Valume and Area

UTKALPRESS

VOLUME & AREA OF SOLID FIGURES — ARITHMETIC HANDBOOK

PAGE 1 — BASIC CONCEPTS, CUBE & CUBOID

A solid figure is a three-dimensional object having length, breadth and height. Examples include cube, cuboid, cylinder, cone, sphere and hemisphere.

Volume = Space occupied by a solid
Surface Area = Total area covered by the surfaces of a solid

1. Important Units

Measurement	Common Units
Length	cm, m, km
Area	cm², m², km²
Volume	cm³, m³, litres
Capacity	mL, L, kL

1 m³ = 1,000 L
1 L = 1,000 cm³
1 mL = 1 cm³
1 m³ = 10,00,000 cm³

2. Cube

A cube has six equal square faces and all its edges are equal. Let the side of the cube be a.

Volume = a³
Total Surface Area (TSA) = 6a²
Lateral Surface Area (LSA) = 4a²
Area of one face = a²
Diagonal of cube = a√3
Total length of all 12 edges = 12a

Example: Find the volume and total surface area of a cube whose side is 8 cm.

Volume = 8³ = 512 cm³
TSA = 6 × 8²
= 6 × 64
= 384 cm²

3. Cuboid

A cuboid is a rectangular solid with length l, breadth b and height h.

Volume = lbh
Total Surface Area = 2(lb + bh + hl)
Lateral Surface Area = 2h(l + b)
Diagonal = √(l² + b² + h²)

Example: A cuboid is 10 cm long, 6 cm broad and 4 cm high. Find its volume and total surface area.

Volume = 10 × 6 × 4 = 240 cm³
TSA = 2(10×6 + 6×4 + 4×10)
= 2(60 + 24 + 40)
= 248 cm²

Remember: Volume is measured in cubic units, while surface area is measured in square units.

UTKALPRESS

VOLUME & AREA OF SOLID FIGURES

PAGE 2 — CYLINDER, PRISM & HOLLOW CYLINDER

1. Cylinder

A cylinder has two equal circular bases and a curved surface. Let radius = r and height = h.

Volume = πr²h
Curved Surface Area (CSA) = 2πrh
Total Surface Area (TSA) = 2πr(h + r)
Area of two circular bases = 2πr²

Example: Find the volume of a cylinder with radius 7 cm and height 10 cm. Use π = 22/7.

Volume = 22/7 × 7 × 7 × 10
= 1540 cm³

2. Open Cylinder

If one circular end is open, only one base is included in the total surface area.

TSA of Cylinder Open at One End = 2πrh + πr²
= πr(2h + r)

If both ends are open:

Surface Area = 2πrh

3. Hollow Cylinder

A hollow cylinder has an outer radius R and inner radius r. If its height is h:

Volume of Material = πh(R² − r²)
Inner Curved Surface Area = 2πrh
Outer Curved Surface Area = 2πRh

Example: A pipe has outer radius 7 cm, inner radius 5 cm and length 10 cm. Find the volume of material used.

Volume = π × 10 × (7² − 5²)
= 10π(49 − 25)
= 240π cm³
≈ 754 cm³

4. Prism

A prism has two identical parallel bases. Its volume is obtained by multiplying the area of the base by its height.

Volume of Prism = Area of Base × Height
V = Bh

For a triangular prism:

Volume = ½ × Base of Triangle × Height of Triangle × Length of Prism

Example: The triangular base of a prism has base 8 cm and height 6 cm. The prism length is 15 cm.

Area of triangular base = ½ × 8 × 6 = 24 cm²
Volume = 24 × 15 = 360 cm³

Key idea: For many solids, remember Volume = Base Area × Height.

UTKALPRESS

VOLUME & AREA OF SOLID FIGURES

PAGE 3 — CONE, SPHERE & HEMISPHERE

1. Cone

A cone has a circular base and a vertex. Let radius = r, vertical height = h, and slant height = l.

Slant Height: l = √(r² + h²)

Volume = ⅓πr²h
Curved Surface Area = πrl
Total Surface Area = πr(l + r)

Example: A cone has radius 7 cm and height 24 cm. Find its volume and slant height.

l = √(7² + 24²)
= √(49 + 576)
= √625 = 25 cm

Volume = ⅓ × 22/7 × 7² × 24
= 1232 cm³

2. Sphere

A sphere is a perfectly round three-dimensional solid. Every point on its surface is at the same distance from its centre.

