Polygons, Solids and Surface Area

The volume of a solid comes from cutting it into pieces you can already measure. The surface area comes from unfolding it flat and adding up the faces. Every formula below is built by one of those two moves.

What makes a quadrilateral special?

A quadrilateral is any shape with four straight sides. It gets a special name when it keeps certain sides or angles equal: a rectangle keeps four right angles, a parallelogram keeps opposite sides parallel, and a trapezium keeps just one pair of parallel sides.

The names nest. A square is a rectangle whose four sides are equal, and a rectangle is a parallelogram whose angles are right angles. Anything true of every parallelogram is therefore true of every rectangle and every square, so one proof covers all three. See Special Quadrilaterals.

A rhombus keeps all four sides equal. A kite keeps two pairs of equal neighboring sides. In both, the diagonals cross at right angles, so the area is half the product of the diagonals: area $=d_1\times d_2$= d₁ × d₂ / 2. That is usually the quickest route through a kite question. See Rhombus and Kite.

How do you find the area of an awkward shape?

Cut the shape into rectangles and triangles you can measure, then add their areas. Or start from a larger shape you can measure and subtract the part that is missing. An L-shape is usually quicker the second way: one big rectangle minus one small one.

Any correct cut gives the same area, so choose the one that needs the fewest unknown lengths. See Area of Composite Shapes.

How do you find the volume of a prism or a cone?

A prism has the same cross section all the way along its length, so it is a stack of identical layers. Its volume is the area of the cross section multiplied by the length. A cylinder is a prism whose cross section is a circle of area $\pi r^2$πr², so its volume is $\pi r^{2}h$πr²h. See Prisms and Cylinders.

A pyramid or a cone tapers to a point. Put it inside the prism with the same base and the same height, and it holds exactly one third of that prism. So the volume of a pyramid or a cone is $\frac{1}{3}\times$1/3 × base area × height, whatever the shape of the base. See Pyramids and Cones.

A sphere has every point of its surface the same distance r from the center. Its volume is $\frac{4}{3}\pi r^3$4/3 πr³. Fit the sphere snugly inside a cylinder of radius r and height 2r, and it fills exactly $\frac{2}{3}$2/3 of the cylinder. Archimedes proved this by slicing: a hemisphere and a cylinder with a cone drilled out have slices of equal area at every height, $\pi \left(r^2-h^2\right)$π(r² − h²), so they have equal volume. The cylinder $\pi r^3$πr³ minus the cone $\frac{\pi r^3}{3}$πr³/3 leaves $\frac{2}{3}\pi r^3$2/3 πr³ for the hemisphere, and double that is the sphere. See Spheres and Archimedes and the Volume of a Sphere.

How do you find surface area?

Volume measures what fits inside a solid. Surface area measures how much paper covers the outside. Unfold the solid into its net so that every face lies flat, then add the areas of the faces.

A cuboid has six faces in three matching pairs, so its surface area is 2(lw + lh + wh). Find the three different faces, add them, and double the total. See Surface Area of a Cuboid.

A cylinder is a tin: a top, a bottom, and a curved label. Peel the label off and it is a rectangle of height h whose width is the circumference, $2\pi r$2πr, so the curved surface is $2\pi$2πrh. Add the two circular ends, $\pi r^2$πr² each: the whole surface is $2\pi r^2+2\pi$2πr² + 2πrh. See Surface Area of a Cylinder.

A cone has two heights. The vertical height h runs straight up the middle from the center of the base to the tip. The slant height l runs from the tip down the sloping side to the rim. The radius, the vertical height and the slant height form a right-angled triangle, so $l^2=h^2+r^2$l² = h² + r² by Pythagoras. A cone with radius 3 and vertical height 4 has slant height 5. See The Slant Height of a Cone.

The curved surface of a cone is $\pi rl$πrl. Cut it up the slant and roll it flat: it is a sector of a circle of radius l whose arc is the base rim $2\pi r$2πr, so it is $\frac{r}{l}$r/l of $\pi l^2$πl², which is $\pi rl$πrl. Add the base $\pi r^2$πr² for the whole cone. A sphere cannot be flattened; its surface area is $4\pi r^2$4πr². See Cones and Spheres.

How do you draw a solid on paper?

Draw what it looks like from three fixed directions. The plan is the view from directly above. The front elevation is the view from straight in front, and the side elevation is the view from the side. Each view is flat, so every length in it can be measured exactly.

The harder task is the reverse: given the three views, work out the solid. One view alone never fixes it, because different solids can share a view. See Plans and Elevations.

On isometric dot paper a solid is drawn in one picture: every edge runs one hop along one of three directions. See Drawing Solids on a Dot Grid.

What happens to area and volume when you scale a shape?

Multiply every length of a shape by k. The area is multiplied by $k^2$k², because an area is two lengths multiplied together. The volume is multiplied by $k^3$k³, because a volume is three lengths multiplied together.

Double every length of a box: it needs 4 times as much paper to cover it and holds 8 times as much. See Scaling Area and Volume.

Two results about solids worth knowing

Euler's formula says that for any solid with flat faces and no holes, vertices − edges + faces = 2. A cube has 8 vertices, 12 edges and 6 faces: 8 − 12 + 6 = 2. Stretching it into a cuboid changes none of the counts, and a square pyramid gives 5 − 8 + 5 = 2. See Euler's Formula for Polyhedra.

A Platonic solid has faces that are all the same regular polygon, with the same number meeting at every corner. At least three faces must meet at a corner, and their angles must add to less than 360°, or the corner lies flat. Equilateral triangles have 60° angles, so 3, 4 or 5 can meet. Squares have 90° angles and regular pentagons 108°, so only 3 of either can meet. Three regular hexagons already make 360°, so hexagons cannot form a corner at all. That leaves exactly five possible corners, and each one builds a solid. See Why There Are Exactly Five Platonic Solids.

How to attack a solid you have never seen

First decide whether the solid has parallel ends. If it does, it is a prism, and its volume is the cross-section area multiplied by the length, however strange the cross section looks. If it tapers to a point, it is a pyramid or a cone, and its volume is one third of the prism around it.

If it is neither, for example a cylinder with a hemisphere on top, split it at the joins and treat each piece separately. Volumes add, because the pieces occupy separate space. Surface areas do not simply add, because joining two pieces hides a face on each of them: the circle where the hemisphere meets the cylinder belongs to neither total.

When one question asks for both, keep the units apart: volume in cm³, surface area in cm². See Combined Volume and Surface Area.

The mistakes worth naming

• Using the slant height for volume, or the vertical height for curved surface area. Convert with Pythagoras first: $l^2=h^2+r^2$l² = h² + r².
• Scaling volume by the length factor. Double the lengths and the volume is 8 times bigger, not 2.
• Forgetting the ends of a cylinder. An open pipe has one surface. A sealed tin has three: the curved side and two circles.
• Missing the one third on a cone or pyramid. A solid that tapers to a point holds one third of the prism around it.

Learn this properly in the app

Each lesson linked above is a lesson in Math Challenge, with a diagram, a worked example and practice questions. This page follows on from shapes, area and perimeter, uses the right-angle work in lines, angles and Pythagoras, and leads into congruence, circle theorems and transformations, where the scaling rule reappears as the ratio of areas and volumes of similar solids.