WBBSE Solutions For Class 10 Maths Mensuration Chapter 3 Right Circular Cone

Mensuration Chapter 3 Right Circular Cone

⇔ The solid generated by the revolution of a right-angled triangle about one of the sides containing the right angle as axis is called a right circular cone.

⇒ If h be the height, r the radius of the base, and l the slant height of a right circular cone, then we have.

1. The area of curved surface = \(\frac{1}{2}\) (circumference of the base) x slant height = πrl sq. unit.
2. l² = h² + r².
3. The area of the whole surface area = πrl + πr² = πr (l + r) sq. units.
4. Volume = \(\frac{1}{3}\)πr²h cubic units.

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Mensuration Chapter 3 Right Circular Cone True Or False

Example 1. If the length of base radius of a right circular cone is decreased by half and its height is increased by twice of it then the volume remains same.

(Hints: Volume = \(\frac{1}{3} \pi\left(\frac{r}{2}\right)^2 \cdot 2 h=\frac{1}{2} \cdot \frac{1}{3} \pi r^2 h \neq \frac{1}{3} \pi r^2 h\))

Solution: False

Example 2. The height, radius and slant height of a right circular cone are always the three sides of a right angled triangle.

Solution: True

Class 10 Maths Mensuration Chapter 3 Solutions

Example 3. Total surface area of a cone which one face open is πrl sq. units.

Solution: True

Example 4. Height of a cone = \(\sqrt{(\text { slant height })^2+(\text { radius })^2}\)

Solution: False

Example 5. Volume of a cone= \(\frac{1}{3}\) x volume of cylinder if heights and length of radii are same for both.

Solution: True

Example 6. The radius and the height of a cone are increased by 20%. Then the volume of the cone is increased by 60%.

Solution: False

Example 7. Curved surface area = \(\pi r \sqrt{r^2-h^2} \text { sq. u. }\)

Solution: True

Combination Of Solids Class 10 Solutions

Example 8. The sharpened end of a pencil is an example of a cone.

Solution: True

Mensuration Chapter 3 Right Circular Cone Fill In The Blanks

Example 1. AC is the hypotenuse of a right-angled triangle ABC, the radius of the right circular cone formed by revolving the triangle once around side AB as axis is _____

Solution: BC

Example 2. If the volume of a right circular cone is V cubic unit and the base area is A sq. unit, then its height is _______

Solution: \(\frac{3V}{A}\)

Example 3. The lengths of the base radii and the heights of a right circular cylinder and a right circular cone are equal. The ratio of their volumes is ________

Solution: 3: 1

Example 4. If the heights of two cones are equal the ratio of their volumes is ________

Solution: (Ratio of radii)²

Class 10 Mensuration Chapter 3 Solved Examples

Example 5. If heights of two cones are equal the ratio of their volumes is _______

Solution: 1: √3

Example 6. Centre of a semicircular paper is O and diameter is AB. A _______ is formed as OA and OB are joined.

Solution: Diameter

Example 7. A _____ is generated by the revolution of a right angled triangle about one of the sides containing the right angle an axis.

Solution: cone

Example 8. The Bare of a right circular cone is ________

Solution: circular

Mensuration Chapter 3 Right Circular Cone Short Answer Type Questions

Example 1. A solid circular cylinder is made by melting a solid circular cone. The radii of both are equal. If the height of the cone is 15 cm, then find the height of the solid cylinder.

Solution: πr²h = \(\frac{1}{3}\)πr² (15)

∴ h = 5 cm

∴ Required height 5 cm.

Example 2. The height of a right circular cone is 12 cm and its volume is 100 π cm³. Write the length of the ratius of the cone.

Solution: \(\frac{1}{3}\)πr²h = 100π

or, \(\frac{1}{3}\).π.r².12 = 100π

or, r = 5 cm.

Example 3. The curved surface area of a right circular cone is √5 times of its base area. Write the ratio of the height and the length of radius of the cone.

Solution: πrl= √5πr²

or, πr \(\sqrt{r^2+h^2}\) = √5πr²

or, r²( r²+ h²) = 5r².r²

or, 4r² = h²

∴ h : r = 2:1

Wbbse Class 10 Mensuration Notes

Example 4. If the volume of a right circular cone is V cubic unit, base area is A sq unit and height is H unit, find the value of \(\frac{AH}{V}\).

Solution: \(\frac{\mathrm{AH}}{\mathrm{V}}=\frac{\pi r^2 \cdot h}{\frac{1}{3} \pi r^2 h}=3\)

Example 5. The numerical values of the volume and the lateral surface area of a right circular cone are equal. If the height and the radius of the H and r respectively, then find the value of \(\frac{1}{h^2}+\frac{1}{r^2}\).

Solution: \(\frac{1}{3} \pi r^2 h=\pi r \sqrt{\left(h^2+r^2\right)}\)

or, \(r^2 h^2=9\left(h^2+r^2\right)\)

or, \(\frac{h^2+r^2}{r^2 h^2}=\frac{1}{9}\)

or, \(\frac{1}{r^2}+\frac{1}{h^2}=\frac{1}{9}\)

Example 6. The ratio of the lengths of the base radii of a right circular cylinder and a right circular cone is 3: 4 and the ratio of their heights is 2 : 3. Write the ratio of the volume of the cylinder and cone.

Solution: \(\frac{\pi r_1{ }^2 h_1}{\frac{1}{3} \pi r_2{ }^2 h_2}=3\left(\frac{3}{4}\right)^2 \cdot\left(\frac{2}{3}\right)\)

= \(3 \times \frac{9}{16} \times \frac{2}{3}=9: 8\)

Example 7. Lateral surface area of a cone is √10 times the length of radius. Find the ratio of height and length of the diameter.

Solution: πr\(\sqrt{r^2+h^2}\) = √10.πr²

or, r² + h²= 10r²

or, 9r²= h²

or, \(\frac{h}{2r}=\frac{3}{2}\) = 3: 2

Example 8. If length of the diameter, height, and slant height of a cone are d, h, and l respectively then what is the relation between d, h, and l?

Solution: We know r² + h² = l²

or, \(\left(\frac{d}{2}\right)^2\) + h² = l²  ⇒ d = \(2 \sqrt{l^2-h^2}\)

Example 9. Length of the diameter and height of a cone are same which is x mt. Find its volume.

Solution: Volume = \(\frac{1}{3}\)πr²h

= \(\frac{1}{3} \pi\left(\frac{x}{2}\right)^2 \cdot x\) cubic mt = \(\frac{\pi x^3}{12}\) cu.mt

Surface Area And Volume Of Combined Solids Class 10

Example 10. Perimeter of the base and height of a cone are \(\frac{660}{7}\) cm and 20 cm respectively. Find its curved surface area.

Solution: 2πr = \(\frac{660}{7}\) ⇒ r = \(\frac{660 \times 7}{7 \times 2 \times 22}\) = 15

l = \(\sqrt{(20)^2+(15)^2}\) cm = 25 cm

πrl = \(\frac{22}{7}\) x 15 x 25 sq. cm = 1178\(\frac{4}{7}\)sq. cm