Volume- Pyramids, Cones - Spheres online exercise for | Live ... - Free Printable

Let’s solve each problem step by step. We’ll use the formulas for volume:

- Pyramid: Volume = (1/3) × Base Area × Height
- Cone: Volume = (1/3) × π × r² × h
- Sphere: Volume = (4/3) × π × r³

We’re told to use π = 3.14 where needed.

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Section A

#### a) Rectangular pyramid: base 6cm × 7cm, height 5cm
Base area = 6 × 7 = 42 cm²
Volume = (1/3) × 42 × 5 = (1/3) × 210 = 70 cm³

#### b) Cone: diameter 10cm → radius = 5cm, height 12cm
Volume = (1/3) × 3.14 × 5² × 12
= (1/3) × 3.14 × 25 × 12
First, 25 × 12 = 300
Then, 3.14 × 300 = 942
Then, 942 ÷ 3 = 314 cm³

#### c) Square pyramid: base 3cm × 3cm, height 4cm
Base area = 3 × 3 = 9 cm²
Volume = (1/3) × 9 × 4 = (1/3) × 36 = 12 cm³

#### d) Rectangular pyramid: base 8cm × 11cm, height 3cm
Base area = 8 × 11 = 88 cm²
Volume = (1/3) × 88 × 3 = (1/3) × 264 = 88 cm³

*(Note: The 3 and 1/3 cancel out — so it’s just 88)*

#### e) Cone: radius 6cm, height 11cm
Volume = (1/3) × 3.14 × 6² × 11
= (1/3) × 3.14 × 36 × 11
First, 36 × 11 = 396
Then, 3.14 × 396 = let’s calculate:
3 × 396 = 1188
0.14 × 396 = 55.44
Total = 1188 + 55.44 = 1243.44
Now divide by 3: 1243.44 ÷ 3 = 414.48 cm³

#### f) Square pyramid: base 8cm × 8cm, height 6cm
Base area = 8 × 8 = 64 cm²
Volume = (1/3) × 64 × 6 = (1/3) × 384 = 128 cm³

#### g) Sphere: radius 8cm
Volume = (4/3) × 3.14 × 8³
8³ = 512
So, (4/3) × 3.14 × 512
First, 3.14 × 512 = let’s compute:
3 × 512 = 1536
0.14 × 512 = 71.68
Total = 1536 + 71.68 = 1607.68
Now multiply by 4/3:
(4 × 1607.68) ÷ 3 = 6430.72 ÷ 3 = 2143.573... ≈ 2143.57 cm³ (to 2 d.p.)

#### h) Sphere: diameter 6cm → radius = 3cm
Volume = (4/3) × 3.14 × 3³
3³ = 27
So, (4/3) × 3.14 × 27
First, 3.14 × 27 = 84.78
Then, (4/3) × 84.78 = (4 × 84.78) ÷ 3 = 339.12 ÷ 3 = 113.04 cm³

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Section B

#### a) Cone: height = 15cm, volume = 270 cm³, find radius x

Formula: V = (1/3)πr²h
Plug in known values:
270 = (1/3) × 3.14 × x² × 15

Simplify right side:
(1/3) × 15 = 5
So: 270 = 5 × 3.14 × x²
→ 270 = 15.7 × x²

Now solve for x²:
x² = 270 ÷ 15.7 ≈ 17.197...

Take square root:
x ≈ √17.197 ≈ 4.147...

Round to 3 significant figures: 4.15 cm

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Final Answer:

Section A:
a) 70
b) 314
c) 12
d) 88
e) 414.48
f) 128
g) 2143.57
h) 113.04

Section B:
a) 4.15