Polygons, Solids and Surface Area

Updated 7 October 2026

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 flat shape with four straight sides, and its angles add to 360°, because a diagonal splits it into two triangles and 2 × 180 = 360. 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. Those properties are what you calculate with. In a parallelogram with an angle of 70°, the neighboring angle is 180 − 70 = 110 degrees, and opposite angles are equal. The parallel sides cause this: each side crosses a parallel pair, making co-interior angles that add to 180°. A name promises only what it guarantees. A shape that looks like a square is not proved to be one until its sides and angles are shown equal, and a trapezium's slanted sides need not match. Test the angles against the total: 70 + 110 + 70 + 110 = 360.

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?

Find the area of an awkward shape by cutting it into rectangles and triangles you can measure, then adding their areas. Or subtract the missing part from a larger shape. An L-shape is usually quicker the second way. Take a 10 m by 8 m rectangle with a 4 m by 3 m corner cut out. The rectangle is 10 × 8 = 80 square meters, the corner is 4 × 3 = 12, and the L-shape is 80 − 12 = 68. Both methods work because pieces that do not overlap and leave no gaps add up to the whole. A slanted edge needs a triangle, half its base times its height, so a triangle with base 6 m and height 4 m adds 6 × 4 ÷ 2 = 12. A missing side often comes from subtracting two others. Cutting the L-shape the other way checks the answer: 10 × 5 + 6 × 3 = 68.

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 along its length, so its volume is the area of the cross section multiplied by the length. A triangular prism with an end of 6 square centimeters and a length of 10 cm holds 6 × 10 = 60 cubic centimeters. A cylinder is a prism with a circular cross section of area $\pi r^2$πr², so its volume is $\pi r^{2}h$πr²h. See Prisms and Cylinders. The rule works because the prism is a stack of identical thin layers. A cone or a pyramid tapers to a point, so its layers shrink, and it holds exactly one third of the prism with the same base and height. A cone with a base of 12 square centimeters and a height of 5 cm holds 12 × 5 ÷ 3 = 20 cubic centimeters. Use the perpendicular height, not the slanted edge. For a check, the matching cylinder holds 12 × 5 = 60, and 3 × 20 = 60.

In symbols, 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 work with water in a tank?

The water in a rectangular tank is itself a cuboid. Its length and width are the tank's, and its height is the depth of the water, so the volume of water is length × width × depth. A liter is 1,000 cubic centimeters, so a tank 80 cm long and 50 cm wide with water 25 cm deep holds 80 × 50 × 25 = 100,000 cubic centimeters, which is 100 liters. Most tank questions ask how far the water rises or falls, and the same formula turned round answers them: the change in depth is the change in volume divided by the area of the base. Pour 20 liters, which is 20,000 cubic centimeters, into that tank, and the water rises by 20,000 ÷ 4,000 = 5 cm, because the base is 80 × 50 = 4,000 square centimeters. Volume of a Cuboid sets up the rule, and Milliliters and Liters the change of unit.

The same rule handles a solid lowered into the water. A stone of 2,000 cubic centimeters that sinks completely pushes the water up by its own volume, so the level rises by 2,000 ÷ 4,000 = 0.5 cm. A tall post that stands on the bottom and sticks out of the water is different, because the water can no longer use the part of the base the post stands on. In a tank 60 cm by 50 cm holding water 20 cm deep, a post with a 20 cm by 25 cm footprint leaves 3,000 − 500 = 2,500 square centimeters for the water, so its 60,000 cubic centimeters now stand 60,000 ÷ 2,500 = 24 cm deep. That answer holds only if the post is taller than 24 cm, so check it. Before any answer, compare the new depth with the height of the tank: if it is taller than the tank, the difference has overflowed. Work in centimeters and cubic centimeters throughout, and change to liters only at the end.

When water moves between tanks, the volume stays the same and only the base changes. Two tanks joined by a pipe at the bottom settle at the same depth, so the depth is the total volume divided by the total base area. A tank with a base of 2,000 square centimeters holding water 30 cm deep, joined to an empty tank with a base of 1,000 square centimeters, holds 60,000 cubic centimeters, and both tanks settle at 60,000 ÷ 3,000 = 20 cm.

A tap fills at a rate, such as 8 liters a minute, and in a tank with straight sides the depth then rises at a steady rate: 8,000 ÷ 4,000 = 2 cm a minute in the first tank. A graph of depth against time is a straight line whose steepness is the rate divided by the base area. Two taps filling together add their rates, a tap filling while another drains subtracts them, and a tank that is narrower at the top fills faster there, so its graph is two straight pieces and gets steeper where the tank narrows. Rates and Line Graphs cover those two ideas on their own.

How do you find surface area?

Surface area is the total area of the faces on the outside of a solid, while volume measures what fits inside. To find it, unfold the solid into its net so every face lies flat, then add the face areas. A cuboid 5 cm long, 3 cm wide and 2 cm high has three pairs of matching faces: 5 × 3 = 15, 5 × 2 = 10 and 3 × 2 = 6 square centimeters. Double their total, since each face has a partner: 2 × (15 + 10 + 6) = 62 square centimeters. Unfolding moves faces without stretching them, so their areas stay the same. A curved face must be unrolled or given its own formula: a cylinder's curved face unrolls into a rectangle, and a sphere cannot be flattened at all. An open box has one face fewer, so leave out the lid. Count the faces as a check: a cuboid has six, and the working above used 2 × 3 = 6.

In letters, a cuboid's surface area is 2(lw + lh + wh). See Surface Area of a Cuboid.

Think of a cylinder as 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?

A solid is drawn on paper as three flat views from fixed directions, or as one picture on isometric paper. 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 its lengths can be measured exactly. For a cuboid 4 cm long, 2 cm wide and 3 cm high, the plan is a 4 by 2 rectangle, the front elevation is 4 by 3, and the side elevation is 2 by 3. Looking straight along one direction hides that depth and leaves two true lengths. Working backward from the views is harder, since a cube and a cylinder can share a square front elevation, so use all three together. Each length appears in two views, which gives a check: the length is 4 in both the plan and the front view, and the plan covers 4 × 2 = 8 square centimeters.

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, and its area is multiplied by $k^2$k² while its volume is multiplied by $k^3$k³. A cube with 2 cm edges has faces of 4 square centimeters and a volume of 8 cubic centimeters. Scale it by 3 and each face becomes 6 × 6 = 36, 9 times as large, while the volume becomes 6 × 6 × 6 = 216, 27 times as large. An area is two lengths multiplied, each grown by k, so the area grows by k × k. A volume is three lengths, so it grows by k × k × k. Given an area ratio, take its square root to get the length ratio; given a volume ratio, take the cube root. Similar jugs with volumes in the ratio 8 : 27 have heights in the ratio 2 : 3. Check against the small cube: 27 × 8 = 216 and 9 × 4 = 36.

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.

This topic 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.