Maths
Maths projects for class 9 and 10: 45 ideas.
Each project below opens to its own page with what it shows, the science behind it, the materials, safety, step-by-step instructions, what you should see and viva questions. Many can be made from things at home; where a kit helps, the page links one.
Other subjects for class 9 and 10: Physics, Chemistry, Biology, Earth and space, Electronics and robotics, Environment, Computing and coding.
Pythagoras theorem model
In a right-angled triangle, the square on the longest side has the same area as the squares on the other two sides together (a² + b² = c²). Cut the two smaller squares into pieces and they fit exactly into the big one.
Area of a circle by cutting sectors
Cut a circle into many equal sectors and arrange them top to tail: they form a shape close to a rectangle with sides πr and r, so the area is πr².
Probability with coins and dice
With only a few throws the results are uneven, but as the number of throws grows, the share of heads gets closer to one half and each face of a die to one sixth.
Clinometer for heights and distances
Measure the angle up to the top of a tree or building and your distance from its base. The height is the distance × tan(angle), plus the height of your eyes above the ground.
Fibonacci numbers in nature
In the sequence 1, 1, 2, 3, 5, 8, 13 and so on, each number is the sum of the two before it, and the ratio of neighbours gets closer to about 1.618, the golden ratio. The spirals on sunflower heads and pine cones often come in Fibonacci numbers.
Napier's bones multiplication rods made from ice-cream sticks
Each rod carries the multiplication table of one digit written in split boxes. Laying rods for 4 and 6 side by side and reading row 7 gives 7 times 46 = 322 by adding along the slanted lines, a 400-year-old calculating tool.
Five Platonic solids folded from cardboard nets
Only five solids can be made with every face the same regular polygon and the same number of faces at every corner: the tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons) and icosahedron (20 triangles). Each is folded from a flat net.
Rice-pouring proof that a cone is one third of a cylinder
A paper cone and a paper cylinder with the same base radius and the same height are filled with rice. Exactly three cones of rice fill the cylinder, showing that the volume of a cone is one third of pi r squared h.
Unit circle trigonometry board with a swinging radius arm
A 10 cm arm pivots at the centre of a circle on a grid. At any angle, the height of its tip above the horizontal is 10 times the sine and the distance along the horizontal is 10 times the cosine, so at 30 degrees the tip is 5 cm up and 8.66 cm across.
Knotted 3-4-5 rope for making right angles like builders
A rope with 13 knots at equal spacing is pulled into a triangle with sides of 3, 4 and 5 spaces. The angle between the 3 and 4 sides is exactly a right angle, because 3 squared plus 4 squared equals 5 squared, a trick used to lay out buildings for thousands of years.
Measure a tree's height from shadows and similar triangles
At the same moment the Sun makes the shadow of a tree and the shadow of a metre stick in the same proportion. If a 1 metre stick casts a 1.5 metre shadow and the tree casts a 9 metre shadow, the tree is 9 divided by 1.5, which is 6 metres tall.
Nail-board Galton machine that drops beads into a bell curve
Beads falling through seven rows of nails bounce left or right at random at each nail. With eight collecting slots at the bottom, the expected share of beads in each slot follows Pascal's triangle, 1, 7, 21, 35, 35, 21, 7, 1 out of 128, which draws a bell-shaped curve.
Birthday paradox survey across the sections of a school
In a group of just 23 people there is a better than even chance (about 50.7 percent) that two share a birthday, and with 30 people the chance is about 70.6 percent. Collecting birthdays from many class sections shows how often a shared birthday actually turns up.
Balance beam that finds the mean of a data set
Equal weights hung from a numbered beam at the data values balance only when the support is at the mean. Weights at 2, 3 and 7 balance at 4, their mean, and moving one weight from 7 to 13 shifts the balance point to 6, showing how an extreme value pulls the mean.
