2D & 3D Mensuration (SSC)

Cheatsheet Content

### 2D Shapes (Area & Perimeter) This section covers essential formulas for calculating the area and perimeter of common two-dimensional geometric shapes. ### Triangle - **Perimeter:** $P = a + b + c$ - **Area:** - $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times b \times h$ - Heron's Formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ (semi-perimeter) - **Equilateral Triangle:** - Area: $A = \frac{\sqrt{3}}{4} a^2$ - Height: $h = \frac{\sqrt{3}}{2} a$ ### Square - **Perimeter:** $P = 4a$ - **Area:** $A = a^2$ - **Diagonal:** $d = a\sqrt{2}$ ### Rectangle - **Perimeter:** $P = 2(l+b)$ - **Area:** $A = l \times b$ - **Diagonal:** $d = \sqrt{l^2 + b^2}$ ### Parallelogram - **Perimeter:** $P = 2(a+b)$ - **Area:** $A = \text{base} \times \text{height} = b \times h$ ### Rhombus - **Perimeter:** $P = 4a$ - **Area:** $A = \frac{1}{2} d_1 d_2$ (where $d_1, d_2$ are diagonals) ### Trapezium (Trapezoid) - **Area:** $A = \frac{1}{2} (a+b) h$ (where $a, b$ are parallel sides) ### Circle - **Circumference:** $C = 2\pi r = \pi d$ - **Area:** $A = \pi r^2$ - **Area of Sector:** $A_{sector} = \frac{\theta}{360^\circ} \pi r^2$ - **Length of Arc:** $L_{arc} = \frac{\theta}{360^\circ} 2\pi r$ ### 3D Shapes (Volume & Surface Area) This section covers essential formulas for calculating the volume and surface area of common three-dimensional geometric solids. ### Cube - **Volume:** $V = a^3$ - **Lateral Surface Area (LSA):** $LSA = 4a^2$ - **Total Surface Area (TSA):** $TSA = 6a^2$ - **Diagonal:** $D = a\sqrt{3}$ ### Cuboid - **Volume:** $V = l \times b \times h$ - **Lateral Surface Area (LSA):** $LSA = 2h(l+b)$ - **Total Surface Area (TSA):** $TSA = 2(lb + bh + hl)$ - **Diagonal:** $D = \sqrt{l^2 + b^2 + h^2}$ ### Cylinder - **Volume:** $V = \pi r^2 h$ - **Curved Surface Area (CSA):** $CSA = 2\pi r h$ - **Total Surface Area (TSA):** $TSA = 2\pi r (r+h)$ ### Cone - **Volume:** $V = \frac{1}{3} \pi r^2 h$ - **Slant Height:** $l = \sqrt{r^2 + h^2}$ - **Curved Surface Area (CSA):** $CSA = \pi r l$ - **Total Surface Area (TSA):** $TSA = \pi r (r+l)$ ### Sphere - **Volume:** $V = \frac{4}{3} \pi r^3$ - **Surface Area:** $SA = 4\pi r^2$ ### Hemisphere - **Volume:** $V = \frac{2}{3} \pi r^3$ - **Curved Surface Area (CSA):** $CSA = 2\pi r^2$ - **Total Surface Area (TSA):** $TSA = 3\pi r^2$ ### Frustum of a Cone - **Volume:** $V = \frac{1}{3} \pi h (R^2 + r^2 + Rr)$ - **Slant Height:** $l = \sqrt{h^2 + (R-r)^2}$ - **Curved Surface Area (CSA):** $CSA = \pi (R+r) l$ - **Total Surface Area (TSA):** $TSA = \pi (R+r) l + \pi R^2 + \pi r^2$