Mathematics
Surface Area of Cone
The surface area of a cone is the total area of all its surfaces, including the base and the curved side. It is calculated using the formula A = πr² + πrl, where A is the surface area, r is the radius of the base, l is the slant height, and π is a constant approximately equal to 3.14159.
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eBook - PDFMathematics for the General Course in Engineering
The Commonwealth and International Library: Mechanical Engineering Division, Volume 2
- John C Moore, N. Hiller, G. E. Walker(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
To find the area of a rectangle we multiply the length by the breadth and so AREA OF CURVED SURFACE OF A CYLINDER = CIRCUMFERENCE x HEIGHT If the diameter of the base circle is D then the base circumference is 7rD. We can write shortly A -TTDH Note the following alternative. If the radius of the base circle is R then the base circumference is InR. We can write shortly A -2TTRH Ex. Find correct to the nearest cubic inch the volume of a right circular cylinder of base diameter 9 in and height 6 in. Find also the curved surface area correct to the nearest square inch. D = 9 and H = 6 V = | 7 r D 2 H = - x 3-1416 x 9 2 x 6 4 - 381-7044 Volume = 382 in 3 (correct to the nearest cubic inch) A = T T DH - 3-1416 x 9 x 6 = 169-6464 Curved surface area = 170 in 2 (correct to the nearest square inch) M E N S U R A T I O N 79 E x . Find correct to the nearest cubic inch the volume of a right circular cylinder of base radius 3-6 in and height 5«2 in. Find also the total surface area correct to the nearest square inch. T H E R I G H T C I R C U L A R C O N E T h e following diagram shows a solid k n o w n as a R I G H T C I R C U L A R C O N E . D DIAGRAM 15 T h e cone is called C I R C U L A R because the base is a C I R C L E and it is called R I G H T because the line in the diagram joining the t o p of the cone to the centre of the base circle is at R I G H T angles to the base. Volume of a Cone Obtaining the formula for the volume of a right circular cone presents quite a problem because the only quick a p p r o a c h involves advanced mathematics. If we restrict ourselves to easy mathematics then it is still possible to produce the formula if we are willing to use the m e t h o d of successive approximations. This extremely i m p o r t a n t method is used in m a n y branches of m a t h e -matics. It is easy to apply but requires patience.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
This property holds for right and oblique cones. Volume of a Cone The volume V of a cone whose base has area A and whose height is h is V Ah = 1 3 . h r A Common Core – Grade 7 Know the formulas for the vol- umes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. T H E O R E M 1 3 . 1 7 For a cone with a circular base of radius r , the volume of the cone is 1 3 2 p r h. Check for Understanding: Exercise/Problem Set A #8–9 ✔ r r 2r Figure 13.74 Volume of a Sphere The volume V of a sphere with radius r is V r = 4 3 3 p . r T H E O R E M 1 3 . 1 8 Section 13.4 Volume 703 Table 13.11 summarizes the volume and surface area formulas for right prisms, right circular cylinders, right regular pyramids, right circular cones, and spheres. The indicated dimensions are the area of the base, A; the height, h; the perimeter or cir- cumference of the base, P or C ; and the slant height, l . By observing similarities, one can minimize the amount of memorization. TABLE 13.11 GEOMETRIC SHAPE SURFACE AREA VOLUME Right prism S A Ph = + 2 V Ah = Right circular cylinder S A Ch = + 2 V Ah = Right regular pyramid S A Pl = + 1 2 V Ah = 1 3 Right circular cone S A Cl = + 1 2 V Ah = 1 3 Sphere S r = 4 2 π V r = 4 3 3 π The remainder of this section presents a more formal derivation of the volume and surface area of a sphere. First, to find the volume of a sphere, we use Cavalieri’s prin- ciple, which compares solids where cross-sections have equal areas. As an aid to recall and distinguish between the formulas for the surface area and volume of a sphere, observe that the r in 4 2 p r is squared, an area unit, whereas the r in 4 3 3 p r is cubed, a volume unit. An ice cream shop has sugar cones with a slant height of 13 cm and the diameter of the base is 10 cm (see Figure 13.75).
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
The segment PO , which joins the vertex to the center of the circular base, is the axis of the cone. If the axis is perpendicular to the plane containing the base, as in Figure 9.31, the cone is a right circular cone. In any cone, the perpendicular segment from the vertex to the plane of the base is the altitude of the cone. In a right circular cone, the length h of the altitude equals the length of the axis. For a right circular cone, and only for this type of cone, any line segment that joins the vertex to a point on the circle is a slant height of the cone; we will denote the length of the slant height by / as shown in Figure 9.31. SURFACE AREA OF A CONE Recall now that the lateral area for a regular pyramid is given by L 5 1 2 /P. For a right circular cone, consider an inscribed regular pyramid as in Figure 9.32. As the number of sides of the inscribed polygon’s base grows larger, the perimeter of the inscribed poly- gon approaches the circumference of the circle as a limit. In addition, the slant height of the congruent triangular faces approaches that of the slant height of the cone. Thus, the lateral area of the right circular cone can be compared to L 5 1 2 /P; for the cone, we have L 5 1 2 /C in which C is the circumference of the base. The fact that C 5 2pr leads to L 5 1 2 /(2pr) so L 5 pr/ SSG EXS. 8, 9 P h O r Figure 9.31 P O Figure 9.30 Theorem 9.3.5 The total area T of a right circular cone with base area B and lateral area L is given by T 5 B 1 L. Alternative Form: Where r is the length of the radius of the base and / is the length of the slant height, T 5 pr 2 1 pr/. Theorem 9.3.4 The lateral area L of a right circular cone with slant height of length / and circumference C of the base is given by L 5 1 2 /C. Alternative Form: Where r is the length of the radius of the base, L 5 pr/. Figure 9.32 The following theorem follows easily from Theorem 9.3.4 and is given without proof.
