Geometry Reference
Every geometry formula you need in one organized table — square, rectangle, triangle and polygon area, circle and ellipse formulas, volume and surface area for cubes, spheres, cylinders, cones and pyramids, the Pythagorean theorem and special right triangles, and coordinate geometry — with one-tap links to our interactive 3D solid visualizer and unit circle so you can explore the shape right away.
2D shapes — area & perimeter
- Square Area=s²
- Square Perimeter=4s
- Rectangle Area=l·w
- Rectangle Perimeter=2(l + w)
- Triangle Area=½·b·h
- Triangle Area (Heron)=√(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
- Parallelogram Area=b·h
- Trapezoid Area=½(a + b)·h
- Rhombus Area=½·d₁·d₂
- Regular polygon (n sides) Area=½·n·s·a = ½·P·a (a = apothem, P = perimeter)
Circles & ellipses
- Circle Circumference=2πr = πd
- Circle Area=πr²
- Arc length=rθ (θ in radians) = 2πr·(θ°/360)
- Sector area=½r²θ = πr²·(θ°/360)
- Ellipse Area=π·a·b (a, b = semi-axes)
3D solids — volume & surface area
- Cube Volume=s³
- Cube Surface area=6s²
- Rectangular prism Volume=l·w·h
- Rectangular prism Surface area=2(lw + lh + wh)
- Sphere Volume=(4/3)πr³
- Sphere Surface area=4πr²
- Cylinder Volume=πr²h
- Cylinder Surface area=2πr(r + h)
- Cone Volume=(1/3)πr²h
- Cone Surface area=πr(r + l), slant l = √(r² + h²)
- Square pyramid Volume=(1/3)·b²·h
- Square pyramid Surface area=b² + 2b·l
- Triangular prism Volume=(½·b·h)·L (L = length)
Right triangles & the Pythagorean theorem
- Pythagorean theorem=a² + b² = c²
- Common triples=3-4-5, 5-12-13, 8-15-17, 7-24-25
- 45-45-90 triangle sides=1 : 1 : √2
- 30-60-90 triangle sides=1 : √3 : 2
- Trig ratios (SOH-CAH-TOA)=sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Coordinate geometry
- Distance d=√((x₂−x₁)² + (y₂−y₁)²)
- Midpoint=((x₁+x₂)/2, (y₁+y₂)/2)
- Slope m=(y₂−y₁)/(x₂−x₁)
- Line (slope-intercept)=y = mx + b
- Line (point-slope)=y − y₁ = m(x − x₁)
These are the standard geometry formulas. Every one holds for all admissible values, so you can check it by plugging in numbers — for example a sphere of radius 3 has volume (4/3)π·3³ = 36π ≈ 113.097. ✓
2D shapes — area & perimeter
- Square Area = s²
- Square Perimeter = 4s
- Rectangle Area = l·w
- Rectangle Perimeter = 2(l + w)
- Triangle Area = ½·b·h
- Triangle Area (Heron) = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
- Parallelogram Area = b·h
- Trapezoid Area = ½(a + b)·h
- Rhombus Area = ½·d₁·d₂
- Regular polygon (n sides) Area = ½·n·s·a = ½·P·a (a = apothem, P = perimeter)
Circles & ellipses
- Circle Circumference = 2πr = πd
- Circle Area = πr²
- Arc length = rθ (θ in radians) = 2πr·(θ°/360)
- Sector area = ½r²θ = πr²·(θ°/360)
- Ellipse Area = π·a·b (a, b = semi-axes)
3D solids — volume & surface area
- Cube Volume = s³
- Cube Surface area = 6s²
- Rectangular prism Volume = l·w·h
- Rectangular prism Surface area = 2(lw + lh + wh)
- Sphere Volume = (4/3)πr³
- Sphere Surface area = 4πr²
- Cylinder Volume = πr²h
- Cylinder Surface area = 2πr(r + h)
- Cone Volume = (1/3)πr²h
- Cone Surface area = πr(r + l), slant l = √(r² + h²)
- Square pyramid Volume = (1/3)·b²·h
- Square pyramid Surface area = b² + 2b·l
- Triangular prism Volume = (½·b·h)·L (L = length)
Right triangles & the Pythagorean theorem
- Pythagorean theorem = a² + b² = c²
- Common triples = 3-4-5, 5-12-13, 8-15-17, 7-24-25
- 45-45-90 triangle sides = 1 : 1 : √2
- 30-60-90 triangle sides = 1 : √3 : 2
- Trig ratios (SOH-CAH-TOA) = sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Coordinate geometry
- Distance d = √((x₂−x₁)² + (y₂−y₁)²)
- Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
- Slope m = (y₂−y₁)/(x₂−x₁)
- Line (slope-intercept) = y = mx + b
- Line (point-slope) = y − y₁ = m(x − x₁)
About the Geometry Reference
This is the geometry cheat sheet students reach for most: area and perimeter for squares, rectangles, triangles, parallelograms, trapezoids, rhombuses and regular polygons; circumference, area, arc length and sector area for circles, plus ellipse area; volume and surface area for cubes, rectangular prisms, spheres, cylinders, cones, square pyramids and triangular prisms; the Pythagorean theorem with common integer triples and the 45-45-90 and 30-60-90 special right triangles; and coordinate geometry — distance, midpoint, slope and the equations of a line. Every formula is written in clean Unicode notation — no images — so it copies cleanly into your notes and is easy for assistive technology and AI to read. What sets this page apart from a static PDF is that it is wired into UnitConv's interactive tools: tap "Visualize 3D volume & surface area" to see a solid change shape as you adjust its dimensions, "Explore the unit circle" or "Trig identities" to connect the Pythagorean ratios to trigonometry, or convert straight into area or volume units for a real-world problem. Sources for these formulas include Paul's Online Math Notes, Mathwords and Wolfram MathWorld. Use it as a homework reference, an exam revision list, or a quick lookup while you are working through a geometry set.
