3D Shapes and Formulas
Solid shapes are three-dimensional figures (3D). We will be studying solids that have faces, meaning they have flat sides such as cubes, rectangular solids, square pyramids, cones, and cylinders. The only solid that we will go over without a face is a sphere. For solids, rather than finding area and perimeter, we find volume. Volume measures how much a solid can hold. The measurement for volume is cubic units.
Tip: answers may vary slightly from other sources/calculators due to the rounding of pi.
Cube
The formula to find the volume for a cube is similar to finding area. Where area is concerned with squaring sides, cubes have edges. Think back to the lesson where we practiced squares and cubes. What number do we use for cube? That’s right, we use 3.
Find the volume of a cube
Volume = Edge³
Volume = 5³
Volume = 125 cm³
Rectangular Solid
Like with cubes, the formula to find the volume for a rectangular solid is similar to finding the area of a rectangle. Where area equals length multiplied by width, add an additional element, height, to find the volume for a rectangular solid.
Find the volume of a rectangular solid
Volume = L * W * H
Volume = 8 * 3 * 2
Volume = 48 cm³
Square Pyramid
Notice the bottom of the pyramid is a square. For the square pyramid, the square bottom is referred to as the base edge. The base edge is squared when finding the volume of a square pyramid. Multiply the squared base edge by 1/3, then by the height of the square pyramid to complete finding the volume.
Find the volume of a square pyramid
Volume = 1/3(Base Edge)² * H
Volume = 1/3(100)² * 120
Volume = 400,000 cm
Cylinder
The formula for the cylinder should be one of the easier to remember if you know the formula for finding the area of a circle. Look at the cylinder below. Note that it is comprised of two circles some distance apart. The distance apart is the height. The volume formula for a cylinder is simply the area of the circle multiplied by its height.
Find the volume of a cylinder
Volume = π(Radius²) * H
Volume = 3.14(2²) * 6
Volume = 75.36 in³
Cone
The volume formula of a cone is built from the cylinder’s volume formula. Use the same formula as the one for the cylinder and multiply by 1/3. Note that 1/3 is also used for the square triangle also.
Find the volume of a cone
Volume = 1/3(π * Radius² * H)
Volume = 1/3(3.14 * 4² * 7)
Volume = 117.23 mm³
Sphere
As mentioned in the first paragraph, the sphere has no faces or edges.
Find the volume of a sphere
Volume = 4/3πr³
Volume =4/3(3.14 *9³)
Volume = 3,052 cm³
