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Published byChaim Levett Modified about 1 year ago

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The amount of space a figure occupies. Units for volume are cubic units: m 3, ft 3, cm 3, etc.

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The total area of all the surfaces of a figure. Surface area formulas come from adding up the areas of each surface, using the usual area formulas to calculate (imagine unfolding a box). Units for surface area are square units: m 2, ft 2, cm 2, etc.

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Volume:

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Surface Area:

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What are the volume and surface area of a rectangular prism with a length of 6 inches, width of 4 inches, and height of 10 inches?

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The cube is just a special case of the box where l=w=h. Since all the sides have equal length, you’ll sometimes see it given as x or s. That means you’re really using the same formulas, but they’ve been modified to reflect that change.

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A cube has a volume of 64 cubic inches. What is the length of each side, and what is the surface area of the cube?

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Volume: Notice the formula comes from the area of the circular base, times the height.

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Surface Area: The surface area formula comes from the idea of peeling the label off a can – the circumference of the circle becomes the top edge of the label.

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That plus the top and bottom circle areas gives the total surface area:

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What are the volume and surface area of the cylinder shown to the right?

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The box and the cylinder are both shapes that stand upright on their base, with the sides perpendicular to the floor. The general name for this type of shape is a right prism, and we can make a general statement about the formula for volume…

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Recall that for the box, V = lwh. Also notice that the area of the base of the box is A = lw. So we can say the volume of a box is given by V=Ah, where A is the area of the base. For the cylinder, we saw V=πr 2 h. But the area of the circular base is A = π r 2. So the volume of a cylinder could be given as V=Ah, where A is the area of the base.

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That turns out to be the general rule – the volume of any right prism can be calculated from V=Ah, as long as we can calculate the area of the base.

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What would the formula be for the volume of a prism with a triangular base, as shown to the right?

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Volume:

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Surface area: Again, surface area comes from picking apart the figure and finding the area of each surface. e is called “slant height” and is the height of the triangle that forms the side of the pyramid

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What are the volume and surface area of the pyramid shown to the right?

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Volume:

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Surface area: As with the pyramid, e is the slant height or edge height, and its formula comes from the Pythagorean Theorem

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What are the volume and surface area of a cone with the dimensions shown in the figure?

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We can make the same observation and generalization about pyramids (and a cone is just a pyramid with a circular base) that we did about cylinders: if we know the area A of the base of the pyramid, the volume can be calculated from

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Volume: Surface area:

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What are the volume and surface area of a sphere with radius 2 ft?

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