Surface Area Formula Calculator
Find the surface area and volume of common 3D shapes with the formula shown, convert between radius, diameter, circumference and area of a circle, work out arc length and sector area, and estimate rates of change and limits.
Surface area formula calculator
Circle calculator and circumference calculator
Arc length and sector area
Average rate of change
Limits calculator
Surface area formulas
| Shape | Surface area | Volume |
|---|---|---|
| Cube (side a) | 6a² | a³ |
| Rectangular prism | 2(ab + ah + bh) | abh |
| Cylinder | 2πr² + 2πrh | πr²h |
| Cone | πr² + πr·s, s = √(r² + h²) | πr²h ÷ 3 |
| Sphere | 4πr² | 4πr³ ÷ 3 |
| Square pyramid | a² + 2a·s, s = √((a/2)² + h²) | a²h ÷ 3 |
A cylinder with radius 3 and height 5 has surface area 2π(9) + 2π(15) = 48π ≈ 150.80 square units and volume 45π ≈ 141.37. Surface area is in square units; volume in cubic units.
Area of a circle formula and circumference
Area = πr² and circumference = 2πr = πd. A circle with radius 5 has area 78.54 and circumference 31.42. Going backwards, r = C ÷ 2π or r = √(A ÷ π). For flat shapes other than circles, see the area calculator.
Arc length formula
Arc length = r × θ with θ in radians, or 2πr × (degrees ÷ 360). A 60° arc of a radius-10 circle is 10π ÷ 3 ≈ 10.47; the sector's area is r²θ ÷ 2 ≈ 52.36 and the chord is 2r sin(θ/2) = 10.
Average rate of change formula
Over an interval [a, b], average rate of change = (f(b) − f(a)) ÷ (b − a), the slope of the line through the two points. For f(x) = x² from 1 to 3: (9 − 1) ÷ 2 = 4.
Limits calculator: how it estimates a limit
It evaluates f just left and right of the point, getting closer each row, and reports the value both sides approach. sin(x)/x → 1 as x → 0, and (x² − 1)/(x − 1) → 2 as x → 1, even though neither is defined at that point. 1/x has no limit at 0 (the sides go to −∞ and +∞). It's a numerical estimate, so check results that matter with algebra (factoring, L'Hôpital's rule).
Common questions
What is the surface area of a cylinder?
2πr² + 2πrh. Radius 3, height 5: 48π ≈ 150.80.
What is the area of a circle formula?
A = πr². Radius 5: 78.54.
How do I find the circumference?
C = 2πr = πd. Radius 5: 31.42.
What is the arc length formula?
s = rθ (θ in radians). A 60° arc of radius 10 is about 10.47.
What is the average rate of change formula?
(f(b) − f(a)) ÷ (b − a).
Sources
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General information, not financial, tax or legal advice. Check important numbers with the official source before acting on them.