Grade 10 Surface Area and Volume: Short Notes

10 ශ්‍රේණිය · 28 පරිච්ඡේදය · පෘෂ්ඨ වර්ගඵලය හා පරිමාව

65%

of Grade 10 term tests include it (35 of 54)

403

marks in past papers (77 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Find the curved and total surface area of a cylinder
Find the surface area of a triangular prism from its faces
Find the volume of a cylinder and of a prism, and convert to litres
Work back from a volume or an area to a missing length
Solve melting and water-level problems using equal volumes

The tank on the roof

A house in Matale has a round water tank on its roof. To paint it you need its surface area. To know if it will last the family through a water cut, you need its volume in litres. One shape, two questions, and both are in this lesson.

Open up a cylinder

A cylinder has two circular faces and a curved surface. Peel off the label of a milk tin and it lays flat as a rectangle: its length is the circumference 2πr and its width is the height h.

🧮 Surface area of a cylinder

Curved surface = the rectangle 2πr × h. A closed cylinder adds two circles, 2πr². A cylinder open at one end (a tank with no lid, a cup) has only one circle: πr² + 2πrh.

Curved = 2πrh Total (closed) = 2πr² + 2πrh

✏️ A closed cylinder

A closed cylinder has radius 7 cm and height 10 cm. Find (i) its curved surface area (ii) its total surface area. (π = 22/7)

1Curved = 2 × 22/7 × 7 × 10 = 440 cm²
2One circle = 22/7 × 7 × 7 = 154 cm², so two circles = 308 cm²
3Total = 440 + 308 = 748 cm²

✅ (i) 440 cm² (ii) 748 cm²

✏️ Rolling a sheet into a tube

A rectangular sheet 44 cm by 20 cm is rolled so that the 44 cm edges just meet, making a tube 20 cm high. Find the radius of the tube.

1The 44 cm edge wraps round once, so it becomes the circumference: 2πr = 44.
22 × 22/7 × r = 44, so 44r/7 = 44
3r = 7 cm

✅ 7 cm

⚠️ Which surfaces are there?

In past papers about a quarter of candidates scored zero on 2πrh questions. Before using a formula, ask:• Curved surface only (label, pipe)? → 2πrh• Closed? → add two circles• Open at the top? → add one circleAnd if a diameter is given, halve it first.

A triangular prism

A prism has the same cross-section all along its length. A triangular prism has 2 congruent triangles and 3 rectangles. Each rectangle is one side of the triangle × the length of the prism.

⚠️ Getting the faces wrong

In O/L papers almost half the candidates picked a wrong shape as a face of a right triangular prism. Remember: only 2 triangles (the two ends) and 3 rectangles. The rectangles are not all the same unless the triangle is equilateral: their widths are the three sides of the triangle.

🧮 Volume

For any prism, volume = area of cross-section × length. A cylinder is a 'circular prism', so V = πr² × h.Capacity: 1000 cm³ = 1 litre, and 1 m³ = 1000 litres.

V = A × l Cylinder: V = πr²h

Melting and pouring: volume stays the same

When a metal solid is melted and recast, or water is poured from one container to another, the volume does not change.So write: volume before = volume after, and solve for the missing length.When a solid is dropped into water, the water rises by a volume equal to the solid's volume.

⚠️ Square units or cube units?

Area is in cm² (two lengths multiplied); volume is in cm³ (three lengths). Mixing them up, or writing 1 litre = 100 cm³, loses easy marks. Also, when solving 22/7 × r² × h = V, cancel step by step and write each line: most lost marks here are simplification slips.

📌 Remember

• Cylinder: curved surface 2πrh; closed total 2πr² + 2πrh; open top πr² + 2πrh.• Triangular prism: 2 triangles + 3 rectangles (side × length).• Volume of a prism = cross-section area × length; cylinder πr²h.• 1000 cm³ = 1 litre.• Melting, pouring, sinking: volume before = volume after.

🧾 Quick revision sheet

Curved surface area of a cylinder	2πrh
Total surface area of a closed cylinder	2πr² + 2πrh
Faces of a triangular prism	2 congruent triangles and 3 rectangles
Volume of a prism	Area of cross-section × length
Volume of a cylinder	πr²h
1 litre = ? cm³	1000 cm³ (and 1 m³ = 1000 litres)
A rectangle is rolled into a cylinder. The edge that wraps round becomes…	the circumference, 2πr