Geometric Probability Worksheet

Fifteen numbered geometry problems, ten of them shaded-area and shaded-length probability targets and five reviewing radicals, composite areas and right-triangle trigonometry, each reprinted with a full worked solution.

An educational teaching resource from Peter Jonnard entitled Geometry Probability Worksheet with Answer Key downloadable at Teach Simple.

Grades: Grades 10–12
Pages: 6
File type: PDF
Answer key: Included

Updated Sep 18, 2021

Subject: Geometry, Math
Topic: Geometry, Probability
Resource types: Worksheets, Worksheets & Printables
Answer key: Included

From the author

About this resource

This review guide comes with an answer key!

What's included

A three-page problem set headed "Geometric Probability Spring 2021", opening with an "Appoint a reader: ______" line and the direction "For full credit, you MUST SHOW the formulas and calculations of all parts"
Problems 1-6: area-based geometric probability figures, including rectangles in a rectangle, a right triangle in a 16 by 20 rectangle, a 6 in circle inscribed in a square, a circle divided into eight congruent sectors, and four circles inside a 24 cm square
Problems 7-9: length-based probability on a 5-12 right triangle's hypotenuse, the crust of a 12 in semicircular pizza, and a 3 m section of segment AC where m AB = 5 m and m BC = 8 m
Problem 10: a square dart board with a 2 cm bullseye inside a 9 cm circle inside a 20 cm square, broken into parts a through g
Problem 11: ten radicals, √40 through √49, to be left in simplified radical form or labelled "ALREADY SIMPLIFIED"
Problems 12-15: unknown sides and areas in two composite rectangle figures, the shaded area of a rhombus, 30-60-90 and 45-45-90 shortcut triangles, and a right triangle solved with trigonometry to the nearest hundredth
A three-page worked version headed "Geometric Probability Spring 2021 Key" reprinting every figure with its completed totals, fractions and decimal values

Resource types: Worksheets, Worksheets & Printables

In the classroom

Teaching tips & learning objectives

Learning objectives

Calculate the probability that a randomly thrown dart lands in a shaded region by forming the ratio of shaded area to total area
Transfer the same ratio idea to one dimension, comparing a shaded length such as a 13-unit hypotenuse or a 6π in arc against a total perimeter
Use the relationship shaded area + unshaded area = total area to reach an answer through whichever region is easier to compute
Write square roots in simplified radical form and recognise which of √40 to √49 are already simplified
Solve a right triangle for unknown sides, area and perimeter using cosine and sine of a given 28° angle, rounded to the nearest hundredth with units

Teaching tips

The sheet opens with a blank "Appoint a reader" line, so decide before handing it out whether groups will name that reader or the line will simply be left unused
Problem 6 carries its own hint, telling students the radius of one of four circles across a 24 cm square is "NOT 12 cm"; it works well as the figure you model on the board first
Item 11 asks for the words "ALREADY SIMPLIFIED" on any radical that cannot be reduced, so remind students that leaving the entry blank is not the same response
Answers on the worked pages are given in exact form and then as a decimal, which lets you accept either 16 - 3π over 16 or 0.411 depending on what the class is practising
Problems 12 through 15 sit on the last page and can be assigned separately as a special-right-triangle and radicals review without touching the probability items

Skills covered

Geometric probability from area — students compute the shaded-to-total ratio for figures such as a 6 in circle inscribed in a square (36π/144) and four circles packed inside a 24 cm square (576 - 108π over 576).
Geometric probability from length — problems 7 to 9 switch the ratio to one dimension: a 13-unit hypotenuse against a 30-unit perimeter, a 6π in arc against a (12 + 6π) in semicircle perimeter, and 6 m of a 13 m segment.
Working with the complement — problem 6 directs students to decide whether the shaded or unshaded region is easier to calculate, then recover the other from the total 576 cm² square.
Simplifying radicals — item 11 has students reduce √40 to 2√10, √44 to 2√11 and √48 to 4√3, and mark the primes such as √41 and √47 as already simplified.
Right-triangle measurement — problems 14 and 15 apply the 30-60-90 and 45-45-90 side ratios and then cos 28° and sin 28° to find a = 6.18 ft, b = 3.29 ft, an area of 10.16 ft² and a perimeter of 16.47 ft.

Good to know

Questions teachers ask about this resource

What does the "Appoint a reader" line at the top of the sheet call for?

A blank line reading "Appoint a reader: ______" heads both the problem set and the worked pages, so the sheet is laid out to be worked by a named group with one student reading the problem text. The direction immediately below it requires the formulas and calculations for all parts to be shown for full credit.

What do the problems after number 10 cover?

Items 11 to 15 review the geometry the probability work leans on: simplifying √40 through √49, finding unknown sides and areas inside two composite rectangle figures, the shaded area of a rhombus marked 20 cm and 8 cm, the 30-60-90 and 45-45-90 shortcut triangles, and a right triangle solved from a 7 ft side and a 28° angle.

Are the solutions written in exact form or as decimals?

Both are given side by side. The worked pages show exact values such as (576 - 108π) cm², 77π/400 and π/(2 + π) together with their decimal equivalents 0.411, 0.605 and 0.611, while problem 15 is specified to the nearest hundredth with units, ending at a = 6.18 ft and p = 16.47 ft.