Geometry formulas
Geometry formulas encode relationships among lengths, angles, areas, surface areas, and volumes. Symbols such as side length, radius, height, and slant height represent measurable distances in a chosen unit system, with dimensional consistency governing which expressions are physically meaningful.
Notation and dimensional consistency
A length has units of L, an area has units of L2, and a volume has units of L3. Expressions that add or equate quantities require matching dimensions (e.g., a perimeter compares to a length, not to an area).
- Radius and diameter: \(d = 2r\).
- Right triangles: \(a^2 + b^2 = c^2\) with \(c\) as the hypotenuse.
- Angle measure for circle arcs: radians satisfy \(2\pi\ \text{rad} = 360^\circ\).
Labeled diagram for common shapes
Plane geometry: perimeter and area
Perimeter describes the total boundary length of a planar figure. Area describes the measure of the enclosed region. Variables below are lengths unless otherwise noted.
| Figure | Perimeter / boundary length | Area | Variable notes |
|---|---|---|---|
| Square | \(P = 4s\) | \(A = s^2\) | \(s\): side length |
| Rectangle | \(P = 2(w+h)\) | \(A = wh\) | \(w\): width, \(h\): height |
| Parallelogram | \(P = 2(a+b)\) | \(A = bh\) | \(b\): base, \(h\): altitude to base; \(a\): adjacent side |
| Triangle | \(P = a+b+c\) | \(A = \tfrac12 bh\) | \(h\): altitude to base \(b\) |
| Heron form (triangle) | \(P = a+b+c\) | \(A = \sqrt{s(s-a)(s-b)(s-c)}\) | \(s = \tfrac12(a+b+c)\) (semi-perimeter) |
| Trapezoid | \(P = a+b+c+d\) | \(A = \tfrac12 (b_1+b_2)h\) | \(b_1, b_2\): parallel bases; \(h\): distance between bases |
| Circle | \(C = 2\pi r = \pi d\) | \(A = \pi r^2\) | \(r\): radius; \(d=2r\): diameter |
| Regular \(n\)-gon | \(P = ns\) | \(A = \tfrac12 aP\) | \(s\): side length; \(a\): apothem (inradius) |
Arc length and sector area
Circle measures involve an angle \(\theta\). Radian measure is standard for direct formula use.
- Arc length: \(s = r\theta\) (with \(\theta\) in radians).
- Sector area: \(A_{\text{sector}} = \tfrac12 r^2 \theta\) (with \(\theta\) in radians).
- Degree conversion: \(\theta_{\text{rad}} = \theta_{\degree}\cdot \tfrac{\pi}{180}\).
Coordinate geometry essentials
Coordinate geometry formulas translate geometric relationships into algebraic expressions on the Cartesian plane. Points are written \((x,y)\).
| Concept | Formula | Interpretation |
|---|---|---|
| Distance between \((x_1,y_1)\), \((x_2,y_2)\) | \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) | Pythagorean theorem applied to the horizontal and vertical differences |
| Midpoint | \(M=\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)\) | Average of coordinates along each axis |
| Slope of a non-vertical line | \(m=\tfrac{y_2-y_1}{x_2-x_1}\) | Rate of change in \(y\) per unit change in \(x\) |
| Point–slope line form | \(y-y_1=m(x-x_1)\) | Line through \((x_1,y_1)\) with slope \(m\) |
| Circle (center-radius form) | \((x-h)^2+(y-k)^2=r^2\) | All points at distance \(r\) from center \((h,k)\) |
Solid geometry: surface area and volume
Surface area measures the total area of boundary faces. Volume measures the three-dimensional content. Dimensions are lengths unless noted.
| Solid | Surface area | Volume | Variable notes |
|---|---|---|---|
| Rectangular prism | \(SA = 2(lw+lh+wh)\) | \(V = lwh\) | \(l,w,h\): edge lengths |
| Right circular cylinder | \(SA = 2\pi r^2 + 2\pi rh\) | \(V = \pi r^2 h\) | \(r\): radius; \(h\): height |
| Right circular cone | \(SA = \pi r^2 + \pi r\ell\) | \(V = \tfrac13 \pi r^2 h\) | \(\ell\): slant height; relation \(\ell=\sqrt{r^2+h^2}\) for a right cone |
| Sphere | \(SA = 4\pi r^2\) | \(V = \tfrac{4}{3}\pi r^3\) | \(r\): radius |
| Pyramid (base area \(B\)) | \(SA = B + \text{lateral area}\) | \(V = \tfrac13 Bh\) | \(h\): perpendicular height to the base plane |
| Prism (base area \(B\)) | \(SA = 2B + Ph\) | \(V = Bh\) | \(P\): base perimeter; \(h\): prism height |
Worked example with units
A rectangle with \(w=8\ \text{cm}\) and \(h=3\ \text{cm}\) has area
\[ A = wh = (8\ \text{cm})(3\ \text{cm}) = 24\ \text{cm}^2. \]
A right circular cylinder with \(r=2\ \text{m}\) and \(h=5\ \text{m}\) has volume
\[ V = \pi r^2 h = \pi(2\ \text{m})^2(5\ \text{m}) = 20\pi\ \text{m}^3. \]
Dimensional checking distinguishes \(20\pi\ \text{m}^3\) (volume) from expressions with units \(\text{m}\) (length) or \(\text{m}^2\) (area).
Common pitfalls
- Radius–diameter confusion: \(A=\pi r^2\) requires the radius; a diameter input must be halved.
- Height versus slant height: cone volume uses perpendicular height \(h\), while lateral surface uses slant height \(\ell\).
- Angle units in circle measure: \(s=r\theta\) and \(A_{\text{sector}}=\tfrac12 r^2\theta\) assume radians.
- Area from side lengths: \(A=s^2\) applies to squares, not to general rectangles or rhombi unless additional constraints hold.
- Coordinate formula domain: slope \(m=\tfrac{y_2-y_1}{x_2-x_1}\) excludes vertical lines where \(x_2=x_1\).