Login
OR
Don't worry. We'll fix this for you!
Mensuration is the branch of mathematics that deals with the measurement of geometric figures and their parameters such as length, area, and volume. It helps us find the size of two-dimensional (2D) shapes like triangles and circles, as well as three-dimensional (3D) solids like cubes and cylinders.
Understanding mensuration is essential not only in academics but also in real-life situations such as construction, packaging, and design. For example, calculating the amount of paint needed to cover a wall requires knowing the wall's area, while determining how much water a tank can hold involves finding its volume.
In this chapter, we will focus on the formulas and methods to calculate areas and volumes of common shapes and solids, using the metric system (centimeters, meters, square meters, cubic meters). This knowledge is vital for competitive exams where quick and accurate problem-solving is required.
The area of a 2D shape is the amount of space enclosed within its boundaries. It is measured in square units such as cm² or m². Let's explore the formulas for common shapes.
Note: Always use the perpendicular height (not slant height) for triangles and parallelograms.
Volume measures the amount of space inside a 3D object and is expressed in cubic units like cm³ or m³. Surface area is the total area of all outer surfaces of the solid, measured in square units.
Note: Surface area includes all outer faces, including top and bottom where applicable.
Step 1: Identify the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Step 2: Substitute the given values:
\[ \text{Area} = \frac{1}{2} \times 10 \times 6 = 5 \times 6 = 30 \text{ cm}^2 \]
Answer: The area of the triangle is 30 cm².
Step 1: Recall the formula for volume of a cylinder:
\[ \text{Volume} = \pi r^2 h \]
Step 2: Substitute the values \(r = 5\) m, \(h = 12\) m:
\[ \text{Volume} = \pi \times 5^2 \times 12 = \pi \times 25 \times 12 = 300\pi \text{ m}^3 \]
Step 3: Use \(\pi \approx 3.14\) for approximate value:
\[ 300 \times 3.14 = 942 \text{ m}^3 \]
Answer: The volume of the cylinder is approximately 942 m³.
Step 1: Use the surface area formula for a cube:
\[ \text{Surface Area} = 6a^2 \]
Step 2: Substitute \(a = 7\) cm:
\[ 6 \times 7^2 = 6 \times 49 = 294 \text{ cm}^2 \]
Answer: The total surface area of the cube is 294 cm².
Step 1: Recall the volume formula for a cone:
\[ \text{Volume} = \frac{1}{3} \pi r^2 h \]
Step 2: Substitute \(r = 3\) m, \(h = 9\) m:
\[ \text{Volume} = \frac{1}{3} \times \pi \times 3^2 \times 9 = \frac{1}{3} \times \pi \times 9 \times 9 = \frac{81}{3} \pi = 27\pi \text{ m}^3 \]
Step 3: Approximate using \(\pi \approx 3.14\):
\[ 27 \times 3.14 = 84.78 \text{ m}^3 \]
Answer: The volume of the cone is approximately 84.78 m³.
Step 1: Calculate the area of the rectangle:
\[ \text{Area}_{rectangle} = 10 \times 6 = 60 \text{ m}^2 \]
Step 2: Calculate the area of the semicircle:
Area of full circle = \(\pi r^2 = \pi \times 3^2 = 9\pi\)
Area of semicircle = \(\frac{1}{2} \times 9\pi = \frac{9\pi}{2}\)
Step 3: Add the two areas:
\[ \text{Total Area} = 60 + \frac{9\pi}{2} \approx 60 + \frac{9 \times 3.14}{2} = 60 + 14.13 = 74.13 \text{ m}^2 \]
Answer: The area of the composite figure is approximately 74.13 m².
When to use: When dimensions are given in mixed units like cm and m.
When to use: When dealing with circle-related problems requiring quick or exact results.
When to use: When calculating volumes of cones or pyramids.
When to use: When dealing with composite figures.
When to use: When asked for surface area vs volume.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|
