Surface Area and Volume
Lessons in this chapter
About Surface Area and Volume
Surface Area and Volume moves from flat 2D area work into three-dimensional figures, covering how much material covers a solid’s outside and how much space fills its inside.
Find the surface area of prisms and find the surface area of pyramids both use the idea of unfolding a solid into its net — the flat 2D shapes that make up each face — then adding up the area of every one of those faces to get the total surface area.
Find the surface area of cylinders applies the same ‘add up every surface’ idea to a curved solid, combining the area of the two circular bases with the area of the curved side, which unrolls into a rectangle.
Find the volume of prisms and pyramids shifts from outside surface to inside space, using base area times height for prisms and one-third of that same formula for pyramids — a relationship kids see demonstrated rather than just told, since three pyramids with a matching base and height exactly fill one prism.
Visualize cross-sections of three-dimensional figures closes the chapter by slicing a 3D solid with a flat plane and identifying the 2D shape that slice reveals — a rectangular prism sliced parallel to its base reveals a rectangle, while a cylinder sliced perpendicular to its base reveals a circle.
Frequently asked questions
A net is a 2D shape you’d get by unfolding a 3D solid flat, showing every one of its faces laid out separately — like an unfolded cardboard box. Find the Surface Area of Prisms uses nets to make it visual why surface area is just ‘add up the area of every face,’ rather than a formula to memorize blindly.
Unfold the prism into its net, find the area of each individual face (rectangles and/or the matching top-and-bottom polygon bases), and add every face’s area together. Find the Surface Area of Prisms works through this face-by-face for rectangular prisms and triangular prisms.
Add the area of the base polygon to the area of every triangular lateral face that meets at the pyramid’s apex. Find the Surface Area of Pyramids introduces slant height specifically for this — the height along a triangular face, which is different from the pyramid’s straight-up vertical height used later for volume.
A cylinder has two circular bases (found with A = πr²) plus one curved lateral surface, which unrolls flat into a rectangle whose length equals the circle’s circumference and whose width equals the cylinder’s height — so Find the Surface Area of Cylinders combines circle-area and circumference formulas from the earlier Circles chapter with this unrolling idea, instead of adding up flat polygon faces like a prism or pyramid does.
Volume of a prism = base area × height, because a prism is essentially the same base shape stacked straight up for the full height — multiplying the flat base area by how many ‘layers’ of that height gives the total space filled. Find the Volume of Prisms and Pyramids starts here before extending to pyramids.
It’s demonstrable, not just a rule to memorize: three identical pyramids with the same base and height exactly fill one prism with that same base and height, which is why the pyramid volume formula is exactly one-third of base area times height — one-third of the matching prism’s volume.