Mensuration - Class 10
Introduction:
Mensuration is a branch of mathematics that deals with the measurement of geometric figures such as lengths, areas, volumes, and surface areas. It is an essential part of geometry and has numerous real-world applications.
Basic Geometric Shapes:
- Rectangle:
- Length (l): Longer side of the rectangle.
- Breadth (b): Shorter side of the rectangle.
- Perimeter (P): P = 2(l + b)
- Area (A): A = l * b
- Square:
- Side (s): All sides of a square are equal.
- Perimeter (P): P = 4 * s
- Area (A): A = s^2
- Triangle:
- Base (b): The bottom side of the triangle.
- Height (h): Perpendicular distance from the base to the opposite vertex.
- Perimeter (P): P = a + b + c (sum of all three sides)
- Area (A): A = 0.5 * b * h
- Circle:
- Radius (r): Distance from the center to any point on the circle.
- Diameter (d): Distance across the circle passing through the center.
- Circumference (C): C = 2 * π * r (π = 22/7 or 3.14 approximately)
- Area (A): A = π * r^2
3D Shapes:
- Cube:
- Edge (a): All edges of a cube are equal.
- Surface Area (SA): SA = 6 * a^2
- Volume (V): V = a^3
- Cuboid:
- Length (l), Width (w), Height (h): Three different dimensions of the cuboid.
- Surface Area (SA): SA = 2(lw + lh + wh)
- Volume (V): V = l * w * h
- Cylinder:
- Radius (r), Height (h): Two dimensions of the cylinder.
- Curved Surface Area (CSA): CSA = 2πrh
- Total Surface Area (TSA): TSA = 2πr(r + h)
- Volume (V): V = πr^2h
- Sphere:
- Radius (r): The distance from the center to any point on the sphere.
- Surface Area (SA): SA = 4πr^2
- Volume (V): V = (4/3)πr^3
Pythagoras Theorem:
In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For a right-angled triangle with sides a, b, and hypotenuse c: c^2 = a^2 + b^2
Trigonometry:
Trigonometry deals with the study of relationships between the angles and sides of triangles. In a right-angled triangle:
- Sine (sin θ) = Opposite side / Hypotenuse
- Cosine (cos θ) = Adjacent side / Hypotenuse
- Tangent (tan θ) = Opposite side / Adjacent side
Mensuration Chart:
| Shape | Area Formula | Perimeter/Circumference Formula | Volume Formula |
|---|---|---|---|
| Rectangle | A = l * b | P = 2(l + b) | - |
| Square | A = s^2 | P = 4 * s | - |
| Triangle | A = 0.5 * b * h | P = a + b + c | - |
| Circle | A = π * r^2 | C = 2 * π * r | - |
| Cube | - | - | V = a^3 |
| Cuboid | - | - | V = l * w * h |
| Cylinder | - | - | V = πr^2h |
| Sphere | - | - | V = (4/3)πr^3 |
Mensuration is a crucial part of mathematics that helps us solve problems related to measurement in various real-life situations, such as construction, architecture, and engineering. Understanding these concepts and formulas will be beneficial in solving complex geometric problems and further studies in mathematics and related fields.