GCSE: Volumes and Surface Area Dr J Frost ([email protected]) www.drfrostmaths.com GCSE Revision Pack Reference: 132, 133, 134, 135, 136i, 136ii, 138 Last modified: 31st August 2015 GCSE Specification 132. Know and use formulae to calculate the surface areas and volumes of cuboids and right-prisms. 133. Find the volume of a cylinder and surface area of a cylinder. 134. Find the surface area and volume of cones, spheres and hemispheres. 135. Find the volume of a pyramid. 136i. Solve a range of problems involving surface area and volume, e.g. given the volume and length of a cylinder find the radius. 136ii. Solve problems in which the surface area or volume of two shapes is equated. 138. Solve problems involving more complex shapes and solids, including (segments of circles and) frustums of cones. ! Dont write these down yet. All the GCSE formulae for 3D shapes (The * indicates ones that wont be in your formula booklet) r r
l h r ? ? ? ? ? * Bro Tip: Roll out the cylinder to work out the area of the curved surface. h ? Bro Tip: The same formula applies to the cone. * Area of curved surface ? SKILL #1: Volumes of Prisms
! Volume of prism = Area of cross section ? length Test Your Understanding ? And what is the surface area? (Hint: youll need Pythagoras) ? ? SKILL #2: Volumes of Cylinders Noting that a cylinder is just a circular prism: ? By making a vertical slit and folding out the curved surface of the cylinder so that it is rectangular:
? ? Test Your Understanding 3 10 Volume = Surface Area = ? ? 6 5 Give your answers in terms of : Volume = ? Surface Area = ? Exercise 1 6 ? ? 4
3 2 1 i) The prism is made of metal of density 6.6g/cm3. Find its mass. ? ? 7 10 2 4 4 8 7 =272 ? 3 3 ? =40=125.66 ?
ii) Surface Area? 5 7 6 To POO [Real world example] A sewage treatment centre fills a cylindrical silo with waste. The diameter is 20m and the height 5m. It is full to the top with 1300kg of waste. Find the density of the waste. ? From Ali 20 Santa Santa wants to wrap a cylindrical present for
Ali, with dimensions as shown above. It costs 0.24p per cm2 of wrapping paper. Determine the cost to wrap the present. 2.85 ? [Edexcel] The pond is completely full of water. Sumeet wants to empty the pond so he can clean it. Sumeet uses a pump to empty the pond. The volume of water in the pond decreases at a constant rate. The level of the water in the pond goes down by 20cm in the first 30 minutes. Work out how much more time Sumeet has to wait for the pump to empty the pond completely. (6 marks) 0.4m3 emptied in first 30 minutes. So 0.8m 3 emptied per hour. Total volume = 1.8m3 ? SKILL #3: Spheres and Hemispheres Give your answers in terms of . Volume = ? 3cm ?
Surface Area = For a Sphere: ? (from formula sheet) ? Test Your Understanding Leave your answers in terms of . 2m 10m ? ? ? ? Exercise 2 Give your answers in terms of unless where specified. 1 2 Mr Wutang and his clan eat from a full (thin) hemispherical bowl of rice, a bowl with diameter 18cm. He eats 400g. What is the density
of the rice to 3sf? (in g/cm3) 6 Volume = 144? Surface Area = 108 ? 4 18cm 6cm Volume = 486 Density = 400 / 486 = 0.262g/cm3 ? A hemispherical bowl with radius 18cm, with a rim of width 6cm. Volume = 3888 1152 = 2736 ? Surface Area = 1296 + 324 144 = 1476
? 5 3 42m Volume = 6174 ? m3 Surface Area = 1764 ? m2 What radius is needed for a hemisphere so that the volume is 18 m3? ? SKILL #4: Volumes of Pyramids In general: ? Bro Exam Tip: This one is not given in the formula booklet!
Quickfire examples: ? = = ? A* Question 50 52 Length of bottom diagonal (by Pythagoras) ? Height of pyramid = (again by Pythagoras) ? Volume =
? Test Your Understanding Q = + = Volume ? Q Determine the volume of a pyramid with a rectangular base of width 6cm and length 8cm, and a slant height of 13cm (your answer should turn out to be a whole number). ? Exercise 3 2 1 3
= ? =? 4 h= ? ? 5
= ? ? The implication is that if we chop a cube across its face diagonals, we have something 6 times as small. 6
? ? ? ? SKILL #4: Cones and Frustums Noting that a cone is just a circular-based pyramid: 1 2 = ? h 3
= ? (where is the slant height) Example ? = = ? 4 3 Frustum ! A frustum is a cone with part of the top chopped off. 1 12 Volume = ? 9 4
Your Go... Volume For a Cone: (Hint: youll need to work out the radius of the top circle, perhaps by similar triangles?) ? 2 8 12 Exercise 4 2 1 3 3 8
12 12 4 5 Volume Surface Area ? ? ? 12 Volume = 96? Surface Area = 96? 6 4 3 The density of ice cream is
1.09g/cm3. I fill a cone with ice cream plus a hemispherical piece on top. What is the mass of the ice cream? 8 5 12 5 6 ? ? 4 ? SKILL #5: Finding values of variables Sometimes the volume and surface area is already given, and you need to find the value of some variable, e.g. radius or height. Example A cylinder has a height of 10m and a volume of . What is its radius?
10 ? Check Your Understanding Q [Edexcel] A dog tin is cylindrical in shape with the indicated lengths. The manufacturer wants to make a new tin with the same volume but a radius of 5.8cm. What height should they make the tin? ? Q The Earth has a volume of What is the radius of the Earth? ? Exercise 5 1 10 2 3
=? SKILL #6: Preserved volume or surface area 3km That aint no moon Chewie Darth Vader decides he doesnt like the shape of his Death Star, so melts it down and rebuilds it using the same amount of material to form a Death Cube. What is the side length x of his Death Cube? 4 3 3 ?3 = 3 Bro Tip: Find the volume of each, equate them, then simplify. Further Example ? Check Your Understanding
Sphere melted to form cone. Express in terms of . ? h This time the surface areas are the same (and should be equated before simplifying) A solid hemisphere with radius has the same surface area as a cylinder with radius and height . Determine the height of the cylinder in terms of . ? Exercise 6 1 A sphere with radius is melted to form a cylinder of radius and height . Determine in terms of . 4
A sphere with radius has the same surface area as a cylinder with radius and height . Find in terms of . ? ? 2 A squared-based pyramid with base of side and height is melted to form a cube of side . Determine in terms of . A hemisphere of radius is melted to form a cone of radius and height . Determine in terms of . ? 3 5 [Edexcel] Pictured are a solid cone and a solid hemisphere. The surface area of the cone is equal to the surface area of the hemisphere. Express in terms of . (Hint: youll need Pythag to find slant height)
? ? GCSE Specification 132. Know and use formulae to calculate the surface areas and volumes of cuboids and right-prisms. 133. Find the volume of a cylinder and surface area of a cylinder. 134. Find the surface area and volume of cones, spheres and hemispheres. 135. Find the volume of a pyramid. 136i. Solve a range of problems involving surface area and volume, e.g. given the volume and length of a cylinder find the radius. 136ii. Solve problems in which the surface area or volume of two shapes is equated. 138. Solve problems involving more complex shapes and solids, including (segments of circles and) frustums of cones.
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