2D Area Formulas (Rectangle, Triangle, Circle, Trapezoid)
The area of a shape measures how much surface it covers, expressed in square units. Here are the essential 2D formulas you need to know.
A = l × w, where l is the length and w is the width. A square is a special case where all sides are equal: A = s².A rectangle with length 8 cm and width 5 cm.
8 × 5 = 40 cm²A = ½ × b × h, where b is the base and h is the perpendicular height (not the slant side).A triangle with base 10 m and height 6 m.
½ × 10 × 6 = 30 m²The height must be measured at a right angle to the base — a common source of errors.
A = πr², where r is the radius.A circle with radius 7.
π(49) ≈ 153.94Remember to use the radius, not the diameter; if given the diameter, divide by 2 first.
A = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the parallel sides and h is the perpendicular distance between them.A trapezoid with parallel sides 5 and 9 and height 4.
½ × (5 + 9) × 4 = 28A = b × h, the same as a rectangle but h must be the perpendicular height, not the slant side.Surface Area of 3D Solids
Surface area measures the total area of all faces of a 3D solid, like the amount of wrapping paper needed to cover it.
Rectangular prism and cube
SA = 2(lw + lh + wh). Cube: SA = 6s².A box with dimensions 3 × 4 × 5.
2(12 + 15 + 20) = 2(47) = 94 square unitsA cube with side 3.
6(9) = 54Cylinder
SA = 2πr² + 2πrh = 2πr(r + h). The first term covers the two circular bases, and the second covers the curved lateral surface.A cylinder with radius 4 and height 10.
2π(4)(4 + 10) = 112π ≈ 351.86Sphere and cone
SA = 4πr².A sphere with radius 6.
4π(36) = 144π ≈ 452.39Notice the sphere formula is exactly 4 times the area of a great circle — a beautiful geometric relationship.
SA = πr² + πrl, where l is the slant height. If you know the cone's height h instead, use l = √(r² + h²).Surface area calculations are essential in manufacturing, packaging design, and any application where material cost depends on how much surface must be covered.
Volume of Prisms and Cylinders
The volume of any prism or cylinder follows a single elegant principle: V = (base area) × height. The base can be any shape — rectangular, triangular, hexagonal — and you simply multiply its area by how tall the solid is.
V = l × w × h. Cube: V = s³.A room that is 12 ft × 10 ft × 8 ft.
960 ft³A cube with side 5.
125V = (½ × b × h_triangle) × L, where b and h_triangle define the triangular base and L is the length (depth) of the prism.A triangular prism with base triangle 6 × 4 and length 10.
½ × 6 × 4 × 10 = 120V = πr²h.A cylinder with radius 3 and height 7.
π(9)(7) = 63π ≈ 197.92The key insight is that prisms and cylinders are "extruded" shapes — you take a 2D cross-section and stretch it along a height. This is why the formula is always base area times height. In calculus, this principle generalizes to finding volumes by integrating cross-sectional areas, but for standard prisms and cylinders, these formulas give instant answers.
Volume of Pyramids and Cones
Pyramids and cones share a fundamental relationship with prisms and cylinders: their volume is exactly one-third of the corresponding prism or cylinder with the same base and height.
V = ⅓ × (base area) × h.Rectangular and square pyramids
V = ⅓ × l × w × h. Square pyramid: V = ⅓ × s² × h.A pyramid with a 6 × 8 rectangular base and height 9.
⅓ × 6 × 8 × 9 = 144The Great Pyramid of Giza has a base side of about 230 m and a height of about 146 m.
⅓ × 230² × 146 ≈ 2,574,467 m³Cone
V = ⅓ × πr² × h.A cone with radius 5 and height 12.
⅓ × π(25)(12) = 100π ≈ 314.16The ⅓ factor is not obvious from intuition — it can be proven using calculus (integrating circular cross-sections), or demonstrated physically by showing that it takes exactly 3 cones of water to fill a cylinder of the same radius and height. This experiment is a classic classroom demonstration.
When solving problems, always identify whether the given height is the perpendicular height (used in the formula) or the slant height (the length along the surface), and convert using the Pythagorean theorem if necessary.
Volume of Spheres
V = (4/3)πr³. This formula, first derived by Archimedes, is one of the most elegant results in geometry.A sphere with radius 6.
(4/3)π(216) = 288π ≈ 904.78A basketball has a radius of about 12 cm.
(4/3)π(1728) ≈ 7,238 cm³V = (2/3)πr³ — simply half the full sphere formula.When working with spheres, pay close attention to whether a problem gives you the radius or the diameter. If given the diameter, divide by 2 before substituting into the formula.
For example, a sphere with diameter 10 has radius 5.
(4/3)π(125) = (500/3)π ≈ 523.60The relationship between a sphere and its circumscribing cylinder is remarkable: the sphere's volume is exactly two-thirds of the cylinder's volume. Archimedes was so proud of discovering this relationship that he requested a sphere inscribed in a cylinder to be engraved on his tombstone.
In real-world applications, the sphere formula is used for calculating tank capacities, planetary volumes, and dosage volumes in medicine.
Quick Reference Table
Here is a summary of every formula covered in this guide.
2D Area
| Shape | Formula |
|---|---|
| Rectangle | A = lw |
| Triangle | A = ½bh |
| Circle | A = πr² |
| Trapezoid | A = ½(b₁ + b₂)h |
| Parallelogram | A = bh |
Surface Area
| Solid | Formula |
|---|---|
| Rectangular prism | SA = 2(lw + lh + wh) |
| Cylinder | SA = 2πr(r + h) |
| Sphere | SA = 4πr² |
| Cone | SA = πr² + πrl |
Volume
| Solid | Formula |
|---|---|
| Rectangular prism | V = lwh |
| Cylinder | V = πr²h |
| Pyramid | V = ⅓ × base area × h |
| Cone | V = ⅓πr²h |
| Sphere | V = (4/3)πr³ |
Tips for remembering: Prisms and cylinders use base area × height. Pyramids and cones are always ⅓ of the corresponding prism or cylinder. Sphere surface area is 4πr² (four circles). Sphere volume is (4/3)πr³.
When in doubt on any geometry problem, snap a photo with Solver AI to see the correct formula applied with full step-by-step calculations including unit conversions and dimensional analysis.