Difficulty: Foundational | Prerequisites: Basic arithmetic, familiarity with shapes
Geometry formulas for perimeter, area, surface area, and volume are prerequisites for Calculus 1. You will need them when setting up integrals for area between curves, volumes of revolution, and surface-area problems. Having these formulas at your fingertips saves time and prevents setup errors on exams.
This covers the standard formulas for perimeter, area, and volume of common 2D and 3D shapes. You will use these as building blocks when computing areas under curves, volumes of solids of revolution, and surface areas in calculus.
Perimeter
The total length around the outside of a two-dimensional shape. Measured in linear units (cm, m, ft).
Think of it as "how far you would walk if you traced the shape's edge."
Area
The amount of two-dimensional space enclosed by a shape. Measured in square units (cm^2, m^2, ft^2).
In simple terms, it is how much flat surface the shape covers.
Surface area
The total area of all the outer faces of a three-dimensional solid. Still measured in square units because each face is a flat surface.
Think of it as the amount of wrapping paper you would need to cover the solid.
Volume
The amount of three-dimensional space enclosed by a solid. Measured in cubic units (cm^3, m^3, ft^3).
In simple terms, it is how much liquid the solid could hold.
Arc length
The distance along the curved part of a circle between two points. For a circle of radius r, the arc length is s = r·theta when theta is in radians.
Think of it as measuring a piece of string laid along the curve.
Sector area
The area of a "pie slice" of a circle. For a circle of radius r, the sector area is A = (theta/2)·r^2 when theta is in radians.
In simple terms, it is the area of the pizza slice.
Shape | Perimeter | Area |
|---|---|---|
Rectangle | P = 2l + 2w | A = lw |
Square | P = 4s | A = s^2 |
Triangle | P = sum of all sides | A = (1/2)bh |
Parallelogram | P = sum of all sides | A = bh |
Trapezoid | P = sum of all sides | A = (1/2)(b_1 + b_2)h |
Circle | C = pi·d = 2·pi·r | A = pi·r^2 |
Arc length (radians): S = r·theta
Arc length (degrees): S = (pi/180)·r·theta
Sector area (radians): A = (theta/2)·r^2
Sector area (degrees): A = (theta/360)·pi·r^2
Note the pattern: radians make the formulas simpler, which is why calculus works exclusively in radians.
Solid | Surface Area | Volume |
|---|---|---|
Rectangular solid | SA = 2lw + 2lh + 2wh | V = lwh |
Cube | SA = 6s^2 | V = s^3 |
Cylinder | SA = 2·pi·r^2 + 2·pi·r·h | V = pi·r^2·h |
Cone | SA = pi·r·s + pi·r^2 (where s = slant height) | V = (1/3)·pi·r^2·h |
Sphere | SA = 4·pi·r^2 | V = (4/3)·pi·r^3 |
Rectangular pyramid | SA = lw + 2ls + 2ws | V = (1/3)·lwh |
The slant height of a cone is related to its radius and height by s = sqrt(r^2 + h^2), from the Pythagorean theorem.
Pythagorean theorem: leg^2 + leg^2 = hypotenuse^2, or a^2 + b^2 = c^2
Distance formula: d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). This is the Pythagorean theorem applied to the coordinate plane.
Area and volume formulas are the foundation of every integration application in Calculus 1. When you compute the volume of a solid of revolution using the disc method, you are stacking infinitely thin cylinders (V = pi·r^2·h, with h becoming dx). The shell method uses infinitely thin cylindrical shells. Knowing these base formulas makes setting up the integral intuitive rather than mysterious.
Students often confuse circumference (C = 2·pi·r) with area (A = pi·r^2). One is a length, the other a surface measure. Watch the units.
Students forget the 1/3 factor for cones and pyramids. A cone is one-third of the cylinder with the same base and height, not half.
Students mix up the slant height (s) and the vertical height (h) of a cone. The slant height runs along the surface; the vertical height goes straight down through the centre. The volume formula uses h; the lateral surface area uses s.
Students sometimes use degrees in the arc-length formula S = r·theta, which requires radians. If theta is in degrees, use S = (pi/180)·r·theta instead.
⚠️ You must be able to write down the area and volume formulas from memory. Calculus exams do not typically provide a formula sheet for these.
⚠️ The disc/washer and shell methods for volumes of revolution are built directly on the cylinder and ring formulas. If the base formula is wrong, the integral is wrong.
⚠️ The distance formula appears in arc-length integrals in Calculus 1 and 2.
True or false: the volume of a cone is half the volume of a cylinder with the same base and height. Answer: False. It is one-third, not one-half.
What is the area of a circle with radius 5? Answer: 25·pi.
Fill in the blank: the surface area of a sphere is ___. Answer: 4·pi·r^2.
True or false: arc length S = r·theta works when theta is in degrees. Answer: False. Theta must be in radians.
Q: A cylinder has radius 3 and height 10. What is its volume?
A: V = pi·r^2·h = pi·(9)·(10) = 90·pi.
Q: A cone has radius 4 and height 6. What is its volume?
A: V = (1/3)·pi·r^2·h = (1/3)·pi·(16)·(6) = 32·pi.
Q: Find the arc length subtended by a central angle of pi/3 radians on a circle of radius 12.
A: S = r·theta = 12·(pi/3) = 4·pi.
Q: Find the distance between the points (1, 2) and (4, 6).
A: d = sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5.
These formulas connect to integration applications in Calculus 1: the disc method stacks circular cross-sections (area = pi·r^2), the washer method subtracts one disc from another (area = pi·(R^2 - r^2)), and the shell method wraps cylindrical shells (circumference = 2·pi·r). Understanding where these formulas come from makes the integral setup transparent.
perimeter, area, surface area, volume, circumference, arc length, sector area, rectangle, square, triangle, parallelogram, trapezoid, circle, cylinder, cone, sphere, cube, rectangular solid, rectangular pyramid, Pythagorean theorem, distance formula, slant height, disc method, washer method, shell method, MATH 140, Calculus 1, geometry review