Learn Perimeter, Area & Volume: Interactive Problems and Worked Examples
Use the question set below to practice the most important perimeter, area, and volume formulas from geometry: perimeter of polygons, circumference of a circle, area formulas (rectangle, triangle, parallelogram, rhombus, circle, semicircle, annulus), and volume & surface area formulas (rectangular prism, cube, cylinder, cone, sphere, pyramid). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Answer the question set and review your mistakes at the end.
How this perimeter, area, and volume practice works
1. Take the practice set: answer the perimeter, area, volume, and surface area questions below.
2. Open the lesson (optional): review the key geometry formulas with clear steps, units, and common mistakes to avoid.
3. Retry: return to the question set and apply the correct formula immediately (and always check units).
What you will learn in the perimeter, area & volume lesson
Foundations & units
Perimeter (distance around): measured in units
Area (space inside): measured in square units (like \(cm^2\))
Volume (space inside 3D): measured in cubic units (like \(cm^3\))
Perimeter & circumference formulas
Rectangle: \(P=2(\ell+w)\) and square: \(P=4s\)
Regular polygon: \(P=ns\) (number of sides \(\times\) side length)
Purpose: Build a clear understanding of perimeter, area, and volume so you can choose the correct formula, compute accurately, and check units in any geometry problem.
Success criteria
Explain the difference between perimeter (distance around), area (space inside), and volume (space inside 3D).
Use correct units: perimeter in units, area in square units (like \(cm^2\)), and volume in cubic units (like \(cm^3\)).
Compute perimeter of rectangles, squares, and regular polygons, and circumference of circles.
Compute area of rectangles, triangles, parallelograms, rhombi, circles, semicircles, and annuli.
Compute volume of rectangular prisms, cubes, cylinders, cones, and spheres.
Compute surface area of rectangular prisms, cubes, cylinders, and square pyramids.
Key vocabulary
Perimeter \(P\): total distance around a 2D shape.
Circumference \(C\): perimeter of a circle (\(C=2\pi r=\pi d\)).
Area \(A\): measure of the region inside a 2D shape (square units).
Volume \(V\): measure of space inside a 3D shape (cubic units).
Surface area \(SA\): total area of the outside surfaces of a 3D shape.
Radius \(r\) and diameter \(d\): \(d=2r\).
Base \(b\), height \(h\), and slant height \(\ell\) (for pyramids/cones).
Quick pre-check
Pre-check 1: Which unit is appropriate for an area measurement?
Hint: Area counts how many unit squares fit inside a region.
Pre-check 2: A circle has radius \(6\). What is its diameter?
Hint: Diameter is twice the radius: \(d=2r\).
Perimeter & Circumference
Perimeter and circumference: distance around
Learning goal: Compute perimeter for polygons and circumference for circles using reliable formulas.
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:\[C=2\pi r=\pi d.\]
Worked example
Example: What is the perimeter of a rectangle with length \(8\) and width \(5\)?
Use \(P=2(\ell+w)\): \[P=2(8+5)=2\cdot 13=26.\]
Try it
Try it 1: What is the perimeter of a regular heptagon with side length \(2\)?
Hint: A heptagon has 7 sides, so \(P=7\cdot 2\).
Try it 2: What is the circumference of a circle with radius \(9\)?
Hint: Use \(C=2\pi r\).
Summary
Perimeter adds side lengths. Regular polygon: \(P=ns\).
Circumference: \(C=2\pi r=\pi d\).
Area of Polygons
Area of common polygons
Learning goal: Use the correct area formula and label your answer in square units.
Key idea
Area measures the amount of surface inside a 2D shape. Common area formulas:\[A_{\text{rectangle}}=\ell w,\quad A_{\text{triangle}}=\frac12 bh,\quad A_{\text{parallelogram}}=bh,\quad A_{\text{rhombus}}=\frac12 d_1d_2.\]
Worked example
Example: What is the area of a triangle with base \(10\) and height \(6\)?
Use \(A=\tfrac12 bh\): \[A=\frac12(10)(6)=30.\]
Try it
Try it 1: What is the area of a parallelogram with base \(12\) and height \(3\)?
Hint: For a parallelogram, \(A=bh\).
Try it 2: What is the area of a rhombus with diagonals \(10\) and \(4\)?
Learning goal: Use circle area formulas, convert between diameter and radius, and handle common composite regions.
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 \(r=\tfrac{d}{2}\).
Worked example
Example: What is the area of a semicircle with diameter \(12\)?
Diameter \(12\) means radius \(r=6\). \[A=\frac12\pi r^2=\frac12\pi(6^2)=\frac12\pi(36)=18\pi.\]
Try it
Try it 1: What is the area of a semicircle with diameter \(10\)?
Hint: Diameter \(10\Rightarrow r=5\). Then \(A=\tfrac12\pi r^2\).
Try it 2: What is the area of an annulus with outer radius \(8\) and inner radius \(5\)?
Convert diameter to radius using \(r=\tfrac{d}{2}\).
Volume of Prisms
Volume of prisms and cubes
Learning goal: Compute volume using the correct dimensions and report answers in cubic units.
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):\[V=s^3.\]
Worked example
Example: What is the volume of a rectangular prism with dimensions \(6\) by \(2\) by \(3\)?
Multiply the three dimensions: \[V=\ell w h = 6\cdot 2\cdot 3=36.\]
Try it
Try it 1: What is the volume of a cube with side length \(5\)?
Hint: Cube volume is \(V=s^3\).
Try it 2: A rectangular prism has dimensions \(2\times 3\times 5\). What is its volume?
Hint: Multiply the three dimensions: \(V=\ell w h\).
Summary
Rectangular prism: \(V=\ell w h\).
Cube: \(V=s^3\).
Volume answers use cubic units.
Cylinder Cone Sphere
Volume of cylinders, cones, and spheres
Learning goal: Recognize which 3D solid you have and apply the correct volume formula (especially the \(\tfrac13\) factor for cones).
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 (\(d=2r\)).
Worked example
Example: What is the volume of a cylinder with radius \(2\) and height \(5\)?
Use \(V=\pi r^2h\): \[V=\pi(2^2)(5)=\pi(4)(5)=20\pi.\]
Try it
Try it: What is the volume of a cone with radius \(2\) and height \(6\)?
Cone: \(V=\tfrac13\pi r^2h\) (do not forget \(\tfrac13\)).
Sphere: \(V=\tfrac43\pi r^3\).
Surface Area
Surface area: add all outer faces
Learning goal: Compute total surface area by adding the areas of every face, and keep units in square units.
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 \(\ell\):\[SA_{\text{square pyramid}}=s^2+2s\ell.\]
Worked example
Example: What is the surface area of a rectangular prism with dimensions \(2\times 3\times 5\)?
Use \(SA=2(\ell w+\ell h+wh)\): \[SA=2(2\cdot 3+2\cdot 5+3\cdot 5)=2(6+10+15)=2(31)=62.\]
Try it
Try it 1: What is the surface area of a square pyramid with base side \(3\) and slant height \(4\)?
Hint: \(SA=s^2+2s\ell\). Here \(s=3\), \(\ell=4\).
Try it 2: What is the surface area of a cube with side length \(5\)?
Next step: Close this lesson and retry your quiz. If you miss a question, reopen the book and review the page for that formula (perimeter, area, volume, or surface area).
