Surface Areas and Volumes · Class 10 Mathematics: 44 flashcards
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1. How do you find the surface area of a combined solid?
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Add the surfaces that remain visible; the faces joined together are not counted.
2. Why must you not add the two total surface areas?
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Because that counts the two faces buried at the join.
3. Does joining two solids raise or lower the total surface area?
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Lowers it — two faces disappear.
4. Give the curved and total surface areas of a cylinder.
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Curved $2\pi rh$; total $2\pi r(r+h)$.
5. Give the curved and total surface areas of a cone.
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Curved $\pi r l$; total $\pi r(l+r)$.
6. How is the slant height found?
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$l = \sqrt{r^2 + h^2}$, by Pythagoras.
7. A cone has radius 7 and vertical height 24. Find its slant height.
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$\sqrt{49 + 576} = 25$.
8. Give the surface area of a sphere.
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$4\pi r^2$.
9. Give the curved and total surface areas of a hemisphere.
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Curved $2\pi r^2$; total $3\pi r^2$.
10. Which hemisphere figure applies when it is stuck onto another solid?
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$2\pi r^2$ — only the dome is visible.
11. Give the surface area of a cube of edge $a$.
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$6a^2$.
12. Give the surface area of a cuboid.
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$2(lb + bh + hl)$.
13. A capsule is a cylinder of radius 3.5 mm, length 8 mm, with hemispherical ends. Find its surface area.
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$2\pi rh + 4\pi r^2 = 176 + 154 = 330$ mm².
14. What do two hemispheres on the ends of a cylinder contribute?
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$4\pi r^2$ — exactly one sphere's surface.
15. A tent is a cylinder r = 7, h = 10 with a cone of height 24 on top. Find the canvas.
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$\pi r l + 2\pi r h = 550 + 440 = 990$ m², with $l = 25$.
16. What happens to the surface area when a hemisphere is scooped out of a face?
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It increases by $\pi r^2$: a circle is lost and a bowl of $2\pi r^2$ is gained.
17. When a cavity is made, is the whole face removed?
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No — only a circle of area $\pi r^2$ is subtracted from it.
18. What three questions should you ask before writing any formula?
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Which surfaces can I touch? Which circles has the join buried? Is anything open?
19. What does the word 'open' in a question change?
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One more circular face is left out of the total.
20. In what units is surface area given?
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Square units — cm², m².
21. Why do questions use radii of 7, 14 or 3.5?
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Because $\frac{22}{7}$ then cancels and the answers are whole numbers.
22. A cone has radius 5 and height 12. Find its curved surface area in terms of π.
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$l = 13$, so the curved surface is $65\pi$.
23. How is the volume of a combined solid found?
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By adding the volumes of the parts — nothing is lost at a join.
24. Why do volumes add while surfaces do not?
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A join hides two faces but occupies no space, so surface is lost and volume is not.
25. Give the volume of a cylinder.
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$\pi r^2 h$.
26. Give the volume of a cone.
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$\frac13 \pi r^2 h$, with $h$ the vertical height.
27. How many cones fill the cylinder on the same base with the same height?
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Exactly three.
28. Give the volume of a sphere.
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$\frac43 \pi r^3$.
29. Give the volume of a hemisphere.
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$\frac23 \pi r^3$.
30. Give the volumes of a cube and a cuboid.
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$a^3$ and $lbh$.
31. Which height does a cone's volume use?
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The vertical height — never the slant height.
32. State the volume ratio of cone, hemisphere and cylinder on the same base and height.
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$1 : 2 : 3$.
33. A capsule is a cylinder r = 3.5 mm, length 8 mm, with hemispherical ends. Find its volume.
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$308 + \frac{539}{3} \approx 487.67$ mm³.
34. A tent is a cylinder r = 7, h = 10 with a cone of height 24 on top. Find its air space.
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$1540 + 1232 = 2772$ m³.
35. How do you find the material left after a cavity is hollowed out?
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Subtract the cavity's volume from the whole solid's.
36. A 7 cm cube has a hemisphere of radius 3.5 cm scooped out. How much wood remains?
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$343 - \frac{539}{6} \approx 253.17$ cm³.
37. What does a cavity do to the volume and to the surface area?
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It lowers the volume but raises the surface area.
38. How many cubic centimetres are in a litre?
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1000.
39. How many litres are in a cubic metre?
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1000.
40. A tank holds 7700 cm³. What is its capacity in litres?
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7.7 litres.
41. Given an outer radius R and an inner radius r, what volume does the material occupy?
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$\pi h (R^2 - r^2)$.
42. For that same container, what is its capacity?
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The inner volume alone, $\pi r^2 h$.
43. In what units is volume measured?
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Cubic units — cm³, m³.
44. Why must a sphere's volume use the radius rather than the diameter?
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Because the formula cubes it; using the diameter makes the answer eight times too large.
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