Volume and Surface Area of Cones and Spheres

What you’ll learn

Choose the correct shape and distinguish curved from total surface area.

ជ្រើសរូបមន្តកោណ និងស្វ៊ែរ ហើយបែងចែកផ្ទៃកោងពីផ្ទៃសរុប។

Meaning and use

A right circular cone has perpendicular height h and slant height l. The slant height is used for curved surface area, while perpendicular height is used for volume. A sphere uses one radius throughout. State whether a surface area includes the base.

កោណរង្វង់ត្រង់មានកម្ពស់កែង h និងកម្ពស់ទ្រេត l។ ផ្ទៃកោងប្រើ l ហើយមាឌប្រើ h។ ត្រូវបញ្ជាក់ថាផ្ទៃសរុបរួមបាតឬអត់។

Formulas

Right circular cone

\[V=\frac13\pi r^2h,\qquad A_{\mathrm{total}}=\pi rl+\pi r^2\]

Sphere

\[V=\frac43\pi r^3,\qquad A=4\pi r^2\]

Symbols and units

Symbol	Meaning	Unit or restriction
r	Radius	Length unit
h	Perpendicular cone height	Same length unit
l	Cone slant height	Same length unit
V, A	Volume and surface area	Cubic and square units, respectively

Conditions and limits

• Cone formulas assume a right circular cone.
• Total cone area includes one circular base; curved area alone is πrl.
• សម្រាប់កោណត្រង់ l²=r²+h² ប៉ុន្តែ h និង l មិនមែនដូចគ្នា។

Worked examples

Cone volume

A cone has radius 3 cm and perpendicular height 4 cm.

1. Use \(V=\frac13\pi r^2h\).
2. Calculate \(V=\frac13\pi\times9\times4=12\pi\) cm³.

\(12\pi\,\mathrm{cm^3}\)

កុំយកកម្ពស់ទ្រេតមកជំនួសកម្ពស់កែង។

Sphere area and volume

A sphere has radius 3 m.

1. Area \(A=4\pi\times3^2=36\pi\) m².
2. Volume \(V=\frac43\pi\times3^3=36\pi\) m³.

Area 36π m²; volume 36π m³

ទោះលេខស្មើគ្នាក្នុងឧទាហរណ៍នេះ បរិមាណ និងឯកតាខុសគ្នា។

Common mistakes

Instead of	Use	Why
Cone volume uses slant height l.	Cone volume uses perpendicular height h.	l is used in curved surface area.
Sphere area = 4πr³.	Sphere area = 4πr².	Area has two powers of length.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Find cone volume for r = 2 cm, h = 3 cm. Leave pi exact.
2. Find sphere area for r = 2 cm.
3. A cone has r = 3 cm and l = 5 cm. Find total surface area.

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 4π cm³

   One third × π × 4 × 3 = 4π.

2. 16π cm²

   4π × 2² = 16π.

3. 24π cm²

   Curved area 15π plus base area 9π.

Quick recap

• Use h for cone volume and l for curved area.
• State whether a base is included.
• Keep square and cubic units distinct.