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Mathematical Architects · Geometry

Connecting Geometric & Algebraic Descriptions — Module Hub & Visuals

TEKS G.10 · G.11 · G.12

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Lesson Slides & PDFs

Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.

📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)

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Start Here — What Module 4 Is About

What this module is for, and what you will be able to do by the end of it.

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.

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Key Vocabulary

The precise words for what you just investigated. The standards this module carries are in the next section.

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 arc it intercepts — the arc that does not contain the vertex: \(\angle PVQ=\tfrac12\,\overset{\frown}{PQ}\). When that arc is the minor arc this equals \(\tfrac12\,\angle POQ\); when it is the major arc, \(\angle PVQ\) is obtuse while \(\angle POQ\) is not.

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: \(V=\pi r^2 h\), \(SA=2\pi r^2+2\pi r h\).

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Standards in This Module

The Texas Essential Knowledge and Skills this module is built to teach. Geometry is assessed locally — there is no state EOC for this course.

G.10A G.10B G.11C G.11D G.12A G.12B G.12C G.12D 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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How to Read It

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

What you're seeing

Scene 1 — Circle on the plane 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: \(\angle PVQ\) is always exactly half the arc it intercepts — the arc \(PQ\) that does not contain \(V\). That arc can be the major one, in which case \(\angle PVQ\) is obtuse and is not half of \(\angle POQ\); the lab highlights whichever arc is actually intercepted so the two numbers can never drift apart.

Scene 2 — Rectangular prism and Scene 3 — Cylinder each draw a real, depth-buffered solid you can rotate by dragging (or with the arrow keys; + / − zoom, 0 resets). The dimension sliders resize it, an exact flat panel beside it carries the measurements — the prism's net, the cylinder's cross-section — and the readout 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

  1. In Scene 1, drag P and Q apart. Confirm the inscribed angle is always half the arc it intercepts — the highlighted arc \(PQ\), the one that does not contain \(V\). That is the Inscribed Angle Theorem. While that arc is the minor arc it is also half of \(\angle POQ\); press Put V on the other arc and watch the two part company.
  2. 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.
  3. 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.
  4. Move the center to \((2,-3)\) and set \(r=4\). Read the equation: \((x-2)^2+(y+3)^2=16\).
  5. Switch to Scene 3 — Cylinder and double the radius. Surface area roughly quadruples while volume grows even faster — watch both numbers.
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Step 5 · 2D Precision Lab

The Circles & Solids Studio

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

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Want to See This in 3D?

Optional, and always second. The lab above is where you measure and calculate; this is the same figure as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.

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Keep Going

Back to the course, or into the planning documents.

Back to Geometry

Return to the full course page — all five modules, materials, and resources.

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Course Syllabus

Policies, the studio learning environment, grading, and the full itinerary.

Read the Syllabus

Pacing Guide

Every module mapped across the grading periods of the scholar calendar.

View the Pacing Guide

Module title, topic structure, and TEKS alignment adapted from TEA Bluebonnet Learning — Secondary Mathematics (Edition 1), itself adapted from Carnegie Learning, licensed CC BY-NC 4.0. Non-commercial classroom use.