How to use this reference
- 1 Find the topic you need in the grouped cards — 2D shapes, circles, 3D solids, right triangles or coordinate geometry.
- 2 Copy the formula in clean Unicode notation straight into your work, or check your own result against it.
- 3 Tap "Visualize 3D volume & surface area" or "Explore the unit circle" to open an interactive tool and see the formula in action.
How these geometry formulas fit together
Geometry formulas split into three families by dimension, and it helps to keep the units straight: perimeter and circumference are lengths (m), area is length squared (m²), and volume is length cubed (m³). Every 2D area formula reduces to "base times height" in some form — a rectangle is l·w, a parallelogram is b·h, and a triangle is half of that, ½·b·h, because a triangle is exactly half a parallelogram sharing the same base and height. Heron's formula, Area = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2, gives the same triangle area from the three side lengths alone with no height needed — for a 3-4-5 right triangle it gives √(6·3·2·1) = 6, matching ½·3·4 = 6 exactly. The circle formulas (circumference 2πr, area πr²) are the limiting case of a regular polygon as the number of sides grows without bound, which is also why arc length rθ and sector area ½r²θ are just the circle's formulas scaled by the fraction of a full turn. In 3D, volume formulas generally multiply a base area by a height (a cylinder is πr²·h, a prism is base area times length), while cones and pyramids pick up a factor of ⅓ because they taper to a point. The Pythagorean theorem a²+b² = c² underlies both the special right triangles (45-45-90 gives sides 1:1:√2, since 1²+1²=2=(√2)²) and the coordinate distance formula, which is the Pythagorean theorem applied to the horizontal and vertical differences between two points.
Frequently asked questions
What is the area of a circle?
The area of a circle is πr², where r is the radius. For a circle with radius 2, the area is π·2² = 4π ≈ 12.566 square units. If you are given the diameter d instead, use r = d/2 first. Circumference, the distance around the circle, is a different formula: 2πr (or πd).
What is the formula for the volume of a sphere?
The volume of a sphere is (4/3)πr³, where r is the radius. For a sphere with radius 3, the volume is (4/3)π·3³ = 36π ≈ 113.097 cubic units. The surface area of the same sphere uses a related but different formula, 4πr² — for r = 3 that gives 4π·3² = 36π ≈ 113.097 square units (the same number here only because r = 3 makes the two formulas coincide numerically; in general they differ).
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the two legs and c is the hypotenuse (the side opposite the right angle). It lets you find any one side when the other two are known: c = √(a²+b²), or a = √(c²−b²). Common integer solutions, called Pythagorean triples, include 3-4-5, 5-12-13, 8-15-17 and 7-24-25 — any multiple of these also works, such as 6-8-10.
How do you find the surface area of a cylinder?
The surface area of a cylinder is 2πr(r + h), where r is the radius and h is the height. This comes from adding the two circular ends, each πr² (together 2πr²), to the curved lateral surface, which unrolls into a rectangle of area 2πr·h. For a cylinder with r = 1 and h = 1, the surface area is 2π·1·(1+1) = 4π ≈ 12.566 square units.
Related tools
Turn any of these formulas into an interactive exploration on UnitConv: visualize how volume and surface area change for cubes, spheres, cylinders, cones and pyramids with the solid geometry tool, see the Pythagorean ratios drive sine and cosine on the unit circle, review the trig identities that connect angles to right triangles, or convert a computed area or volume straight into the units you need.