G.8(E)·Geometry·Area & Circle MeasurementSupporting

Geometric Probability

Use area models to find probabilities.

Students find geometric probabilities from area ratios and spinner-sector angles, using relationships among unequal sectors instead of assuming every labeled outcome is equally likely.

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 practice45 minutes

Learn the concept

Probability measures a share of the whole

When a point is selected uniformly from a region, $P(\text{event})=\frac{\text{favorable area}}{\text{total area}}$ . Larger regions are more likely. This model requires equal-area pieces to have equal chances of being selected.

Spinner angles replace sector areas

For an unbiased spinner with sectors sharing one center and radius, $P(\text{sector})=\frac{\theta}{360}$ . Sector area is $\frac{\theta}{360}\pi r^2$ , so the common circle area cancels. Twice the sector area means twice the central angle.

Use the entire circle

All central angles sum to $360^\circ$ . Subtract known angles, then divide the remaining angle according to any given ratio. For an event containing several nonoverlapping sectors, add their angles or their probabilities.

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 45 · 1 point

A circular spinner divided into Sectors 1, 2, 3, and 4, with an arrow from the center. Sectors 2 and 4 each have a right-angle mark at the center. Sectors 1 and 3 are unequal in size. — Four-sector spinnerOriginal visual from the released STAAR test.

Worked example

An unbiased spinner has four sectors. Blue measures 90° and green measures 120°. Red has twice the area of yellow. Find the probability of red.

Red and yellow together measure 360 − 90 − 120 = 150 degrees.
Let yellow measure x degrees. Red then measures 2x degrees.
Solve x + 2x = 150, giving x = 50 and red = 100 degrees.
The probability of red is $\frac{100}{360}=\frac5{18}$ .

$\frac5{18}$

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 Predictions Using Theoretical Probability

Five Minute Math · Aaron Daffern, Dallas ISD

Review theoretical probability as a favorable share of the sample space before applying the lesson's unequal-sector area models.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. An unbiased spinner has sectors A, B, and C. C is a semicircle, and A has three times B's area. Find P(A).
Show answer and reasoning
$\frac38$
A and B share the remaining $180^\circ$ in a 3:1 ratio. A measures $\frac34(180)=135^\circ$ , so $P(A)=\frac{135}{360}=\frac38$ .
2. A point is selected uniformly from a 12-by-8 rectangular target. A nonoverlapping collection of favorable regions has total area 30 square units. Find the probability of selecting a favorable point.
Show answer and reasoning
$\frac5{16}$
The target's area is $12(8)=96$ , so the probability is $\frac{30}{96}=\frac5{16}$ .

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 45 · 1 point

In the spinner modeled, Sector 1 has twice the area of Sector 3. If the arrow is spun once, what is the probability that the arrow will land in Sector 1?

$\frac{1}{3}$
$\frac{1}{4}$
$\frac{1}{6}$
$\frac{2}{3}$

Show STAAR answer and explanation

$\frac{1}{3}$

Sectors 2 and 4 are each right-angle sectors, leaving 180° for sectors 1 and 3. With sector 1 twice sector 3, their angles are 120° and 60°. The probability is $120/360=1/3$ .

$\frac{1}{4}$ : The sectors are not all equal; only sectors 2 and 4 each occupy one quarter.

$\frac{1}{6}$ : This is the probability of sector 3, half the size of sector 1.

$\frac{2}{3}$ : This treats sector 1 as two-thirds of the full circle rather than two-thirds of the remaining half.

Ask the tutor about this question

Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Express geometric probability as favorable area divided by total area.
Use central angles and area relationships to compare spinner sectors.
Find a missing sector probability using a whole-circle constraint.

Instructional sequence

1
Engage
5 min
Show a spinner with four unequal sectors. Ask whether four labels guarantee a one-fourth probability for each label.
2
Direct instruction
10 min
State the uniform-selection assumption and connect sector area fractions to central-angle fractions.
3
Guided practice
12 min
Translate marked right angles, semicircles, and sector-area ratios into equations whose angles sum to 360 degrees.
4
Area models
12 min
Find probabilities for combined favorable regions and rectangular targets, checking that the numerator describes the event.
5
Closure
6 min
Complete the exit ticket and explain why counting sectors would give the wrong result.

Misconceptions to address

““Four sectors means every probability is one-fourth.””

Teacher move: That is true only when the sectors have equal angles and the spinner is unbiased. Unequal sectors have different probabilities.

““Twice another sector means a probability of two-thirds.””

Teacher move: Two-thirds would describe a whole consisting only of those two sectors. Include every sector when determining the total.

Materials

•Unequal-sector spinner diagrams
•Protractors
•Colored pencils
•Area-model cards

Teaching tips

State that the spinner is unbiased so equal angle intervals are equally likely.
Write all sector angles around one 360-degree total before solving.
Check that probabilities are between 0 and 1 and that all sector probabilities sum to 1.

Exit ticket · 3–5 min

An unbiased spinner has sectors of 80°, 100°, and 180°. Find the probability of landing in the 100° sector.

Show expected answer

$P=\frac{100}{360}=\frac5{18}$ . The number of sectors is not the denominator because their sizes differ.

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

Probability measures a share of the whole

Spinner angles replace sector areas

Use the entire circle

Study released STAAR visuals

Archived released STAAR Geometry EOC · 2013 · Item 45 · 1 point

Worked example

Watch the lesson

7th Grade Math Predictions Using Theoretical Probability

Try it yourself

1. An unbiased spinner has sectors A, B, and C. C is a semicircle, and A has three times B's area. Find P(A).

2. A point is selected uniformly from a 12-by-8 rectangular target. A nonoverlapping collection of favorable regions has total area 30 square units. Find the probability of selecting a favorable point.

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 45 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Area models

Closure

Misconceptions to address