Experiments
Interactive classroom tools built at Riverside.
3D Right Angles
Rotate cuboids, wedges and pyramids to reveal the right-angled triangles hidden inside them.
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Elevation & Depression
Move students around a beach lighthouse and see the angle of elevation up to the light equal the angle of depression back down.
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Angle of Elevation in 3D
Rotate the prism from the question and build the angle of elevation of E from A, one right triangle at a time.
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Motion Graphs
Distance, speed, and acceleration visualised across live motion scenarios.
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What is a Radian?
Drag to open an angle: when the arc is as long as the radius, the angle is one radian. Roll the radius around and a whole turn comes to 2π.
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Arc Length and Sector Area
A sector is a fraction of the whole circle. Drag the angle and see the arc and the area as that fraction of 2πr and πr², then switch to radians and watch the 2π cancel to rθ and ½r²θ.
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Perpendicular Bisector and Angle Bisector
The two compass constructions, step by step. Equal-radius arcs cross to bisect a segment at a right angle, or an angle into two equal halves. The compass point lights up at each step.
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Draw the Bisectors
Your turn with the compass. Plant the point, sweep the arcs, then join the crossings. The crossings only appear if your arcs really cross, so an opening that is too small leaves nothing to join.
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Congruence Tests
Congruent means one triangle fits exactly onto the other. Pick a test (SSS, SAS, AAS, RHS), press Match, and watch one triangle slide and turn to land on the other.
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Similar Triangles
Same shape, different size. See why two triangles are similar (AA, SSS, SAS, enlarge to match), then use the equal ratios of matching sides to find a missing length.
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Area and Volume of Similar Figures
Enlarge by scale factor k and every length grows k times, but area grows k² and volume grows k³. Count the unit squares and cubes to see why: square the ratio for area, cube it for volume.
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Water in a Cone
A part-filled cone, where the water is itself a cone similar to the container. One-eighth full puts the surface at half the depth, not an eighth. Drag the surface, or work back from the volume with a cube root.
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Fold the Circle
All four symmetric properties as a single fold. Fold along the right line through the centre and the two halves land on each other, which is where every one of them comes from.
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Angle at the Centre
The angle at the centre is twice the angle at the circumference on the same arc. Drag the points and the other three angle properties fall out of that one line, reflex case included.
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Proving the Angle Properties
All four angle properties, proved a step at a time from a blank figure. Every proof starts by joining a point to the centre, because OA = OP = OB is the only fact doing any work.
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Speed
Read speed and motion from distance–time graphs across three scenes.
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Alternate Segment Theorem
Drag the points around a circle to see the tangent-chord angle equal the angle in the alternate segment, then step through the proof.
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Midpoint Theorem
Drag a triangle and watch the segment joining two midpoints stay parallel to the third side and exactly half its length, then step through the proof.
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Kinematics
Drive a car along a road and see how displacement s, velocity v and acceleration a connect, and why v = 0 is a turning point.
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Area by Integration
Find area to the x-axis, the y-axis, or between a curve and a line, then shift it across the axis to see which cases turn negative and which do not.
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Increasing & Decreasing
Drag a point along a curve to see where the function is increasing (dy/dx > 0) and decreasing (dy/dx < 0), and read off the ranges of x.
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Maximum, Minimum & Inflexion
Drag a point to find where dy/dx = 0, and classify each stationary point as a maximum, minimum, or point of inflexion with the second derivative.
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Tangents & Normals
Drag a point along a curve to see the tangent and the normal, and why the normal gradient is −1 over the tangent gradient.
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First Principles
Drag a point along a curve and shrink h to watch the chord PQ become the tangent, so its gradient closes in on dy/dx.
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The Gradient Function
Sweep along a curve and plot its gradient at every point. Those gradients trace out the derivative, showing why x² gives 2x and sin x gives cos x.
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Solving Trig Equations
Drag a line across the curve and every crossing lights up as a solution, so students find all the answers in the range, not just the one the calculator gives.
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Trig Ratios of Any Angle
Drag a point round the unit circle, where cos θ and sin θ are simply its coordinates. The acute reference angle gives the size and ASTC gives the sign, so tan of an obtuse or reflex angle stops being a guess.
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Sine, Cosine & Tangent Curves
Watch each wave draw itself as a point sweeps the unit circle. sin is the height, cos is the base, and tan is the gradient of the line to the point.
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Transforming Trig Graphs
Drag a, b and c to stretch, squash and lift a sine, cosine or tangent graph, and read the amplitude, period and midline straight off the equation.
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Linear Law
Watch a curved scatter straighten when you plot the right axes, then read the constants off the gradient and intercept.
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Perpendicular Gradients
Drag two lines to a right angle, then watch a quarter turn show why their gradients multiply to make −1.
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Perpendicular Bisector
Drag P until it is equidistant from A and B to trace the bisector, then step through finding the line's equation from the midpoint and perpendicular gradient.
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Equation of a Circle
Drag the centre and a point on the circle; (x − a)² + (y − b)² = r² falls out of Pythagoras on the radius, with the general form on a second tab.
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Tangent to a Circle
Drag the point of contact around the circle; the tangent stays perpendicular to the radius, and its negative-reciprocal gradient gives the tangent's equation.
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Chord and Centre
Two points on a circle share a perpendicular bisector that runs through the centre. Slide the centre along it to see that two points alone don't fix a circle.
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Circle Through Three Points
Three points fix a circle. Step through the perpendicular bisectors of two chords; where they cross is the centre, and the radius gives the equation.
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Two Points and a Line
A circle through two points, with its centre on a given line. Solve the perpendicular bisector of the chord and the line together to find the centre, then the equation.
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Nature of Roots
Drag a parabola up and down and watch b² − 4ac decide it: cutting the x-axis twice, just touching, then missing it entirely.
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Exponentials & Logarithms
Drag a point on y = aˣ and watch its mirror image land on y = logₐ x: a log is just the inverse of an exponential.
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Binomial Theorem
Watch Pascal's triangle build from the sum of pairs, and click a cell to see where each term of the expansion comes from.
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Remainder & Factor Theorem
The remainder when you divide by (x − a) is just P(a), the height of the curve. Drag the marker until the height hits zero to find a factor.
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Line Meets Curve
Drag a line across a curve. Two crossings slide into one, then none, exactly as the substituted quadratic runs out of roots.
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Completing the Square
Cut the bx rectangle in half, slide one piece under the square, and the corner that finishes it is exactly what you take back off.
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