G.8(A)·Geometry·Area & Circle MeasurementReadiness

Areas of Polygons and Composite Figures

Find areas of regular polygons, circles, and composite figures.

Students find regular-polygon areas using apothems, decompose composite polygons, and subtract circular regions from squares, including reading lengths from coordinates.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Geometry released practice50 minutes

Learn the concept

Regular polygons use the apothem

Connecting a regular polygon's center to its vertices creates triangles with a common height a, the apothem. Adding their areas gives $A=\frac12 aP$ . For a regular hexagon with side s, $a=\frac{\sqrt3}{2}s$ , so $s=\frac{2a}{\sqrt3}$ and $P=6s$ .

Composite polygons

Divide a composite figure into nonoverlapping familiar shapes. A rectangle has area $bh$ and a triangle has area $\frac12 bh$ , with the height perpendicular to the selected base. Find missing heights by subtracting given lengths when the diagram supports that relationship.

Subtract excluded regions

For a shaded remainder, use containing area minus excluded area. Two congruent semicircles with disjoint interiors have combined area $\pi r^2$ . On a coordinate grid, find horizontal or vertical lengths by subtracting endpoint coordinates. For nested polygons, calculate both areas before subtracting.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 15 · 1 point

A smaller regular hexagon is inscribed in a larger regular hexagon, with each vertex of the smaller hexagon at the midpoint of a side of the larger one. The six triangular regions between them are shaded. A dotted segment b runs from center C perpendicular to a side of the smaller hexagon, and a dotted segment a runs from C perpendicular to a side of the larger hexagon. — Two regular hexagons with center COriginal visual from the released STAAR test.

Archived released STAAR Geometry EOC · 2013 · Item 33 · 1 point

A banner made of a rectangle with a congruent triangle attached to each short side. The rectangle is 17 centimeters long and 10 centimeters tall. A dotted height drawn from the tip of the right triangle to the rectangle's side is 12 centimeters, with a right-angle mark. — Banner of a rectangle and two trianglesOriginal visual from the released STAAR test.

Archived released STAAR Geometry EOC · 2013 · Item 50 · 1 point

Coordinate grid. Square ABCD has vertices A at negative 4, 4; D at 2, 4; C at 2, negative 2; and B at negative 4, negative 2. Point E at negative 4, 1 and point F at 2, 1 are centers of two congruent semicircles that bulge into the square and touch at G, at negative 1, 1. The regions of the square outside both semicircles are shaded. — Square with two tangent semicirclesOriginal visual from the released STAAR test.

Worked example

A square has opposite corners at (−5, −2) and (5, 8), with sides parallel to the axes. Two semicircles lie inside it with the left and right sides as diameters. Find the area outside both semicircles but inside the square.

The side length is 5 − (−5) = 10 units; the vertical difference is also 10.
Each diameter is 10 units, so each radius is 5 units. The semicircles meet only at the square's center.
The square's area is $10^2=100$ . The combined semicircle area is $2\left(\frac12\pi(5)^2\right)=25\pi$ .
Subtract: $100-25\pi\approx21.5$ .

$100-25\pi\approx21.5$ square units

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

7th Grade Math Area of Composite Figures (Additive Property - updated)

Five Minute Math · Aaron Daffern, Dallas ISD

Add areas of nonoverlapping pieces in composite figures.

Watch on YouTube

Texas public school video

7th Grade Math Area of Composite Figures (Sections Removed)

Five Minute Math · Aaron Daffern, Dallas ISD

Find a remaining area by subtracting removed regions.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. A regular hexagon has apothem $5\sqrt3$ centimeters. Find its area.
Show answer and reasoning
$150\sqrt3$ square centimeters
Use $s=\frac{2(5\sqrt3)}{\sqrt3}=10$ , so $P=60$ . Then $A=\frac12(5\sqrt3)(60)=150\sqrt3$ .
2. A decorative panel consists of a 14-by-6 rectangle and two exterior triangles, each attached along a 6-unit side. Each triangle extends 4 units perpendicularly from that side. Find the total area.
Show answer and reasoning
108 square units
The pieces have disjoint interiors, so add: $14(6)+2\left(\frac12(6)(4)\right)=84+24=108$ .
3. A smaller regular hexagon of area 54 square units is entirely inside a larger regular hexagon with apothem 6 units and perimeter 30 units. Find the area between them.
Show answer and reasoning
36 square units
The larger area is $\frac12(6)(30)=90$ . Subtract the inner area: $90-54=36$ .

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 15 · 1 point

Two regular hexagons with center $C$ and apothems $a$ and $b$ are shown in the figure. Each vertex of the smaller hexagon is a midpoint on a side of the larger hexagon. If $a = 12\sqrt{3}$ cm and $b = 18$ cm, what is the total area of the shaded regions?

