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FrustumsAn online exercise that focuses on calculating volume and surface area of frustums using similar shapes. |
This is level 5: Calculate the surface area of a frustum by finding the areas of the top, bottom and sloping sides.. You will be awarded a trophy if you get at least 3 answers correct and you do this activity online.
InstructionsTry your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file. |
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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician? Comment recorded on the 3 October 'Starter of the Day' page by S Mirza, Park High School, Colne: "Very good starters, help pupils settle very well in maths classroom." Comment recorded on the 1 February 'Starter of the Day' page by M Chant, Chase Lane School Harwich: "My year five children look forward to their daily challenge and enjoy the problems as much as I do. A great resource - thanks a million." |
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Go MathsLearning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school. Maths MapAre you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic. | ||
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If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows: |
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❎Level 1 - Use similar triangles to find the missing height or length of the original cone or pyramid before the top was cut off.
Level 2 - Calculate the volume of a frustum when the top dimensions, bottom dimensions and vertical height are all given.
Level 3 - Find the volumes of frustums with different base shapes, including cones, square-based pyramids and rectangular-based pyramids.
Level 4 - Work backwards from the volume of a frustum to find a missing length, height or radius.
Level 5 - Calculate the surface area of a frustum by finding the areas of the top, bottom and sloping sides.
Level 6 - Solve real-life problems involving frustums, including containers, cups, plant pots, lampshades and similar shapes.
Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).
More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.
Some questions have a hint that can help you find a way to solve the problem. If you see a button, click it to reveal the hint.
Internationally the two main approaches for finding the area of a frustum are:
1. Difference of cones/pyramids
Find the volume of the complete cone/pyramid and subtract the small one cut off:
$$V = \frac{1}{3}A_1H - \frac{1}{3}A_2h$$
where \(H\) and \(h\) are the heights of the large and small shapes respectively and \(A_1\) is the area of the base and \(A_2\) is the area of the top.
2. Standard frustum formula $$V = \frac{H}{3}(A_1 + A_2 + \sqrt{A_1 A_2})$$ where \(A_1\) is the area of the base and \(A_2\) is the area of the top and \(H\) is the vertical height of the frustum. For a cone this becomes: $$V = \frac{\pi H}{3}(R^2 + Rr + r^2)$$ where \(R\) is the radius of the base and \(r\) is the radius of the top.
Both give the same answer. Formula 1 is probably the most intuitive since it uses methods you already know. Formula 2 is the most commonly seen in textbooks.
Here are some more formulas that might help you with the questions about surface area.
The area of a trapezium is: $$A = \frac{1}{2}(a + b)h$$
where \(a\) and \(b\) are the lengths of the two parallel sides and \(h\) is the shortest (perpendicular) distance between them.
The surface area of a cone has two parts:
Curved (lateral) surface area $$A_{curved} = \pi r l$$
Total surface area (including the circular base) $$A_{total} = \pi r l + \pi r^2 = \pi r(l + r)$$
where \(r\) is the base radius and \(l\) is the slant height. If you are given the vertical height \(h\) instead of the slant height, use Pythagoras to find \(l\) first: $$l = \sqrt{r^2 + h^2}$$
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.
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