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Connecting Geometric & Algebraic Descriptions — Visual Lab

Module 4 is where pictures become equations. In Circles, drag points around a circle and watch the central angle, the inscribed angle, the arc, and the equation \((x-h)^2+(y-k)^2=r^2\) update together — you'll see why an inscribed angle is always half its central angle. In 3-D Solids, rotate a box or a cylinder, resize it, and watch its volume and surface area recompute live. Everything here is hand-built in the studio — no Desmos or GeoGebra account needed.

TEKS G.10 · G.11 · G.12

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The Circles & Solids Studio

Two live tools in one. Switch between them, then drag, slide, and rotate — watch the math move.

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How to Read It

What's on the screen, and a few experiments worth running.

What you're seeing

Circle mode draws a circle on the coordinate plane. Drag the center and the radius handle to change the equation \((x-h)^2+(y-k)^2=r^2\) live. Two points, P and Q, sit on the circle; drag them to set an arc. The central angle \(\angle POQ\) opens from the center, while the inscribed angle \(\angle PVQ\) opens from a third point V on the circle. Watch the readout: the inscribed angle is always exactly half the central angle that subtends the same arc.

3-D Solid mode draws a rectangular prism or a cylinder you can rotate by dragging. The dimension sliders resize it, and the panel shows volume and surface area with the formula filled in. Notice volume grows far faster than surface area as the solid scales — the heart of the surface-area-to-volume idea.

Try this

In Circle mode, drag P and Q apart. Confirm the inscribed angle is always half the central angle — the Inscribed Angle Theorem.
Drag the inscribed point V anywhere along the major arc. The inscribed angle does not change — every inscribed angle on the same arc is equal.
Drag P and Q until they are opposite ends of a diameter. The inscribed angle snaps to \(90^{\circ}\) — an angle inscribed in a semicircle is a right angle.
Move the center to \((2,-3)\) and set \(r=4\). Read the equation: \((x-2)^2+(y+3)^2=16\).
Switch to 3-D Solid, choose Cylinder, and double the radius. Surface area roughly quadruples while volume grows even faster — watch both numbers.

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Key Vocabulary & Standards

The words a Mathematical Architect uses for what you just did.

Equation of a circle

A circle with center \((h,k)\) and radius \(r\) is the set of points satisfying \((x-h)^2+(y-k)^2=r^2\).

Radius & diameter

The radius \(r\) runs from center to edge; the diameter \(d=2r\) runs all the way across through the center.

Central angle

An angle with its vertex at the center. Its measure equals the measure of the arc it intercepts.

Inscribed angle

An angle with its vertex on the circle. It measures half the central angle that intercepts the same arc: \(\angle PVQ=\tfrac12\,\angle POQ\).

Arc & arc length

An arc is a portion of the circle. Arc length \(=\dfrac{\theta}{360^\circ}\cdot 2\pi r\), where \(\theta\) is the central angle.

Volume & surface area

Prism: \(V=\ell wh\), \(SA=2(\ell w+\ell h+wh)\). Cylinder: \(V=\pi r^2 h\), \(SA=2\pi r^2+2\pi r h\).

G.10A G.10B G.11C G.11D G.12A G.12B G.12C G.12E

G.10 — identify, describe, and connect two- and three-dimensional figures, including cross-sections and solids of revolution. G.11 — apply surface-area and volume formulas to solve problems. G.12 — apply circle relationships: central and inscribed angles, arcs, arc length, and the equation of a circle.

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