Perimeter and circumference: distance around

Key idea

Perimeter is the total distance around a 2D shape. Add all side lengths. For common shapes: \[ P_{\text{rectangle}}=2(\ell+w),\quad P_{\text{square}}=4s,\quad P_{\text{regular }n\text{-gon}}=ns. \] The circumference is the perimeter of a circle: \[ C=2\pi r=\pi d. \]

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Summary

• Perimeter adds side lengths. Regular polygon: \(P=ns\).
• Circumference: \(C=2\pi r=\pi d\).

Area of common polygons

Key idea

Area measures the amount of surface inside a 2D shape. Common area formulas: \[ A_{\text{rectangle}}=\ell w,\quad A_{\text{triangle}}=\frac12 bh,\quad A_{\text{parallelogram}}=bh,\quad A_{\text{rhombus}}=\frac12 d_1d_2. \]

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Summary

• Triangle: \(A=\tfrac12 bh\). Parallelogram: \(A=bh\).
• Rhombus (diagonals): \(A=\tfrac12 d_1d_2\).
• Area answers use square units.

Area of circles, semicircles, and annuli

Key idea

Circle area depends on the radius: \[ A_{\text{circle}}=\pi r^2. \] A semicircle is half a circle: \[ A_{\text{semicircle}}=\frac12\pi r^2. \] An annulus (a ring) subtracts the inner circle from the outer circle: \[ A_{\text{annulus}}=\pi(R^2-r^2). \] Remember: if you are given a diameter \(d\), then \(r=\tfrac{d}{2}\).

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Summary

• Circle: \(A=\pi r^2\). Semicircle: \(A=\tfrac12\pi r^2\).
• Annulus: \(A=\pi(R^2-r^2)\).
• Convert diameter to radius using \(r=\tfrac{d}{2}\).

Volume of prisms and cubes

Key idea

Volume measures how much 3D space a solid contains. A common structure is: \[ V=\text{(area of base)}\times \text{height}. \] For a rectangular prism: \[ V=\ell w h. \] For a cube (all sides equal): \[ V=s^3. \]

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Summary

• Rectangular prism: \(V=\ell w h\).
• Cube: \(V=s^3\).
• Volume answers use cubic units.

Volume of cylinders, cones, and spheres

Key idea

For curved solids, the radius is critical: \[ V_{\text{cylinder}}=\pi r^2h,\quad V_{\text{cone}}=\frac13\pi r^2h,\quad V_{\text{sphere}}=\frac43\pi r^3. \] Always check whether you are given radius or diameter (\(d=2r\)).

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Summary

• Cylinder: \(V=\pi r^2h\).
• Cone: \(V=\tfrac13\pi r^2h\) (don’t forget \(\tfrac13\)).
• Sphere: \(V=\tfrac43\pi r^3\).

Surface area: add all outer faces

Key idea

Surface area is the total area of the outside surfaces of a 3D shape. Common formulas: \[ SA_{\text{rectangular prism}}=2(\ell w+\ell h+wh),\quad SA_{\text{cube}}=6s^2. \] For a square pyramid with base side \(s\) and slant height \(\ell\): \[ SA_{\text{square pyramid}}=s^2+2s\ell. \]

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Summary

• Rectangular prism: \(SA=2(\ell w+\ell h+wh)\).
• Cube: \(SA=6s^2\). Square pyramid: \(SA=s^2+2s\ell\).
• Surface area answers use square units.

Choose the right formula (and check your units)

Fast strategy

• Circle the target: Does the problem ask for around (perimeter), inside (area), fill (volume), or cover (surface area)?
• Confirm dimensions: perimeter needs 1D lengths, area needs 2D measurements, volume needs 3D measurements.
• Check units: \(cm\), \(cm^2\), \(cm^3\).
• Use exact \(\pi\) unless the question requests an approximation.

Worked example: quick area

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Fun facts (a little history)

• \(\pi\) appears in circle and sphere formulas because circles and spheres are based on radius.
• Scaling: if you double a length, area scales by \(4\) and volume scales by \(8\).
• Real life: perimeter helps with fencing, area helps with flooring/painting, and volume helps with filling tanks and containers.

Final recap

• Perimeter: rectangle \(2(\ell+w)\), regular \(n\)-gon \(ns\), circle \(C=2\pi r=\pi d\).
• Area: rectangle \(\ell w\), triangle \(\tfrac12 bh\), parallelogram \(bh\), rhombus \(\tfrac12 d_1d_2\), circle \(\pi r^2\).
• Volume: rectangular prism \(\ell w h\), cube \(s^3\), cylinder \(\pi r^2h\), cone \(\tfrac13\pi r^2h\), sphere \(\tfrac43\pi r^3\).
• Surface area: rectangular prism \(2(\ell w+\ell h+wh)\), cube \(6s^2\), square pyramid \(s^2+2s\ell\).