Volume of a cone · 3 marks

A right triangle ABC with sides 5 cm, 12 cm and 13 cm is rotated through 360° about the side with length 12 cm. Find the volume of the solid so obtained.

Answer: The solid is a cone with height 12 cm and radius 5 cm: volume = 13π × 5² × 12 = 100π cm³ (≈ 314.3 cm³ with π = 227).

Step-by-step solution

Given: Right triangle with sides 5 cm, 12 cm, 13 cm; Rotated about the 12 cm side
To find: Volume of the solid formed

Idea: Turning a right triangle about one of its perpendicular sides sweeps out a cone: that side becomes the height, the other perpendicular side becomes the radius, and the hypotenuse becomes the slant height.

5² + 12² = 25 + 144 = 169 = 13², so the triangle is right-angled with perpendicular sides 5 cm and 12 cm and hypotenuse 13 cm.1 mark
Rotating about the 12 cm side makes a right circular cone with height h = 12 cm and radius r = 5 cm.1 mark
Volume = 13πr²h = 13 × π × 5² × 12 = 100π cm³ (≈ 314.29 cm³).1 mark

Volume = 100π cm³ ≈ 314.29 cm³.

Check: 13 × 25 × 12 = 100 ✓; 100 × 227 ≈ 314.29 ✓.

Answer to write in the exam

5² + 12² = 169 = 13² ⇒ right angle between 5 cm and 12 cm sides

Cone: h = 12 cm, r = 5 cm

V = 13π × 5² × 12

∴ V = 100π cm³ ≈ 314.29 cm³

Common mistakes that cost marks

Taking the radius as 12 and height as 5: the side you rotate about is the height.
Using 13 cm (the hypotenuse) as the height; it is the slant height.

How this can come in the exam

MCQ (1 mark)

A right triangle with legs 3 cm and 4 cm is rotated about the 3 cm side. The volume of the solid is

12π cm³
16π cm³
48π cm³
9π cm³

Show answer

(B) 16π cm³
Cone r = 4, h = 3: 13π × 16 × 3 = 16π cm³.

Try one yourself

A right triangle with sides 7 cm, 24 cm, 25 cm is rotated about the 24 cm side. Find the volume. (π = 227)