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Perimeter, Area, and Volume

The problems in this section explore the numerical and/or physical
properties of perimeter, area, and/or volume, which are among the most
important (and practical!) topics in a standard geometry curriculum. Some problems deal
with classic geometric relations, but many incorporate practical
applications. Heron's formula would be an appropriate search topic as
a special subject.
 Related Resources

Interactive resources from our Math Tools project:
Geometry: Perimeter, Area, & Volume
The closest match in our Ask Dr. Math archives:
High School: Euclidean/Plane Geometry
NCTM Standards:
Geometry Standard for Grades 912
Access to these problems requires a Membership.

Added Areas
 Annie Fetter

Geometry, difficulty level 1. A circle and an isosceles triangle are inscribed in a square. If the areas of the three figures added together is 28 square units, what's the edgelength of the square?
... more>>

Aim at This Target!
 Annie Fetter

Geometry, difficulty level 4. Figure out the areas of different parts of this target, and the relation between the different areas. Can you explain this relation?
... more>>

All Around the World
 Terry Trotter & Annie Fetter

Geometry, difficulty level 2. If a wire wrapped around the equator of the Earth is lengthened by 100 meters, how far above the surface of the Earth would the wire now be if lifted off the surface an equal distance all the way around?
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Approximating Pi
 Annie Fetter

Geometry, difficulty level 3. Approximate the value of pi using polygons instead of a circle.
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Approximating the Circumference and Area of a Circle
 Annie Fetter

Geometry, difficulty level 2. Take a circle and circumscribe a square and inscribe a square. Construct a square between the other two squares, and compare the area and the perimeter of the middle square with the area and circumference of the circle.
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Area of a Rectangle
 Annie Fetter

Geometry, difficulty level 2. Given rectangle ABCD with point E on CD. AE is 3, BE is 4, and AE is perpendicular to BE. What's the area of ABCD?
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Area of a Rhombus
 Annie Fetter

Geometry, difficulty level 2. Find the area of a rhombus, given one angle measure and the length of
the longer diagonal.
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Assessing Seismic Risk
 Craig Foster and Annie Fetter

Geometry, difficulty level 3. Figure out how likely it would be for an earthquake to occur within
certain distances of a building.
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Building a Dog Enclosure
 Annie Fetter

Geometry, difficulty level 3. Two people are arguing about whether three sections of fencing 6, 8, and 10 feet long will enclose more or less area than three sections 6, 8, and 12 feet long.
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Building a Rain Gauge
 Annie Fetter

Geometry, difficulty level 3. Given the depth of the water in a bottle, figure out how much rain really fell, considering that the opening of the bottle is much smaller than the main part.
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Buying Paint and Shingles
 Annie Fetter

Geometry, difficulty level 2. Figure out how many gallons of paint and how many bundles of shingles I need to buy to finish my new shed.
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Clever Cuts
 Annie Fetter

Geometry, difficulty level 2. Find three different ways to cut a hexagonal cake into six congruent
pieces.
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Comparing the Areas of Two Squares
 Annie Fetter

Geometry, difficulty level 3. Given two squares, one with the diameter of a circle as an edge, the
other inscribed in the circle. How do the areas of the squares
compare?
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Congruence and Area
 Annie Fetter

Geometry, difficulty level 3. "If two triangles have the same area, then they are congruent." Is this a true statement?
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Congruent Chords
 Annie Fetter

Geometry, difficulty level 3. Two congruent circles are drawn, and four congruent chords are drawn, two in each circle, all perpendicular to the diameter through both circles. The distance between the two furthest chords is 20, and the distance between two chords of the same circle is 8. What's the area of one of the circles?
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Congruent Rectangles
 Annie Fetter

Geometry, difficulty level 1. Seven congruent rectangles are arranged to form a larger rectangle. If the area of the large rectangle is 756 units^2, what's the perimeter of the large rectangle?
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Congruent Rectangles Help
 Annie Fetter

Geometry, difficulty level 2. Find some possible areas for the large rectangle shown in the
given picture.
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Cutting Corners
 Annie Fetter

Geometry, difficulty level 1. Some election signs must have corners cut off them to comply with
local election rules.
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Descartes' Triangle?
 Annie Fetter

Geometry, difficulty level 3. Two vertices of a triangle are located at (0,6) and (0,12). If the area of the triangle is 12 square units, where can the third vertex be? What if the triangle is also isosceles?
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Different Views
 Annie Fetter

Geometry, difficulty level 3. Given a front, top, and side view of an object, find the volume of that object.
... more>>

Divided Rectangle
 Annie Fetter

Geometry, difficulty level 1. A rectangle is divided into four rectangles with areas 45, 25, 15, and x. Find x.
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Equal Area & Congruence
 Annie Fetter

Geometry, difficulty level 2. If two triangles are congruent, they have the same area. Is the converse true?
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Equilateral Triangle Areas
 Annie Fetter & Steve Weimar (extra)

