume of a cube with side of 8 cm. 8 cm 8 cm 8 cm9. Find the volume of a triangular chocolate box. h = 7 cm 4 cm3 cm 11

10. Find the volume of a camping tent in a square pyramid shape with a side of the base as 5 cm and the height of a triangle as 7 cm. h = 8.5 cm 6.3 cm 6.3 cm11. A pyramid has a rectangular base whose length and width are 15.5 cm and 3.3 cm respectively. The height of the pyramid is 4 cm. Find its volume. h = 4 cm w = 3.3 cm l = 15.5 cm12. Find the volume of a tetra pack juice drink in triangular pyramid shape with the given dimensions. h = 10 cm (pyramid) h = 5 cm (base) 12