Geometry Formula Sheet
Every geometry formula middle school actually uses — area and perimeter of the 2-D figures, volume and surface area of prisms, cylinders, cones and spheres, plus the Pythagorean theorem — on one printable page. Each formula ships with a worked example, because a formula you have never watched produce a number is a formula you will freeze on mid-test.
Perimeter and Area (2-D figures)
Perimeter walks around the outside — think fencing a garden. Area covers the inside — think carpeting it. On a test the units hand you the answer: plain units (ft, in, cm) mean perimeter, square units (sq ft) mean area.
| Shape | Formula | Worked example |
|---|---|---|
| Rectangle Perimeter | P = 2(l + w) | l = 5 ft, w = 3 ft → P = 2(5 + 3) = 16 ft |
| Rectangle Area | A = lw | l = 4 ft, w = 3 ft → A = 4 × 3 = 12 sq ft |
| Square Perimeter | P = 4s | s = 6 in → P = 4 × 6 = 24 in |
| Square Area | A = s² | s = 6 in → A = 6² = 36 sq in |
| Triangle Area | A = ½bh | b = 10 cm, h = 4 cm → A = ½ × 10 × 4 = 20 sq cm |
| Parallelogram Area | A = bh | b = 8 cm, h = 5 cm → A = 8 × 5 = 40 sq cm |
| Trapezoid Area | A = ½(b₁ + b₂)h | b₁ = 6 in, b₂ = 10 in, h = 4 in → A = ½(6 + 10) × 4 = 32 sq in |
| Circle Circumference | C = 2πr | r = 3 in → C = 6π ≈ 18.85 in |
| Circle Area | A = πr² | r = 3 in → A = 9π ≈ 28.27 sq in |
Surface Area and Volume (3-D figures)
Volume fills — how much water the shape holds. Surface area wraps — how much paper covers it. The prism and cylinder formulas are worth deriving once with a cereal box (unfold it) instead of memorizing: the "2 of" pieces in SA = 2(lw + lh + wh) are the three pairs of opposite faces.
| Shape | Formula | Worked example |
|---|---|---|
| Rectangular prism Volume | V = lwh | 5 × 3 × 2 → V = 30 cu in |
| Rectangular prism Surface area | SA = 2(lw + lh + wh) | 2(15 + 10 + 6) = 62 sq in |
| Cylinder Volume | V = πr²h | r = 2, h = 5 → V = 20π ≈ 62.83 cu in |
| Cylinder Surface area | SA = 2πr² + 2πrh | r = 2, h = 5 → SA = 28π ≈ 87.96 sq in |
| Cone Volume | V = ⅓πr²h | r = 3, h = 6 → V = 18π ≈ 56.55 cu in |
| Sphere Volume | V = (4/3)πr³ | r = 3 → V = 36π ≈ 113.10 cu in |
| Sphere Surface area | SA = 4πr² | r = 2 → SA = 16π ≈ 50.27 sq in |
Right Triangles (Pythagorean theorem)
Only right triangles get the theorem, and c is always the hypotenuse — the side across from the right angle, and the longest side. The triples 3-4-5, 5-12-13 and 8-15-17 show up everywhere; recognizing them on sight saves the arithmetic.
| Shape | Formula | Worked example |
|---|---|---|
| Right triangle Hypotenuse | a² + b² = c² | legs 3 and 4 → 9 + 16 = 25 → c = 5 |
| Right triangle Missing leg | a² = c² − b² | c = 13, b = 5 → 169 − 25 = 144 → a = 12 |
Why every formula carries a worked example
Most formula sheets are lists, and lists get memorized as shapes — students recognize "A = ½(b₁ + b₂)h" without being able to run it. The example column fixes that: cover it, compute from the given numbers, uncover to check. Thirty seconds a row converts the sheet from reference into self-quizzing study material, which is the difference between a sheet that helps and a sheet that decorates a binder.
The sheet is organized the way the courses sequence it. Perimeter and area are grade 6-7 work (with circles as the pi milestone), volume of prisms arrives in grade 5-7, and the Pythagorean theorem is the grade 8 standard — so the same page serves a three-year span: early grades use the top sections, grade 8 uses all of it. Most state end-of-course exams hand students a reference sheet anyway; the point of owning these formulas is speed and confidence, not surviving without paper.
Teaching tips
Let the units pick the formula
Test questions announce what they want: answer in square inches means area, plain inches means perimeter or circumference, cubic inches means volume. Students who read units first never compute the wrong thing with the right numbers.
Write the formula before substituting
The habit that survives every math audit: copy "A = ½bh", replace letters with numbers, then compute. Substituting straight into the calculator skips the step where mistakes are catchable — and where partial credit lives.
Watch the radius-diameter trap
Half the circle errors on any given quiz come from plugging a diameter into r. Before touching a circle problem, underline what is given and write r = or d = next to it. Ten seconds that saves the whole problem.
See the cone-cylinder third
A cone is a cylinder with one end pinched to a point, which is why its volume is exactly one-third: V = ⅓πr²h. Fill a cone with water, pour it into a same-size cylinder, repeat three times. Students who have watched it never forget which formula has the ⅓.
Frequently asked questions
What is on this geometry formula sheet?
Three groups: perimeter and area of rectangles, squares, triangles, parallelograms, trapezoids and circles (including circumference); volume and surface area of rectangular prisms, cylinders, cones and spheres; and the Pythagorean theorem in both directions — finding the hypotenuse and finding a missing leg.
What grade is a geometry formula sheet for?
Grades 6 through 8 are the core: area and perimeter are grade 6-7 standards, volume of prisms and cylinders lands in grades 5-7, and the Pythagorean theorem is the grade 8 standard. High school geometry students use it as a prerequisite review sheet.
Should students memorize all of these formulas?
Not by brute force. Triangle area, rectangle area and the Pythagorean theorem get memorized simply by use. The rest is fair to look up — most state end-of-course exams provide a reference sheet, because the skills being tested are reasoning and substitution, not recall. What students must know cold is which formula the situation calls for.
Why does the trapezoid formula work?
A trapezoid is two triangles sharing the slanted sides, one on each base. Half the sum of the bases times the height is exactly the average of the two triangle areas — the formula is the "average base" shortcut for the same computation.
How do I print the sheet?
The print button above prints it on one letter page in landscape orientation, no download needed. The worked-example column prints with the formulas; cover it with a folded scrap of paper to turn the sheet into a quiz.