Volumes of Solids
Aligned to G-GMD.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
Volume formulas: cylinder = pi*r^2*h; cone = (1/3)*pi*r^2*h; pyramid = (1/3)*base area*height; sphere = (4/3)*pi*r^3. The cone and pyramid each hold one-third of the prism or cylinder with the same base and height.
Worked examples
Example 1
Find the volume of a cone with radius 3 and height 10. Use V = (1/3)*pi*r^2*h. Substitute: V = (1/3)*pi*(3)^2*(10) = (1/3)*pi*9*10 = (1/3)*90*pi = 30*pi. Numerically that is about 94.2 cubic units. Note this is exactly one-third of a cylinder with the same base and height (90*pi).
Example 2
Find the volume of a sphere of radius 6. Use V = (4/3)*pi*r³: V = (4/3)*pi*(6)³ = (4/3)*pi*216 = 288*pi, about 904.8 cubic units. Compare with the cylinder that exactly contains it (radius 6, height 12): pi*36*12 = 432*pi. The sphere fills exactly two-thirds of its cylinder.
Example 3
Common mistake: cubing only part of the expression, or forgetting the fraction. For a sphere of radius 3, writing (4/3)*pi*3*3 uses r² instead of r³ and gives 12*pi instead of 36*pi. And dropping the (1/3) from a cone triples the answer. A quick check: a cone, a sphere and a cylinder of the same radius and matching height are in the ratio 1 : 2 : 3. If your three answers are not in that ratio, a factor went missing.
Practice questions
- Find the volume of a cylinder with radius 4 and height 7.
- Find the volume of a sphere with radius 6.
- A pyramid has a square base of side 5 and height 12. Find its volume.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Volúmenes de sólidos
Fórmulas de volumen: cilindro = pi*r^2*h; cono = (1/3)*pi*r^2*h; pirámide = (1/3)*área de la base*altura; esfera = (4/3)*pi*r^3. El cono y la pirámide contienen cada uno un tercio del prisma o cilindro con la misma base y altura.
Ejemplo 1: Halla el volumen de un cono con radio 3 y altura 10. Usa V = (1/3)*pi*r^2*h. Sustituye: V = (1/3)*pi*(3)^2*(10) = (1/3)*pi*9*10 = (1/3)*90*pi = 30*pi. Numéricamente es unos 94.2 unidades cúbicas. Nota que es exactamente un tercio de un cilindro con la misma base y altura (90*pi).
Ejemplo 2: Halla el volumen de una esfera de radio 6. Usa V = (4/3)*pi*r³: V = (4/3)*pi*(6)³ = (4/3)*pi*216 = 288*pi, unas 904.8 unidades cúbicas. Compáralo con el cilindro que la contiene exactamente (radio 6, altura 12): pi*36*12 = 432*pi. La esfera llena exactamente dos tercios de su cilindro.
Ejemplo 3: Error común: elevar al cubo solo parte de la expresión, u olvidar la fracción. Para una esfera de radio 3, escribir (4/3)*pi*3*3 usa r² en vez de r³ y da 12*pi en lugar de 36*pi. Y quitar el (1/3) de un cono triplica la respuesta. Una comprobación rápida: un cono, una esfera y un cilindro del mismo radio y altura correspondiente están en razón 1 : 2 : 3. Si tus tres respuestas no guardan esa razón, se perdió un factor.
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