Volume Formulas for K-12 Students
Elementary School Level (K-5)
Cube
Volume = side length × side length × side length
In mathematical notation:
\(V = s^3\)
where \(s\) is the length of a side of the cube.
Rectangular Prism (Box)
Volume = length × width × height
In mathematical notation:
\(V = l \times w \times h\)
where \(l\) is length, \(w\) is width, and \(h\) is height.
Middle School Level (6-8)
Cylinder
Volume = area of the base × height
In mathematical notation:
\(V = \pi r^2 h\)
where \(r\) is the radius of the base, and \(h\) is the height.
Sphere
Volume = four-thirds of pi times the radius cubed
In mathematical notation:
\(V = \frac{4}{3} \pi r^3\)
where \(r\) is the radius of the sphere.
Cone
Volume = one-third of the area of the base × height
In mathematical notation:
\(V = \frac{1}{3} \pi r^2 h\)
where \(r\) is the radius of the base, and \(h\) is the height.
Square Pyramid
Volume = one-third of the area of the base × height
In mathematical notation:
\(V = \frac{1}{3} \times s^2 \times h\)
where \(s\) is the side length of the base, and \(h\) is the height.
Triangular Prism
Volume = area of the triangular base × height of prism
In mathematical notation:
\(V = \frac{1}{2} \times b \times h_t \times l\)
where \(b\) is the base of the triangle, \(h_t\) is the height of the triangle, and \(l\) is the length of the prism.
High School Level (9-12)
Rectangular Pyramid
Volume = one-third of the area of the base × height
In mathematical notation:
\(V = \frac{1}{3} \times l \times w \times h\)
where \(l\) is the length and \(w\) is the width of the rectangular base, and \(h\) is the height.
Ellipsoid
Volume = four-thirds of pi times the product of the three semi-axes
In mathematical notation:
\(V = \frac{4}{3} \pi abc\)
where \(a\), \(b\), and \(c\) are the three semi-axes of the ellipsoid.
Frustum of a Cone
Volume = one-third of pi times height times the sum of the areas of both bases and the square root of the product of the areas of both bases
In mathematical notation:
\(V = \frac{1}{3} \pi h (R^2 + Rr + r^2)\)
where \(h\) is the height, \(R\) is the radius of the base, and \(r\) is the radius of the top.
Frustum of a Square Pyramid
Volume = one-third of the height times the sum of the areas of both bases and the square root of the product of the areas of both bases
In mathematical notation:
\(V = \frac{1}{3} h (A_1 + A_2 + \sqrt{A_1 A_2})\)
where \(h\) is the height, \(A_1\) is the area of the base, and \(A_2\) is the area of the top.
Torus (Ring Donut)
Volume = pi squared times the product of the minor radius squared and the major radius
In mathematical notation:
\(V = 2\pi^2 Rr^2\)
where \(R\) is the major radius (from the center of the tube to the center of the torus), and \(r\) is the minor radius (the radius of the tube).
General Formula Using Integration
For a solid of revolution, volume can be calculated using integral calculus
In mathematical notation:
\(V = \pi \int_{a}^{b} [f(x)]^2 dx\)
where \(f(x)\) is the function being rotated around the x-axis from \(x = a\) to \(x = b\).
Important Note for Students
Remember these key points:
- Volume is measured in cubic units (e.g., cm³, m³, ft³)
- Volume formulas help us calculate the space occupied by a three-dimensional object
- Always check your units and make sure they are consistent
- For irregular shapes, you may need to break them down into regular shapes and add their volumes