6.5 Circles, Polygon Angle Measures, and Composite Figures

6.5 Circles, Polygon Angle Measures, and Composite Figures

On the Praxis 5164, Geometry and Measurement items frequently ask you to combine polygon angle formulas with circle circumference and area. The ETS Study Companion explicitly flags composite problems such as finding the difference between the area of a square and the area of a circle inscribed in that square. The good news: the key formulas appear on the ETS 5164 math reference sheet. Your job on test day is not to invent them from scratch—it is to locate them quickly and apply them accurately.

Formulas on the ETS 5164 Reference Sheet

Practice opening the on-screen reference sheet and finding these three relationships until the motion is automatic:

A regular polygon has all sides congruent and all interior angles congruent. Once you know the sum S, each interior angle of a regular n-gon is S/n = 180°(n−2)/n.

Exterior angles (useful companion fact)

At each vertex of a convex polygon, an interior angle and an adjacent exterior angle form a straight line, so they sum to 180°. For any convex polygon, the sum of one exterior angle at each vertex is 360°. For a regular n-gon, each exterior angle measures 360°/n. Many Praxis items let you choose whether to work from the interior-sum formula on the sheet or from the exterior-angle shortcut—both are valid if applied carefully.

Worked Example 1: Regular Polygon Interior Angles

Problem. What is the measure of each interior angle of a regular octagon?

Solution. An octagon has n = 8 sides. From the reference sheet,

S = 180°(8 − 2) = 180° · 6 = 1080°.

Because the octagon is regular, each interior angle is 1080°/8 = 135°.

Check with exterior angles. Each exterior angle of a regular octagon is 360°/8 = 45°, so each interior angle is 180° − 45° = 135°. Same answer, different path.

Worked Example 2: Circumference and Area from the Sheet

Problem. A circular garden has radius 7 meters. Find its circumference and area in terms of π.

Using C = 2πr and A = πr²,

C = 2π(7) = 14π meters, and A = π(7)² = 49π square meters.

Praxis tip. Leave answers in terms of π unless the item asks for a decimal approximation. If a decimal is required, use the on-screen graphing calculator and watch the radius-versus-diameter trap: C = 2πr = πd, so doubling a radius/diameter error doubles the circumference error and quadruples the area error.

Worked Example 3: Square with an Inscribed Circle (ETS-style Composite)

Problem. A circle is inscribed in a square with side length 10 cm. Find (a) the area of the square, (b) the area of the circle, and (c) the area of the region inside the square but outside the circle.

When a circle is inscribed in a square, the circle touches all four sides, so the circle’s diameter equals the side of the square.

• Side of square: s = 10 cm → diameter d = 10 cm → radius r = 5 cm.
• Area of square: 10 × 10 = 100 cm².
• Area of circle: A = πr² = π(5)² = 25π cm².
• Shaded “corners” region: 100 − 25π cm².

If the item asks for a numerical approximation, 25π ≈ 78.54, so the difference is about 21.46 cm². Prefer the exact form 100 − 25π when it is an option.

Circumscribed versus inscribed (a high-frequency language trap)

• Circle inscribed in a square: diameter = side of square.
• Square inscribed in a circle: diagonal of square = diameter of circle.

Swapping those two phrases is one of the most common Praxis geometry errors. Always sketch before substituting into C = 2πr or A = πr².

Worked Example 4: Composite Perimeter

Problem. A running track consists of a rectangle that is 100 m by 60 m with a semicircle attached to each 60-m end (two semicircles make one full circle of diameter 60 m). What is the perimeter of the outer boundary?

The two lengths of 100 m remain part of the outer path. The two widths are replaced by the curved ends. Those two semicircles form one full circle of radius r = 30 m, so the curved distance is C = 2π(30) = 60π meters. Total perimeter:

100 + 100 + 60π = 200 + 60π meters.

Do not add the 60-m widths; those segments are interior to the track outline once the semicircles are attached.

Teaching Scenario

A seventh-grade student is finding each interior angle of a regular hexagon. The student correctly recalls “something with n − 2” but writes S = 180°(n + 2), gets S = 1440°, and concludes each angle is 240°. Another student uses S = 180°(n − 2) = 720° correctly, then forgets to divide by 6 and reports 720° as “the” interior angle.

Instructional move. Have both students open the Praxis-style formula S = 180(n−2) and label S as a sum, not a single angle. Then require a two-line template: (1) compute S; (2) if the polygon is regular, divide by n. Asking “Is 240° plausible for an interior angle of a convex hexagon?” builds estimation habits—convex interior angles are less than 180°.

Praxis Traps to Avoid

1. Memorizing instead of using the sheet. The Study Companion advises familiarity with what is given. During practice, force yourself to look up C = 2πr, A = πr², and S = 180(n−2) rather than relying on memory alone.
2. Radius/diameter mix-ups. Area uses r²; if a problem gives diameter 12, then r = 6 and A = 36π, not 144π.
3. Forgetting to divide the angle sum. S is the total; regular polygons need S/n.
4. Composite shading errors. Decide clearly whether you need square minus circle, circle minus square, or a ring/annulus. Sketch and shade before computing.
5. Perimeter double-counting. In composite outlines, omit shared internal segments.

Mastering these sheet-based formulas with composite reasoning prepares you for the exact Geometry and Measurement tasks emphasized in the Praxis 5164 Study Companion.