Knowledge organisers

Surface Areas and Volumes: CBSE Class 10 Maths knowledge organiser

Everything to know about surface areas and volumes on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Combination of solids: A solid made by joining simple solids such as a cone on a cylinder.
Total surface area: The area of every surface you can touch on the outside of the solid.
Volume of a combination: The sum of the volumes of the parts, minus any part scooped out.

Key formulas

Cuboid \(l,b,h\)	\(\text{TSA}=2(lb+bh+hl),\) \(\text{LSA}=2h(l+b),\) \(V=lbh\)
Cube, edge \(a\)	\(\text{TSA}=6a^2,\) \(V=a^3\)
Cylinder	\(\text{CSA}=2\pi rh,\) \(\text{TSA}=2\pi r(r+h),\) \(V=\pi r^2h\)
Cone, \(l=\sqrt{r^2+h^2}\)	\(\text{CSA}=\pi rl,\) \(\text{TSA}=\pi r(l+r),\) \(V=\tfrac13\pi r^2h\)
Sphere	\(\text{SA}=4\pi r^2,\) \(V=\tfrac43\pi r^3\)
Hemisphere	\(\text{CSA}=2\pi r^2,\) \(\text{TSA}=3\pi r^2,\) \(V=\tfrac23\pi r^3\)
Combined solids: add only the surfaces you can see; volumes simply add (or subtract)

Worked example

A toy is a cone of radius 4.2 cm on a hemisphere of the same radius. The total height is 9.8 cm. Find its total surface area \(\left(\pi=\frac{22}{7}\right)\).

Cone height \(=9.8-4.2=5.6\), \(l=\sqrt{5.6^2+4.2^2}=7\). TSA \(=\pi rl+2\pi r^2=92.4+110.88=203.28\ \text{cm}^2\).

Common mistakes

Adding the total surface areas of the parts and so counting the hidden joined faces.
Using the vertical height \(h\) in \(\pi rl\) instead of the slant height.
Using the diameter as the radius, or mixing units (cm and m, mm and cm).
For a vessel, "inner surface area" excludes the open top.

You should be able to…

Use the surface area and volume formulae of the cube, cuboid, cylinder, cone, sphere and hemisphere.
Find the surface area and volume of solids made from two shapes, deciding which faces are exposed.
Write complete 5-mark answers on combined solids and real objects (tents, toys, vessels) with units and conversions.
Handle drilled holes, hollow objects and the 'largest cone in a cube' questions that decide 95+.

The printable sheet

Surface Areas and Volumes knowledge organiser for CBSE Class 10 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Surface Areas and Volumes knowledge organiser (CBSE Class 10 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 10 Maths topics: Real Numbers · Polynomials · Pair of Linear Equations in Two Variables · Quadratic Equations · Arithmetic Progressions · Triangles · Coordinate Geometry · Introduction to Trigonometry · Some Applications of Trigonometry · Circles · Areas Related to Circles · Statistics · Probability · All CBSE Class 10 Maths organisers