$648\sqrt{3}$ cm²
$36\sqrt{3}$ cm²
$216\sqrt{3}$ cm²
$1{,}512\sqrt{3}$ cm²

Show STAAR answer and explanation

$216\sqrt{3}$ cm²

A regular hexagon with apothem r has area $2\sqrt{3}r^2$ . The larger area is $864\sqrt{3}$ and the smaller area is $648\sqrt{3}$ , so the shaded difference is $216\sqrt{3}$ cm².

$648\sqrt{3}$ cm²: This is the smaller hexagon area, not the shaded difference.

$36\sqrt{3}$ cm²: This is one of the six congruent shaded triangles, not all six.

$1{,}512\sqrt{3}$ cm²: This adds the two hexagon areas instead of subtracting them.

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Archived released STAAR Geometry EOC · 2013 · Item 33 · 1 point

A banner is composed of two congruent triangles and a rectangle, as shown. What is the total area of the banner in square centimeters?

Show STAAR answer and explanation

290

The rectangle area is $17(10)=170$ . Each triangle has base 10 and perpendicular height 12, so each area is $\frac12(10)(12)=60$ . The total is $170+60+60=290$ cm².

410: This omits the one-half factor for both triangular areas.

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Archived released STAAR Geometry EOC · 2013 · Item 50 · 1 point

Points $A$ , $B$ , $C$ , and $D$ are the vertices of a square. Points $E$ and $F$ are the centers of two congruent semicircles that are tangent to each other at point $G$ . Which value is closest to the area of the shaded regions?

7.7 units²
4.3 units²
17.2 units²
64.3 units²

Show STAAR answer and explanation

7.7 units²

The grid shows a square of side 6. Each semicircle has radius 3, so together they have area $9\pi$ . The shaded area is $36-9\pi\approx7.7$ square units.

4.3 units²: This is not the square area minus the area of the two radius-3 semicircles.

17.2 units²: This is approximately $64-\pi(4)^2$ , using side 8 and radius 4 rather than the grid measurements.

64.3 units²: This is approximately $36+9\pi$ , adding instead of subtracting the unshaded semicircles.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Use an apothem and perimeter to find a regular polygon's area.
Add nonoverlapping polygon areas using perpendicular heights.
Subtract semicircular or polygonal regions to find shaded area.

Instructional sequence

1
Engage
5 min
Partition a regular hexagon into six congruent triangles. Ask which triangle dimension is the apothem.
2
Direct instruction
12 min
Derive A = one-half aP. Use a 30°-60°-90° triangle to connect a regular hexagon's apothem and side length when needed.
3
Guided decomposition
12 min
Find areas of panels formed from rectangles and triangles, distinguishing each triangle's perpendicular height from a slanted side.
4
Shaded regions
14 min
Subtract nested regular-polygon areas and solve square-minus-semicircles problems. Read coordinate differences before identifying radii.
5
Closure
7 min
Complete the exit ticket and identify whether addition or subtraction models the shaded region.

Misconceptions to address

““The apothem runs from the center to a vertex.””

Teacher move: The apothem meets a side at a right angle. The center-to-vertex segment is a radius and is longer.

““Use the semicircle's diameter as its radius.””

Teacher move: Identify the straight diameter first, then divide by 2 before squaring the radius.

““Add all visible areas.””

Teacher move: Decide what is shaded. Add disjoint included pieces, but subtract excluded regions from a containing shape.

Materials

•Regular-hexagon diagrams
•Composite-region cards
•Coordinate grids
•Formula chart and calculator

Teaching tips

Mark right angles on apothems and triangle heights before substituting.
Write an area expression before using a calculator.
Keep pi and radicals exact until the final approximation.

Exit ticket · 3–5 min

A regular octagon has perimeter 48 centimeters and apothem 7 centimeters. Find its area.

Show expected answer

$A=\frac12 aP=\frac12(7)(48)=168$ square centimeters.

Sources & alignment

TEA 2013 Geometry released test TEA 2013 Geometry answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Regular polygons use the apothem

Composite polygons

Subtract excluded regions

Study released STAAR visuals

Archived released STAAR Geometry EOC · 2013 · Item 15 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 33 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 50 · 1 point

Worked example

Watch the lesson

7th Grade Math Area of Composite Figures (Additive Property - updated)

7th Grade Math Area of Composite Figures (Sections Removed)

Try it yourself

1. A regular hexagon has apothem 535\sqrt353​ centimeters. Find its area.

2. A decorative panel consists of a 14-by-6 rectangle and two exterior triangles, each attached along a 6-unit side. Each triangle extends 4 units perpendicularly from that side. Find the total area.

3. A smaller regular hexagon of area 54 square units is entirely inside a larger regular hexagon with apothem 6 units and perimeter 30 units. Find the area between them.

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 15 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 33 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 50 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided decomposition

Shaded regions

Closure

Misconceptions to address

1. A regular hexagon has apothem $5\sqrt3$ centimeters. Find its area.