Geometry Formulas

Cheatsheet Content

### 2D Shapes - **Perimeter (P):** Total length of the boundary of a closed figure. - **Area (A):** The amount of surface enclosed by a closed figure. #### Square - **Side:** $a$ - **Perimeter:** $P = 4a$ - **Area:** $A = a^2$ - **Diagonal:** $d = a\sqrt{2}$ #### Rectangle - **Length:** $l$ - **Breadth:** $b$ - **Perimeter:** $P = 2(l + b)$ - **Area:** $A = l \times b$ - **Diagonal:** $d = \sqrt{l^2 + b^2}$ #### Circle - **Radius:** $r$ - **Diameter:** $d = 2r$ - **Circumference (Perimeter):** $C = 2\pi r = \pi d$ - **Area:** $A = \pi r^2$ - **Area of Sector (angle $\theta$ in degrees):** $A = \frac{\theta}{360^\circ} \times \pi r^2$ - **Arc Length (angle $\theta$ in degrees):** $L = \frac{\theta}{360^\circ} \times 2\pi r$ #### Triangle - **Base:** $b$ - **Height:** $h$ - **Area:** $A = \frac{1}{2} \times b \times h$ - **Semi-perimeter (s):** $s = \frac{a+b+c}{2}$ (for sides a, b, c) - **Heron's Formula (Area):** $A = \sqrt{s(s-a)(s-b)(s-c)}$ #### Equilateral Triangle - **Side:** $a$ - **Area:** $A = \frac{\sqrt{3}}{4} a^2$ - **Height:** $h = \frac{\sqrt{3}}{2} a$ #### Parallelogram - **Base:** $b$ - **Height:** $h$ - **Area:** $A = b \times h$ - **Perimeter:** $P = 2(a+b)$ (where a, b are adjacent sides) #### Rhombus - **Side:** $a$ - **Diagonals:** $d_1, d_2$ - **Area:** $A = \frac{1}{2} d_1 d_2$ - **Perimeter:** $P = 4a$ #### Trapezium (or Trapezoid) - **Parallel sides:** $a, b$ - **Height:** $h$ - **Area:** $A = \frac{1}{2}(a+b)h$ ### 3D Shapes - **Volume (V):** The amount of space occupied by a 3D object. - **Total Surface Area (TSA):** The sum of the areas of all faces/surfaces. - **Curved Surface Area (CSA) / Lateral Surface Area (LSA):** The area of the curved surface or the sum of the areas of the lateral faces (excluding bases). #### Cube - **Side:** $a$ - **Volume:** $V = a^3$ - **Lateral Surface Area (LSA):** $4a^2$ - **Total Surface Area (TSA):** $6a^2$ - **Diagonal:** $d = a\sqrt{3}$ #### Cuboid - **Length:** $l$ - **Breadth:** $b$ - **Height:** $h$ - **Volume:** $V = l \times b \times h$ - **Lateral Surface Area (LSA):** $2h(l+b)$ - **Total Surface Area (TSA):** $2(lb + bh + hl)$ - **Diagonal:** $d = \sqrt{l^2 + b^2 + h^2}$ #### Cylinder - **Radius:** $r$ - **Height:** $h$ - **Volume:** $V = \pi r^2 h$ - **Curved Surface Area (CSA):** $2\pi r h$ - **Total Surface Area (TSA):** $2\pi r h + 2\pi r^2 = 2\pi r(h+r)$ #### Cone - **Radius:** $r$ - **Height:** $h$ - **Slant Height:** $l = \sqrt{r^2 + h^2}$ - **Volume:** $V = \frac{1}{3} \pi r^2 h$ - **Curved Surface Area (CSA):** $\pi r l$ - **Total Surface Area (TSA):** $\pi r l + \pi r^2 = \pi r(l+r)$ #### Sphere - **Radius:** $r$ - **Volume:** $V = \frac{4}{3} \pi r^3$ - **Surface Area (TSA = CSA):** $A = 4\pi r^2$ #### Hemisphere - **Radius:** $r$ - **Volume:** $V = \frac{2}{3} \pi r^3$ - **Curved Surface Area (CSA):** $2\pi r^2$ - **Total Surface Area (TSA):** $2\pi r^2 + \pi r^2 = 3\pi r^2$ #### Frustum of a Cone - **Radii of bases:** $R$ (larger), $r$ (smaller) - **Height:** $h$ - **Slant Height:** $l = \sqrt{h^2 + (R-r)^2}$ - **Volume:** $V = \frac{1}{3} \pi h (R^2 + Rr + r^2)$ - **Curved Surface Area (CSA):** $\pi (R+r)l$ - **Total Surface Area (TSA):** $\pi (R+r)l + \pi R^2 + \pi r^2$ #### Prism - **Area of Base:** $A_b$ - **Perimeter of Base:** $P_b$ - **Height:** $h$ - **Volume:** $V = A_b \times h$ - **Lateral Surface Area (LSA):** $P_b \times h$ - **Total Surface Area (TSA):** $P_b \times h + 2A_b$ - **Specific forms:** - **Triangular Prism:** $A_b = \frac{1}{2} \times \text{base} \times \text{height of triangle}$ - **Square Prism:** (Same as Cuboid if base is square) - **Hexagonal Prism:** $A_b = \frac{3\sqrt{3}}{2} a^2$ (a = side of hexagon) #### Pyramid - **Area of Base:** $A_b$ - **Perimeter of Base:** $P_b$ - **Height:** $h$ - **Slant Height (for regular pyramid):** $l$ (height of triangular face) - **Volume:** $V = \frac{1}{3} A_b \times h$ - **Lateral Surface Area (LSA, for regular pyramid):** $\frac{1}{2} P_b \times l$ - **Total Surface Area (TSA, for regular pyramid):** $\frac{1}{2} P_b \times l + A_b$ - **Specific forms:** - **Square Pyramid:** $A_b = a^2$ (a = side of base), $P_b = 4a$, $l = \sqrt{h^2 + (\frac{a}{2})^2}$ - **Triangular Pyramid:** $A_b = \frac{1}{2} \times \text{base} \times \text{height of triangle}$