Pascal's triangle cup pyramid with hidden number patterns
Paper cups stacked in a triangle carry Pascal's numbers: each cup holds the sum of the two cups above it. Row sums double each time (1, 2, 4, 8, 16, 32), the second diagonal counts 1, 2, 3, 4, and the third diagonal holds the triangular numbers 1, 3, 6, 10, 15.
Golden rectangle and spiral built from compass squares
Squares of side 1, 1, 2, 3, 5, 8 and 13 cm fit together into a rectangle of 13 cm by 21 cm, and a quarter circle in each square joins into a spiral. The ratio of the sides, 21 divided by 13, is about 1.615, close to the golden ratio 1.618, and the ratio gets closer as more squares are added.
Estimate pi by scattering rice over a square and its circle
Rice grains scattered evenly over a square land inside the circle drawn within it with probability equal to the circle's share of the area, which is pi divided by 4. Counting grains inside and outside and multiplying the inside fraction by 4 estimates pi, a Monte Carlo method used in real computer simulations.
Cardboard algebra tiles that factorise x squared plus 5x plus 6
Square tiles of side x, strips of x by 1 and unit squares are arranged into a rectangle. One x squared tile, five x strips and six unit tiles fit exactly into a rectangle with sides x + 2 and x + 3, so x squared + 5x + 6 = (x + 2)(x + 3).
Pinboard coordinate plane with string lines, slopes and intercepts
Pins at grid points on a four-quadrant board let a string be stretched through any two points to show a straight line. The string through (0, 1) and (2, 5) rises 4 for every 2 across, slope 2, and crosses the y axis at 1, so its equation is y = 2x + 1.
Matchstick Nim game board with its unbeatable strategy
Two players take 1, 2 or 3 sticks in turn from a pile of 21 and the player who takes the last stick wins. Whoever leaves a multiple of 4 after their move (20, 16, 12, 8, 4) can always win, so the second player wins a game of 21 by always making the two moves in a round add to 4.
Tower of Hanoi from bottle caps with the 2 to the n minus 1 rule
Discs of different sizes must be moved from one peg to another, one at a time, never putting a larger disc on a smaller one. The least number of moves is 1, 3, 7, 15, 31 and 63 for 1 to 6 discs, which is 2 to the power n minus 1, so 64 discs would take more than 18 quintillion moves.
Day-of-the-week wheel that dates any day with the doomsday rule
In any year the dates 4/4, 6/6, 8/8, 10/10, 12/12, 9/5, 5/9, 7/11, 11/7 and the last day of February all fall on the same weekday, called the doomsday. A wheel finds the doomsday for the year, and counting from the nearest of these dates gives the weekday of any date, such as 15 August 1947, a Friday.
Vedic maths flip board for squaring numbers ending in 5
To square a number ending in 5, multiply the tens digit by the next number up and write 25 after it: 35 squared is 3 times 4 = 12 followed by 25, so 1,225. The flip board shows each step and the algebra behind it: (10a + 5) squared = 100a(a + 1) + 25.
Nikhilam slider that multiplies numbers near 100 in one line
To multiply 97 by 96, write how far each is below 100 (3 and 4). Subtract crosswise, 97 minus 4 = 93, and multiply the deficits, 3 times 4 = 12, to get 9,312. The slider lines up the two numbers and their deficits so the answer appears in two steps.
String art parabola made from straight lines only
Threads joined between two scales at right angles, the first point on one to the last on the other and so on, form no curve themselves, yet their edges outline a smooth parabola. The curve is the envelope of the lines, touched by every one of them.
Seven bridges of Konigsberg board with removable bridges
Four land areas joined by seven bridges cannot be walked in one trip crossing every bridge once, because every land area has an odd number of bridges. Lifting any one bridge off the board leaves only two odd areas and the walk becomes possible, starting at one odd area and ending at the other.
Width of a pond or river from a baseline and the sine rule
You can find a distance you cannot measure directly by measuring a baseline and two angles. The sine rule in the triangle then gives the width.
