Grade 11 Surface Area of Solids: Short Notes

11 ශ්‍රේණිය · 1 වාරය · 4 පරිච්ඡේදය · ඝන වස්තුවල පෘෂ්ඨ වර්ගඵලය

18%

of Grade 11 term tests include it (11 of 61)

marks in past papers (14 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Find the total surface area of a right pyramid with a square base
Find the curved and total surface area of a right circular cone
Find the radius and height of a cone made by folding a sector
Find the surface area of a sphere and a hemisphere
Use Pythagoras to find a slant height or a vertical height when needed

Painting the dagoba

Before Vesak, the dayakayas of a village temple want to whitewash the dome of the small dagoba. The paint seller asks, 'How many square metres?' The dome is close to half a sphere. Today you'll learn to find the outside area of domes, cones and pyramids, and the dagoba gets its paint by the end.

What is surface area?

The surface area of a solid is the total area of all its outside surfaces, in cm² or m².• Flat faces: find each area and add (think of the solid's net opened flat).• A cone, sphere or hemisphere has a curved surface, with its own formula.Total surface area = curved part + any flat faces.

A square-based pyramid

Base: a square of side a. Four equal triangular faces. l = slant height (height of a triangular face). h = vertical height from the top to the centre O of the base. Surface area uses l, not h.

🧮 Total surface area of a square pyramid

Square base plus four triangles, each ½ × base × slant height. If only h is given, use the right triangle with sides h, a/2 and l: l² = h² + (a/2)².

A = a² + 4 × ½ × a × l = a² + 2al

✏️ A Vesak kudu with a pyramid top

The top of a Vesak lantern is a square pyramid with base side 10 cm and slant height 13 cm. Find (i) the area of paper on the whole pyramid, base included, (ii) its vertical height.

1Base = 10 × 10 = 100 cm².
2Four triangles = 4 × ½ × 10 × 13 = 260 cm².
3(i) Total = 100 + 260 = 360 cm².
4(ii) h² = 13² − 5² = 169 − 25 = 144, so h = 12 cm.

✅ (i) 360 cm² (ii) 12 cm

A right circular cone

r = base radius, h = vertical height, l = slant height (top to the edge of the base). They form a right triangle, so l² = r² + h².

🧮 Surface area of a cone

Curved surface = πrl. A solid cone also has a circular base, so total = πrl + πr². An open cone (a party hat, a funnel) has only the curved part.

πrl πrl + πr²

✏️ A solid cone

A solid cone has base radius 7 cm and vertical height 24 cm. Find its total surface area. (π = 22/7)

1l² = 7² + 24² = 49 + 576 = 625, so l = 25 cm.
2Curved = 22/7 × 7 × 25 = 550 cm².
3Base = 22/7 × 7 × 7 = 154 cm².
4Total = 550 + 154 = 704 cm².

✅ 704 cm²

Folding a sector into a cone

Roll a paper sector until its two straight edges meet.• The sector's radius becomes the cone's slant height l.• The sector's arc goes round the base: arc length = 2πr.• The sector's area becomes the curved surface area.So find the arc, set it equal to 2πr, and solve for r.

⚠️ The sector's radius is not the base radius

In one O/L paper only 13% of candidates found the radius of a cone made from a sector. Most took the sector's radius as the cone's radius. Sketch the cone first: the sector's radius runs from the tip to the base edge, so it is the slant height. The base radius comes from arc = 2πr.

🧮 Sphere and hemisphere

That is a whole sphere. A hemisphere's curved surface is half: 2πr². A solid hemisphere also has a flat circle, πr², so its total is 3πr². A hollow bowl has the curved part only.

A = 4πr²

⚠️ Which surfaces are really outside?

Before adding, ask which surfaces you can actually touch.• Solid hemisphere: 3πr², not 2πr².• A cone sitting on a hemisphere: the joining circle is inside, so add only πrl + 2πr².• Never use the vertical height h in πrl or ½ × a × l: you need the slant height.

📌 Remember

• Square pyramid: a² + 2al (l = slant height).• Cone: curved πrl, total πrl + πr², with l² = r² + h².• Sector to cone: sector radius = l, arc = 2πr.• Sphere 4πr²; hemisphere curved 2πr², solid 3πr².• Add only the surfaces on the outside.

🧾 Quick revision sheet

Total surface area of a square pyramid	a² + 2al
Curved surface area of a cone	πrl
Total surface area of a solid cone	πrl + πr²
Link between r, h and l in a cone	l² = r² + h²
Surface area of a sphere	4πr²
Total surface area of a solid hemisphere	3πr² (curved 2πr² + flat πr²)
Sector folded into a cone: what becomes the slant height?	The sector's radius. Its arc becomes the base circumference.