Surface Area of Sphere = 4πr²
Volume of Sphere = 4/3 πr³
Diameter = 2r

Example: Find the surface area and volume of a sphere of radius 3 cm.

Surface Area = 4π × 3² = 36π cm²
Volume = 4/3 × π × 3³
= 36π cm³

3. Hemisphere

A hemisphere is half of a sphere.

Curved Surface Area = 2πr²
Total Surface Area = 3πr²
Volume = 2/3πr³

Example: Find the volume of a hemisphere of radius 6 cm.

Volume = 2/3 × π × 6³
= 2/3 × π × 216
= 144π cm³

4. Important Comparison

Solid	Volume	Important Area
Cube	a³	6a²
Cuboid	lbh	2(lb+bh+hl)
Cylinder	πr²h	2πrh
Cone	⅓πr²h	πrl
Sphere	4/3πr³	4πr²
Hemisphere	2/3πr³	2πr² CSA

Important: In a cone, height h and slant height l are different quantities. Use h in the volume formula and l in the curved-surface formula.

UTKALPRESS

FORMULA SHEET • SOLVED EXAMPLES • PRACTICE

PAGE 4 — COMPLETE FORMULA REVISION

``` ```

Solid	Volume	Surface Area
Cube	a³	TSA = 6a² LSA = 4a²
Cuboid	lbh	TSA = 2(lb+bh+hl) LSA = 2h(l+b)
Cylinder	πr²h	CSA = 2πrh TSA = 2πr(h+r)
Cone	⅓πr²h	CSA = πrl TSA = πr(l+r)
Sphere	4/3πr³	4πr²
Hemisphere	2/3πr³	CSA = 2πr² TSA = 3πr²

SOLVED REVISION EXAMPLES

Example 1: A cube has volume 729 cm³. Find its side.

a³ = 729
a = ∛729
a = 9 cm

Example 2: Find the volume of a cuboid measuring 12 m × 5 m × 3 m.

V = 12 × 5 × 3
= 180 m³

Example 3: Find the curved surface area of a cylinder with radius 7 cm and height 15 cm. Use π = 22/7.

CSA = 2πrh
= 2 × 22/7 × 7 × 15
= 660 cm²

Example 4: Find the total surface area of a sphere with radius 7 cm.

TSA = 4πr²
= 4 × 22/7 × 49
= 616 cm²

PRACTICE QUESTIONS

1. Find the volume of a cube whose side is 6 cm.

2. Find the total surface area of a cube whose side is 10 cm.

3. Find the volume of a cuboid measuring 15 cm × 8 cm × 4 cm.

4. Find the volume of a cylinder with radius 7 cm and height 12 cm. Use π = 22/7.

5. Find the curved surface area of a cylinder with radius 5 cm and height 14 cm. Use π = 22/7.

6. Find the volume of a cone with radius 7 cm and height 12 cm. Use π = 22/7.

7. Find the surface area of a sphere of radius 7 cm. Use π = 22/7.

8. Find the volume of a hemisphere of radius 3 cm. Express the answer in terms of π.

9. A cuboid has length 20 cm, breadth 10 cm and height 5 cm. Find its total surface area.

10. A cylinder has radius 7 cm and height 10 cm. Find its total surface area. Use π = 22/7.

ANSWER KEY

``` 1. 216 cm³
2. 600 cm²
3. 480 cm³
4. 1848 cm³
5. 440 cm²
6. 616 cm³
7. 616 cm²
8. 18π cm³
9. 700 cm²
10. 748 cm² ```

EXAM SHORTCUTS & KEY POINTS

1. Volume is always expressed in cubic units such as cm³ or m³.

``` 2. Surface area is expressed in square units such as cm² or m².

3. If diameter is given, first calculate r = d/2.

4. For a cone, remember l² = r² + h².

5. For a hemisphere, curved surface area is 2πr², while total surface area is 3πr².

6. Always check whether the question asks for CSA, LSA, TSA or Volume.

7. Convert all dimensions to the same unit before applying a formula. ```