geometry formulaes

Sum of Interior Angles of a Triangle

The sum of all three interior angles in any planar triangle is always $180^\circ$. Hinglish recall: Teeno interior angles ko jodo, yog hamesha $180^\circ$ aayega.

Basic Area Formula for Any Triangle

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$, where height represents the perpendicular altitude dropped onto the chosen base. Hinglish mnemonic: Base aur perpendicular height ke gunaa ka aadha (samosa ka aadha).

Perimeter and Semi-Perimeter of a Triangle

Perimeter is the total boundary distance: $P = a + b + c$. Semi-perimeter ($s$) is exactly half of the total boundary: $s = \frac{a + b + c}{2}$. Hinglish: Puri boundary ka aadha.

Heron's Formula for Triangle Area

$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$, used whenever the lengths of all three sides are known directly. For example, a triangle with sides $13$, $14$, and $15$ yields $s = 21$ and an area of $84$.

Area of an Equilateral Triangle

$\text{Area} = \frac{\sqrt{3}}{4}a^2$, where $a$ represents the side length. Hinglish mnemonic: 'Root-3 by 4, a-square'.

Height (Altitude) of an Equilateral Triangle

$h = \frac{\sqrt{3}}{2}a$. Note the denominator distinction: height uses $2$, whereas area uses $4$ (area requires a larger denominator). Hinglish: Height me $2$, Area me $4$.

Equilateral Triangle — Perimeter and Interior Angles

Perimeter equals $3a$ because all three sides are equal. Each individual interior angle measures $\frac{180^\circ}{3} = 60^\circ$. Hinglish: Teen judwa bhai, sab barabar.

Pythagoras Theorem for Right Triangles

$P^2 + B^2 = H^2$, where $P$ is the perpendicular height, $B$ is the base, and $H$ is the hypotenuse located directly opposite the $90^\circ$ angle.

Area of a Right-Angled Triangle

$\text{Area} = \frac{1}{2} \times B \times P$, calculated by multiplying the two mutually perpendicular legs and halving the product. Hinglish: Dono perpendicular sides ka gunaa, phir aadha.

Fundamental Pythagorean Triplets (Top 6)

The six most common integer sets satisfying $a^2 + b^2 = c^2$: $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$, $(7, 24, 25)$, $(20, 21, 29)$, and $(9, 40, 41)$. Any scalar multiple also forms a valid triplet, such as $(6, 8, 10)$, $(14, 48, 50)$, and $(12, 16, 20)$.

Classification of Triangles by Angles

Acute: all three interior angles are $< 90^\circ$. Right: exactly one angle measures $90^\circ$. Obtuse: exactly one angle measures $> 90^\circ$.

Classification of Triangles by Sides

Scalene: all three sides have different lengths. Isosceles: exactly two sides are equal in length. Equilateral: all three sides are equal. Hinglish memory trick: S-I-E = 'Sab alag, Ik jodi same, Ek jaise teeno'.

Square — Area, Perimeter, and Diagonal

For side length $a$: $\text{Area} = a^2 = \frac{d^2}{2}$; $\text{Perimeter} = 4a$; $\text{Diagonal } d = a\sqrt{2}$. Shortcut: If diagonal contains $\sqrt{2}$, removing it instantly reveals the side length $a$.

Rectangle — Area, Perimeter, and Diagonal

For length $l$ and breadth $b$: $\text{Area} = l \times b$; $\text{Perimeter} = 2(l + b)$ (semi-perimeter is $l + b$); $\text{Diagonal } d = \sqrt{l^2 + b^2}$. Tip: Use triplets for diagonals (e.g., sides $12$ and $5$ give diagonal $13$).

Rhombus — Area, Side Length, and Perimeter

For diagonals $d_1$ and $d_2$: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$; $\text{Side } a = \frac{1}{2}\sqrt{d_1^2 + d_2^2}$; $\text{Perimeter} = 4a$. Diagonals bisect each other perpendicularly at $90^\circ$. Hinglish: Tedha square jisme chaaron sides barabar hoti hain.

Parallelogram — Area, Perimeter, and Diagonal Theorem

$\text{Area} = \text{base} \times \text{perpendicular height}$ (Trap: Never use $l \times b$ because the slant side is not perpendicular height); $\text{Perimeter} = 2(a + b)$; Diagonal relation: $d_1^2 + d_2^2 = 2(a^2 + b^2)$ ('Do ka double' rule).