Sine wave drawn by a turning wheel and a moving roll of paper
The height of a point on a turning wheel, measured above the centre, follows a sine curve in time. Tracing it on moving paper gives amplitude, period and a smooth wave.
Arm span against height: scatter plot, mean, spread and best-fit line
Two measurements on the same person can be linked. Collecting data for a class and drawing a scatter plot, a line of best fit and a correlation shows how statistics finds a pattern in data.
Area of a leaf by random dots: a Monte Carlo estimate with a check
The area of an odd shape can be estimated by throwing random points on a square that contains it. The share of points that land inside the shape estimates the share of the square it covers.
Monty Hall with three cups: should you switch? 100 trials and the proof
In the three-door puzzle, switching wins two times out of three, though many people feel that it is a fifty-fifty choice. Playing it many times and listing all the outcomes shows why.
How long is a game of snakes and ladders? 100 simulated games
The number of turns to finish a game is random but has a stable average. Playing or simulating many solo games gives the mean game length and shows how the snakes and ladders change it.
Balancing a cardboard triangle at its centroid, checked by coordinates
The point where the three medians of a triangle meet is its centroid, and a flat triangle balances there. Its coordinates are the average of the three vertex coordinates.
Taxicab geometry on a city grid: distances, midpoints and square circles
On a street grid you cannot travel diagonally, so the distance is the sum of the horizontal and vertical steps. With this rule a circle looks like a diamond, which shows that geometry depends on how distance is defined.
Bouncing ball: a geometric sequence of heights and its infinite sum
A ball that bounces back to a fixed fraction of its previous height gives a geometric sequence of heights. The total distance travelled until it stops is the sum of a geometric series.
Straw and pipe-cleaner dome that shows why triangles are rigid
A four-sided frame can collapse but a triangle cannot change shape without changing the length of a side. A dome of triangles is strong and light.
Straw and protractor clinometer to find building height by trigonometry
The height of a tall object can be found from the angle of elevation and the distance to its base using tan of the angle. The clinometer measures that angle.
Decoding the home electricity bill: units, slabs and appliance costs
A bill is calculated from units used, in kilowatt hours, at slab rates plus fixed charges. By computing the energy of each appliance and checking it against the bill, you apply percentages, multiplication and a piecewise function.
Piggy bank mathematics: simple against compound interest
Money in a deposit can earn simple interest, only on the original amount, or compound interest, on the growing total. Tables and graphs for Rs 1,000 over ten years show how the two grow apart.
Is a 1 kg packet really 1 kg? Sampling and averages with a kitchen scale
No two packets weigh exactly the same, so averages, ranges and standard deviation describe how close the packets are to the printed weight. You weigh sample packets and decide whether the pack is accurate.
String wound on a half ball: surface area of a sphere is four circles
The string that covers the curved surface of a hemisphere covers exactly two flat circles of the same radius, so a hemisphere has an area 2 pi r squared. A whole sphere is therefore four circles, 4 pi r squared.
Cone, hemisphere and cylinder of equal radius: volumes in ratio 1 to 2 to 3
A cone, a hemisphere and a cylinder with the same radius, and height equal to the radius for the cone and cylinder, have volumes in the ratio 1 to 2 to 3. Pouring rice from a cone and a hemisphere together exactly fills the cylinder.
Mirror on the ground: finding the height of a tree or building by similar triangles
A small mirror on the ground, a tape and your own eye height are enough to find the height of a pole, tree or building without climbing it. Light reflecting from the mirror forms two similar right triangles, so the heights are in proportion to the distances.
Pin and thread board for the circle theorem: angle at the centre is double
On a circle, an arc makes an angle at the centre exactly twice the angle it makes at any point on the remaining circumference. Moving the pin around the circle shows that the angle at the edge does not change.
One 40 cm thread loop, many shapes: which encloses the most area?
A closed loop of thread of fixed length can be made into a triangle, rectangle, square, hexagon or circle. The enclosed area grows as the shape gets more regular and is largest for the circle, so for a fixed perimeter the area can be very different.