A right triangle with sides 6 cm, 8 cm and 10 cm is rotated through 360° about the side of 8 cm. Find the volume and the curved surface area of the solid so formed.
Step-by-step solution
To find: Volume and CSA of the solid
Idea: Rotating about the 8 cm side makes a cone: 8 cm is the height, the other leg 6 cm is the radius, and the hypotenuse 10 cm is the slant height.
- 62 + 82 = 100 = 102, so the right angle is between the 6 cm and 8 cm sides. Cone: h = 8 cm, r = 6 cm, l = 10 cm.1 mark
- Volume = 13πr2h = 13 × π × 36 × 8 = 96π ≈ 96 × 3.14 = 301.44 cm3.1 mark
- CSA = πrl = π × 6 × 10 = 60π ≈ 188.4 cm2.1 mark
Check: With π = 227: V ≈ 301.71 cm3 and CSA ≈ 188.57 cm2, close to the values above ✓.
Answer to write in the exam
Cone: h = 8 cm, r = 6 cm, l = 10 cm
V = 13π × 62 × 8 = 96π ≈ 301.44 cm3
CSA = π × 6 × 10 = 60π
∴ V ≈ 301.44 cm3, CSA ≈ 188.4 cm2
Common mistakes that cost marks
- Taking r = 8 and h = 6 (rotating about the wrong side).
- Using 10 cm as the height.
How this can come in the exam
Assertion (A): Rotating a right triangle with legs 6 cm and 8 cm about the 6 cm leg gives a larger cone than rotating it about the 8 cm leg.
Reason (R): The volume of a cone depends on the square of its radius but only on the first power of its height.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
About 6 cm: r = 8, h = 6, V = 128π. About 8 cm: V = 96π. A is true, and R explains why the larger leg should be the radius.
Try one yourself
A right triangle with sides 9 cm, 12 cm, 15 cm is rotated about the 12 cm side. Find the volume and curved surface area in terms of π.
Show answer
r = 9, h = 12, l = 15: V = 13π × 81 × 12 = 324π cm3; CSA = π × 9 × 15 = 135π cm2.
More questions like this
- Suppose you have a cup in the shape of a right circular cone. Fill it with water to half the depth of the cone. What fraction of the volume of the cup is occupied by the water?
- But you may be able to find your own derivation of the formula by connecting it with the formula for volume of a cone; namely, by writing it as
V = 13 × (4πr2) × r,
i.e., Volume of sphere = 13 × surface area of sphere × radius of sphere.
Try to work out the connecting links on your own. - Take a rubber ball and drive a nail into it. Using the nail for support, wind a string tightly around the ball. When you have reached the ‘fullest’ part of the ball, use pins to keep the string in place, and continue to wind the string around the remaining part of the ball, till you have completely covered it; see the figure. Mark the starting and finishing points on the string, and unwind the string from the surface of the ball. Now, measure the diameter of the ball and get its radius. On a sheet of paper, draw four circles with a radius equal to the radius of the ball. Fill the circles with the string you had wound around the ball (as shown in the figure).
- A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?
- A ball bearing has a radius of 0.7 cm. Find its surface area.