Trapezium — Area Formula

$\text{Area} = \frac{1}{2} \times (a + b) \times h$, where $a$ and $b$ are the parallel sides and $h$ is the perpendicular distance between them. Hinglish intuition: Parallel sides ka average multiplied by height.

Circle — Diameter, Circumference, Area, and Ratio

$\text{Diameter } D = 2R$; $\text{Circumference } C = 2\pi R = \pi D$; $\text{Area} = \pi R^2$. Ratio of circumference to area: $\frac{C}{\text{Area}} = \frac{2\pi R}{\pi R^2} = \frac{2}{R}$.

Semicircle — Area and Boundary Perimeter

$\text{Area} = \frac{\pi R^2}{2}$; $\text{Perimeter} = \pi R + 2R = \frac{36}{7}R$ when using $\pi = \frac{22}{7}$. Common trap: Never forget to add the flat straight diameter edge $2R$. Hinglish: Chintu ki aadhi roti ka kinara.

Circular Ring (Annulus) or Path — Area

$\text{Area} = \pi(R^2 - r^2) = \pi(R + r)(R - r)$, where $R$ is the outer radius and $r$ is the inner radius. Hinglish: Difference of squares ($a^2 - b^2$) ka algebraic jaadu.

Circle Benchmark Values for Standard Radii

When $\pi = \frac{22}{7}$: For $R = 3.5$: $C = 22$, $\text{Area} = 38.5$. For $R = 7$: $C = 44$, $\text{Area} = 154$. For $R = 14$: $C = 88$, $\text{Area} = 616$. For $R = 21$: $C = 132$, $\text{Area} = 1386$. For $R = 28$: $C = 176$, $\text{Area} = 2464$.

Optimal 2D Shape for Maximum Area at Fixed Perimeter

The circle. Given a fixed perimeter or length of wire, a circle encloses a strictly greater area than any polygon.

Cube — Volume, Surface Areas, Diagonal, and Ratios

For side length $a$: $\text{Volume} = a^3$; Lateral surface area (4 walls) $= 4a^2$; Total surface area (6 faces) $= 6a^2$; Space diagonal $= a\sqrt{3}$; Total length of $12$ edges $= 12a$; Ratio of $\text{Volume} : \text{TSA} = a : 6$.

Cuboid — Volume, Surface Areas, and Diagonal

For dimensions $l$, $b$, and $h$: $\text{Volume} = l \times b \times h$; Lateral surface area (4 walls) $= 2(l + b)h$ (perimeter of the floor $\times$ height); Total surface area $= 2(lb + bh + hl)$; Internal space diagonal $= \sqrt{l^2 + b^2 + h^2}$. Triplets: $(3, 4, 12) \rightarrow 13$, $(12, 16, 21) \rightarrow 29$.

Hollow Box of Thickness $t$ — Internal Dimensions

Closed box: inner dimensions are $(l - 2t)$, $(b - 2t)$, and $(h - 2t)$. Open box (open top): subtract $2t$ from length and breadth, but subtract only $t$ from height: $(h - t)$.

Three Cubes Recast into One — Side Length Identity

$3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216 = 6^3$. Melting three metallic cubes of edges $3$, $4$, and $5$ produces a single large cube of edge length $6$.

Right Circular Cylinder — Volume, CSA, TSA, and Ratios

For radius $R$ and height $H$: $\text{Volume} = \pi R^2H$; Curved surface area (can label) $= 2\pi RH$; Total surface area $= 2\pi R(H + R)$. Ratio of $\text{CSA} : \text{TSA} = H : (H + R)$; Ratio of $\frac{\text{Volume}}{\text{CSA}} = \frac{R}{2}$.

Hollow Cylindrical Pipe — Volume of Material

$\text{Volume} = \pi(R_1^2 - R_2^2)h$, where $R_1$ is the external radius, $R_2$ is the internal radius, and $h$ is the cylinder height. Hinglish: Bada cylinder volume minus khaali inner core.

Right Circular Cone — Slant Height, Volume, CSA, and TSA

Slant height $L = \sqrt{H^2 + R^2}$; $\text{Volume} = \frac{1}{3}\pi R^2H$ (one-third of an equivalent cylinder); Curved surface area (tent canvas) $= \pi RL$; Total surface area $= \pi R(L + R)$; Ratio of $\text{CSA} : \text{TSA} = L : (L + R)$.

Height Relationship for Cylinder Recast into a Cone of Equal Radius

Since volume is conserved and cylinder volume is $3$ times cone volume for equal height, $\text{Cone Height} = 3 \times \text{Cylinder Height}$. Conversely, melting a cone into a cylinder of identical radius yields $\text{Cylinder Height} = \frac{\text{Cone Height}}{3}$.

