Mensuration: Chapter Explanation & Basic Problems (10th Grade)
Mensuration is the branch of mathematics that studies the measurement of geometric figures and their parameters like length, volume, shape, surface area, and perimeter. For 10th-grade students, the focus is primarily on 3D solid shapes and combinations of these solids.
1. Key Formulas for 3D Shapes
- Cuboid: Length $l$, Breadth $b$, Height $h$.
Volume = $l \times b \times h$
Total Surface Area (TSA) = $2(lb + bh + hl)$ - Cube: Side $a$.
Volume = $a^3$
Total Surface Area (TSA) = $6a^2$ - Right Circular Cylinder: Base Radius $r$, Height $h$.
Volume = $\pi r^2 h$
Curved Surface Area (CSA) = $2\pi rh$
Total Surface Area (TSA) = $2\pi r(r + h)$ - Right Circular Cone: Base Radius $r$, Height $h$, Slant Height $l = \sqrt{r^2 + h^2}$.
Volume = $\frac{1}{3}\pi r^2 h$
Curved Surface Area (CSA) = $\pi rl$
Total Surface Area (TSA) = $\pi r(r + l)$ - Sphere: Radius $r$.
Volume = $\frac{4}{3}\pi r^3$
Surface Area = $4\pi r^2$ - Hemisphere: Radius $r$.
Volume = $\frac{2}{3}\pi r^3$
Curved Surface Area (CSA) = $2\pi r^2$
Total Surface Area (TSA) = $3\pi r^2$
2. Detailed Solutions to Basic Problems
Problem 1: Find the volume and total surface area of a cylinder with a base radius of 7 cm and a height of 10 cm. (Use $\pi = \frac{22}{7}$)
Solution:
Given:
Radius ($r$) = 7 cm
Height ($h$) = 10 cmStep 1: Calculate Volume
Formula: $V = \pi r^2 h$
$V = \frac{22}{7} \times (7)^2 \times 10$
$V = 22 \times 7 \times 10 = 1540$
The volume is 1540 cm³.Step 2: Calculate Total Surface Area (TSA)
Formula: $TSA = 2\pi r(r + h)$
$TSA = 2 \times \frac{22}{7} \times 7 \times (7 + 10)$
$TSA = 44 \times 17 = 748$
The total surface area is 748 cm².
Problem 2: A cone has a base radius of 6 cm and a height of 8 cm. Find its slant height and curved surface area. (Use $\pi = 3.14$)
Solution:
Given:
Radius ($r$) = 6 cm
Height ($h$) = 8 cmStep 1: Calculate Slant Height ($l$)
Formula: $l = \sqrt{r^2 + h^2}$
$l = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm.Step 2: Calculate Curved Surface Area (CSA)
Formula: $CSA = \pi rl$
$CSA = 3.14 \times 6 \times 10 = 188.4$
The curved surface area is 188.4 cm².
Problem 3: Find the volume of a sphere whose radius is 3 cm. (Use $\pi = 3.14$)
Solution:
Given:
Radius ($r$) = 3 cmStep 1: Calculate Volume
Formula: $V = \frac{4}{3}\pi r^3$
$V = \frac{4}{3} \times 3.14 \times (3)^3$
$V = \frac{4}{3} \times 3.14 \times 27$
$V = 4 \times 3.14 \times 9 = 113.04$
The volume of the sphere is 113.04 cm³.
Problem 4: A toy is in the form of a cone mounted on a hemisphere of the same radius. The radius of the base is 3.5 cm and the total height of the toy is 15.5 cm. Find the total surface area of the toy. (Use $\pi = \frac{22}{7}$)
Solution:
Given:
Radius of hemisphere ($r$) = Radius of cone ($r$) = 3.5 cm
Total height of toy = 15.5 cmStep 1: Find the height of the cone ($h$)
Height of cone = Total height - Radius of hemisphere
$h = 15.5 - 3.5 = 12$ cmStep 2: Find the slant height of the cone ($l$)
$l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + (12)^2}$
$l = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5$ cmStep 3: Calculate Total Surface Area of the Toy
TSA of toy = CSA of cone + CSA of hemisphere
$TSA = \pi rl + 2\pi r^2$
$TSA = \pi r(l + 2r)$
$TSA = \frac{22}{7} \times 3.5 \times (12.5 + 2 \times 3.5)$
$TSA = \frac{22}{7} \times \frac{7}{2} \times (12.5 + 7)$
$TSA = 11 \times 19.5 = 214.5$
The total surface area of the toy is 214.5 cm².