Bihar Board Class 10 · Maths · Matric exam 2027
Chapter 13: Surface Areas and Volumes — notes
The chapter’s 5 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. Combining basic solids
Okay, so, from Class IX, you already know a few basic solids.
- • Basic solids: cuboid, cone, cylinder, sphere, hemisphere
- • Combination solids: made of two or more basic solids
- • Example: oil tanker = cylinder + 2 hemispheres
- • Example: test tube = cylinder + hemisphere
2. Surface area of a combination of solids
Okay, so when two solids are joined together, something interesting happens.
- • TSA of combination = sum of CURVED surfaces only
- • Tanker: TSA = CSA hemisphere + CSA cylinder + CSA hemisphere
- • Toy (cone+hemisphere): TSA = CSA hemisphere + CSA cone
- • Rasheed's top: height 5 cm, diameter 3.5 cm, r = 1.75 cm
- • Cone height = 5 − 1.75 = 3.25 cm, l = 3.7 cm (approx)
- • Area to colour ≈ 39.6 cm²
3. Volume of a combination of solids
Okay, so let's talk about volume now.
- • Volume of combination = sum of volumes of parts (nothing is removed)
- • Shed: volume = volume of cuboid + 1/2 volume of cylinder
- • Cuboid: 15 m x 7 m x 8 m; half-cylinder: diameter 7 m, length 15 m
- • Volume of air (empty shed) = 1128.75 m³
- • Machinery = 300 m³, Workers = 20 x 0.08 = 1.6 m³
- • Air with machinery & workers = 1128.75 − 301.6 = 827.15 m³
4. Converting one solid shape into another
Okay, so, sometimes we melt a solid and reshape it into another shape.
- • When reshaped: volume before = volume after
- • Example 8: cone (h = 24 cm, r = 6 cm) melted into a sphere
- • Volume of cone = (1/3)πr²h
- • Volume of sphere = (4/3)πr³
- • r³ = 3x3x24 = 3³x2³ , so r = 6
- • Radius of new sphere = 6 cm
5. Frustum of a cone
Okay, so take a right circular cone.
- • Frustum: cone sliced by plane parallel to base, top cone removed
- • Has two circular ends of different radii r1, r2
- • l = √(h² + (r1 − r2)²)
- • Volume = (1/3)πh(r1²+r2²+r1r2)
- • CSA = π(r1+r2)l
- • TSA = πl(r1+r2) + πr1² + πr2²
Summary
Okay, so let's look back at everything we covered in this chapter. See, we started by noticing that everyday objects, like oil tankers and test tubes, aren't just one shape. They're combinations of basic solids, like cuboids, cones, cylinders, spheres and hemispheres. Then we found the total surface area of such a combination. We only add up the curved surfaces we can actually see, because the joined flat faces disappear. You saw that in Rasheed's top, and in the decorative block too. Now, here's something interesting — volume doesn't behave like that. For volume, we simply add up the volumes of all the parts, since no space gets lost when solids are joined. Remember Shanta's shed, and the solid toy? Both used this idea. After that, we looked at converting one solid shape into another, like reshaping a clay cone into a sphere, and the key rule there was — the volume stays exactly equal before and after. And finally, we met the frustum of a cone. You get it by cutting a cone with a plane parallel to its base, and removing the smaller cone on top. We also learned its formulas, for volume, curved surface area, and total surface area. So, tell me — out of all these, which part did you find the most interesting?
AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 13. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Surface Areas and Volumes" notes as a PDF?
- Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
- Where are these notes written from?
- From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.
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