eBook - PDF- John Bird(Author)
- 2017(Publication Date)
- Routledge(Publisher)
The volume of a frustum of a pyramid or cone is given by the volume of the whole pyramid or cone minus the volume of the small pyramid or cone cut off. The surface area of the sides of a frustum of a pyra- mid or cone is given by the surface area of the whole pyramid or cone minus the surface area of the small 190 Engineering Mathematics pyramid or cone cut off. This gives the lateral surface area of the frustum. If the total surface area of the frus- tum is required then the surface area of the two parallel ends are added to the lateral surface area. There is an alternative method for finding the volume and surface area of a frustum of a cone. With reference to Fig. 20.12: h R r I Figure 20.12 Volume = 1 3 πh(R 2 + Rr + r 2 ) Curved surface area = π l (R + r) Total surface area = π l (R + r) + π r 2 + π R 2 Problem 16. Determine the volume of a frustum of a cone if the diameter of the ends are 6.0 cm and 4.0 cm and its perpendicular height is 3.6 cm Method 1 A section through the vertex of a complete cone is shown in Fig. 20.13 Using similar triangles AP DP = DR BR Hence AP 2.0 = 3.6 1.0 from which AP = (2.0)(3.6) 1.0 = 7.2cm The height of the large cone = 3.6 + 7.2 = 10.8 cm. Volume of frustum of cone = volume of large cone − volume of small cone cut off = 1 3 π(3.0) 2 (10.8) − 1 3 π(2.0) 2 (7.2) = 101.79 − 30.16 = 71.6 cm 3 4.0cm P E A D R B Q C 2.0cm 3.6cm 1.0cm 3.0cm 6.0cm Figure 20.13 Method 2 From above, volume of the frustum of a cone = 1 3 πh(R 2 + Rr + r 2 ) where R = 3.0cm, r = 2.0cm and h = 3.6cm Hence volume of frustum = 1 3 π(3.6)[(3.0) 2 + (3.0)(2.0) + (2.0) 2 ] = 1 3 π(3.6)(19.0) = 71.6 cm 3 Problem 17. Find the total surface area of the frustum of the cone in Problem 16 Method 1 Curved surface area of frustum = curved surface area of large cone—curved surface area of small cone cut off.
eBook - PDF- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
If a cone has a diameter of 80 mm and a perpendicular height of 120 mm, calculate its volume in cm 3 and its curved surface area 2. A square pyramid has a perpendicular height of 4 cm. If a side of the base is 2.4 cm long find the volume and total surface area of the pyramid 3. A sphere has a diameter of 6 cm. Determine its volume and surface area 4. A pyramid having a square base has a perpendicular height of 25 cm and a volume of 75 cm 3 Determine, in centimetres, the length of each side of the base 5. A cone has a base diameter of 16 mm and a perpendicular height of 40 mm. Find its volume correct to the nearest cubic millimetre 6. Determine (a) the volume, and (b) the surface area of a sphere of radius 40 mm 7. The volume of a sphere is 325 cm 3 . Deter- mine its diameter 8. Given the radius of the earth is 6380 km, calculate, in engineering notation (a) its surface area in km 2 and (b) its volume in km 3 9. An ingot whose volume is 1.5 m 3 is to be made into ball bearings whose radii are 8.0 cm. How many bearings will be produced from the ingot, assuming 5% wastage? 10. A spherical chemical storage tank has an internal diameter of 5.6 m. Calculate the storage capacity of the tank, correct to the nearest cubic metre. If 1 litre = 1000 cm 3 , determine the tank capacity in litres 14.3 Summary of volumes and surface areas of common solids A summary of volumes and surface areas of regular solids is given in Table 14.1. 14.4 Calculating more complex volumes and surface areas Here are some worked problems involving more complex and composite solids. Problem 16. A wooden section is shown in Figure 14.14 on page 152. Find (a) its volume, in m 3 , and (b) its total surface area (a) The section of wood is a prism whose end comprises a rectangle and a semicircle. Since the radius of the semicircle is 8 cm, the diameter is 16 cm. Hence the rectangle has dimensions 12 cm by 16 cm.
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Determine its volume. Round the answer to 2 decimal places. LATERAL AREAS AND SURFACE AREAS OF REGULAR PYRAMIDS AND RIGHT CIRCULAR CONES If necessary, use the tables in Appendix A for equivalent units of measure. Solve these problems. Where necessary, round the answers to 2 decimal places unless otherwise specified. 7. Find the lateral area of a regular pyramid that has a base perimeter of 92.0 inches and a slant height of 20.0 inches. 8. Find the lateral area of a right circular cone with a slant height of 1.80 meters and a base perimeter of 6.40 meters. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Unit 29 PYRAMIDS AND CONES: VOLUMES, SURFACE AREAS, AND WEIGHTS 727 9. A regular pyramid has a base perimeter of 58.4 centimeters, a slant height of 17.8 centimeters, and a base area of 213.16 square centimeters. Round the answers to a and b to 3 significant digits. a. Compute the lateral area of the pyramid. b. Compute the total surface area of the pyramid. 10. A right circular cone has a slant height of 4.250 feet, a base circumference of 18.84 feet, and a base area of 28.26 square feet. a. Find the lateral area of the cone. b. Find the total surface area of the cone. 11. A regular pyramid with a square base has a slant height of 10.00 inches. Each side of the base is 8.00 inches long. a. Find the lateral area of the pyramid. b. Find the total surface area of the pyramid. 12. A right circular cone with a slant height of 14.05 centimeters has a base diameter of 9.72 centimeters. Round the answers to a and b to 3 significant digits.
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