Volume: Step-by-Step Guide
Volume is the amount of space inside a 3D shape. It is measured in cubic units. For prisms and cylinders, volume is the area of the base times the height. Pyramids and cones hold one third as much as a prism or cylinder with the same base and height. A sphere has volume (4/3) x pi x r^3.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Identify the solid. Decide if the shape is a cube, a box, a cylinder, a sphere, a cone, or a pyramid.\[\text{cube},\;\text{prism},\;\text{cylinder},\;\text{sphere},\;\text{cone},\;\text{pyramid}\]
- Write the volume formula. Box: V = l x w x h. Cube: V = s^3. Cylinder: V = pi x r^2 x h. Sphere: V = (4/3) x pi x r^3. Cone: V = (1/3) x pi x r^2 x h. Pyramid: V = (1/3) x B x h, where B is the area of the base.\[V_{\text{box}} = lwh,\; V_{\text{cyl}} = \pi r^2 h,\; V_{\text{sph}} = \tfrac{4}{3}\pi r^3,\; V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h\]
- Put in the measurements. Replace the letters with numbers. If the formula needs the base area, find that first.\[V = lwh\]
- Multiply and write the units. Work out the answer and write it in cubic units, like cm^3 or m^3.\[V = \text{value}\;\text{units}^3\]
Worked example
A cylinder has a radius of 3 cm and a height of 10 cm. Find its volume.
Step 1: The solid is a cylinder with radius 3 cm and height 10 cm.
Step 2: The formula is V = pi x r^2 x h.
Step 3: Square the radius: 3^2 = 9.
Step 4: Multiply by the height: 9 x 10 = 90.
Step 5: Multiply by pi: V = 90 x pi.
Step 6: For a decimal, use 3.14159: V = about 282.74.
Step 7: Add the units: the volume is 90 x pi cubic cm, or about 282.74 cubic cm.
Check: A box 6 by 6 by 10 would hold 360, and a cylinder fits inside it, so something a bit smaller like 283 makes sense.
Tips and common mistakes
- Volume is in cubic units (cm^3). Surface area is in square units.
- For prisms and cylinders, volume = base area x height. Find the base area first if it is not given.
- A cone holds exactly one third as much as a cylinder with the same base and height. A pyramid holds one third as much as a prism with the same base and height.
- The 4/3 in the sphere formula belongs only to spheres.
- Use r^3 (r cubed) for a sphere, but r^2 (r squared) for a cylinder or cone.
- Make sure all the measurements are in the same unit before you multiply.
- Leave your answer in terms of pi unless the problem asks for a decimal.