Frustum of a Right Circular Cone — Volume Formula

$\text{Volume} = \frac{1}{3}\pi h(R^2 + Rr + r^2)$, where $h$ is the vertical height, $R$ is the larger base radius, and $r$ is the smaller top radius. Represents a cone with its top peak sliced off (bucket shape).

Sphere — Volume and Surface Area

$\text{Volume} = \frac{4}{3}\pi R^3$; Surface Area $= 4\pi R^2$. For a complete sphere, Curved Surface Area equals Total Surface Area ($\text{CSA} = \text{TSA} = 4\pi R^2$) because it has no flat base boundaries.

Hemisphere — Volume, CSA, TSA, and Height

For radius $R$: $\text{Volume} = \frac{2}{3}\pi R^3$; Curved surface area $= 2\pi R^2$; Solid total surface area $= 3\pi R^2$ (curved bowl $2\pi R^2$ plus flat circular base $\pi R^2$). The vertical height of any hemisphere is exactly equal to its radius: $h = R$.

Number of Small Spheres Formed by Melting a Large Sphere

$n = \left(\frac{R}{r}\right)^3$, where $R$ is the radius of the original large sphere and $r$ is the radius of each new small sphere. Example: If $R = 6$ and $r = 1$, $n = 6^3 = 216$ balls.

Composite Solid: Cylinder Surmounted by a Hemisphere — Volume

$\text{Total Volume} = \pi R^2\left(h + \frac{2}{3}R\right)$, where $h$ is the height of the cylindrical section alone and $R$ is the common radius. If total height is $H_{\text{total}}$, then $h = H_{\text{total}} - R$.

Cost Estimation: Boundary Fencing vs Surface Finishing

Linear boundary work (fencing, wiring, border edging) $= \text{Perimeter} \times \text{Rate per unit length}$. Surface work (painting, flooring, turfing, gardening) $= \text{Area} \times \text{Rate per unit area}$.

Conservation Invariance: Melting vs Wire Reshaping

Melting and recasting: Volume remains constant and invariant, while total surface area changes. Reshaping a continuous wire into a new closed geometry: Perimeter remains constant and invariant, while enclosed surface area changes.

Uniform Pathway Around a Rectangle — Area Formulas

Pathway of width $w$ outside a rectangle of length $l$ and breadth $b$: $\text{Area} = 2w(l + b + 2w)$. Pathway inside the rectangle: $\text{Area} = 2w(l + b - 2w)$. For a pathway inside a square of side $a$: $\text{Area} = 4w(a - w)$. Hinglish: Bahar ke path me plus $2w$, andar ke path me minus $2w$.

Net Percentage Change in Area Formulas

When both radius or side change by $+x\%$: $\text{Net Area Change} = \left(2x + \frac{x^2}{100}\right)\%$. For example, $+50\%$ gives $2(50) + \frac{2500}{100} = 125\%$, while $+100\%$ gives $300\%$. For two independent dimensions changing by $x\%$ and $y\%$: $\text{Net Area Change} = \left(x + y + \frac{xy}{100}\right)\%$.

Scaling Laws for Geometrically Similar 3D Solids

If every linear dimension (length, radius, height) is multiplied by a scaling factor $k$: all surface areas multiply by $k^2$, and the volume multiplies by $k^3$.

Painted Cube Cut into $n^3$ Smaller Unit Cubes

Exactly $3$ painted faces (located at corners): always $8$. Exactly $2$ painted faces (along the edges): $12(n - 2)$. Exactly $1$ painted face (center of each face): $6(n - 2)^2$. Zero painted faces (internal core): $(n - 2)^3$. Total surface area of all small cubes combined increases by a factor of $n$.

Transverse Cuts on a Cylinder and Tank Filling Time

Making $n$ parallel cuts transverse to the circular base divides the cylinder into pieces and introduces $2(n - 1)$ new circular faces. Time needed to fill a tank is given by $\text{Time} = \frac{\text{Tank Volume}}{\text{Volumetric Flow Rate}}$. Hinglish: Har cut se do nayi circular satah judti hain.

Algebraic Shortcut: Finding Two Numbers from Their Sum and Difference

Given sum $S$ and difference $D$: $\text{Larger Number} = \frac{S + D}{2}$, and $\text{Smaller Number} = \frac{S - D}{2}$. Hinglish: Bada nikalne ke liye jod kar aadha, chhota nikalne ke liye ghata kar aadha.