Geometry, difficulty level 3. If two equilateral triangles have edgelengths of 9 cubits and 12
cubits, what's the edgelength of the equilateral triangle whose area
is equal to the sum of the areas of the other two?
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Equilateral Triangles and Circles
 Annie Fetter

Geometry, difficulty level 2. Two vertices of an equilateral triangle are on the diameter of a circle whose area is 49 pi. The other vertex is on the circumference of the circle. What's the largest possible area of the triangle?
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Filling the Hot Tub
 Annie Fetter and Steve Risberg

Geometry, difficulty level 3. Given the dimensions of a hot tub and its submerged woodburning
stove, figure out how long it will take to fill the tub using a
garden hose.
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Finding the Base of a Trapezoid
 Annie Fetter

Geometry, difficulty level 3. Given trapezoid ABCD, with AD one base and BC the other and a 60 degree angle at CDA. If CD is 26sqrt3/3 meters, BC is 7 meters, and the area of the trapezoid is 130 square meters, what's the length of AD?
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Finding the Volume of a Box
 Annie Fetter

Geometry, difficulty level 2. If the areas of the top and front of a box are 252 and 105 square inches, respectively, and the surface area of the box is 834 square inches, what's the volume of the box?
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Find the Area of This Square
 Annie Fetter

Geometry, difficulty level 2. Given a square split into four pieces, one with an area of 35 and the other with an area of 8, find the area of the whole square.
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Find the Largest Equilateral Triangle
 Annie Fetter

Geometry, difficulty level 3. Two vertices of an equilateral triangle lie on a diameter of a circle whose area is 36pi cm^2, and the third vertex lies on the circle. What is the largest possible area of the triangle?
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Find the Volume of a Box
 Annie Fetter

Geometry, difficulty level 2. Given the surface area of three sides, find the dimensions and volume of a rectangular box.
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Fitting In
 Annie Fetter

Geometry, difficulty level 3. Figure out which peg takes up a larger percentage of the corresponding
hole.
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Framed!
 Annie Fetter

Geometry, difficulty level 3. Given a triangular painting with dimensions 20 x 14 x 10, and three different ways you could frame it, which frame will take the least amount of material?
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A Gardening Puzzle
 Annie Fetter

Geometry, difficulty level 2. If a rectangular garden were 2 feet wider and 3 feet longer, it would be 64 square feet larger. If it were 3 feet wider and 2 feet longer, it would be 68 square feet larger. What are the dimensions of the garden?
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Geoboard Gymnastics
 Annie Fetter

Geometry, difficulty level 3. Find squares with different areas on a geoboard.
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Geometric Algebra?
 Annie Fetter

Geometry, difficulty level 2. Given a geometric picture that illustrates the fact that a(a+b)=a^2+ab, explain what the second picture illustrates.
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Going Global
 Annie Fetter & NCTM's WLME

Geometry, difficulty level 2. Look at Shakespeare's Globe Theater from a mathematical standpoint  figure out the interior angle of an icosagon and calculate how many people could stand in the "field" in front of and around the stage.
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A Hexagon and a Triangle in a Circle
 Annie Fetter

Geometry, difficulty level 2. A regular hexagon and an equilateral triangle are both inscribed in the same circle so that the hexagon and the triangle share three vertices. The radius of the circle is 10 units. What is the area of the region between the two polygons?
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Hippocrates' Lunes
 Annie Fetter

Geometry, difficulty level 3. ABC is half a square inscribed in a semicircle. Then a semicircle is constructed on AB. BD is then constructed, the perpendicular bisector of AC, and triangle ABD is shaded, as is the part of the outer semicircle that's not part of the original semicircle that's not part of the original semicircle. What did Hippocrates prove about the two shaded regions?
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How Deep Is It?
 Annie Fetter

Geometry, difficulty level 2. How deep will the water be when this container is resting its largest
face?
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In a Pickle
 Lillian Ray

Geometry, difficulty level 2. Figure out whether or not there will be enough pickle juice to make
the tartar
sauce.
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An Inscribed Circle
 Annie Fetter

Geometry, difficulty level 2. A circle is inscribed in a triangle with sides of 6, 6, and 4. What's the radius of the circle?
... more>>

Is the Fair Fair?
 Annie Fetter

Geometry, difficulty level 3. Figure out the probability of winning at this popular midway game.
... more>>

Journey Through Outer Space
 Annie Fetter

Geometry, difficulty level 2. How many Earths can fit in Uranus?
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Learning About GPS
 Annie Fetter

Geometry, difficulty level 4. What percentage of the surface of the earth can be seen by a satellite orbiting at an altitude of 20,200 km? Are the Global Positioning System satellites "geosynchronous"?
... more>>

Lotsa Popcorn!
 Annie Fetter

Geometry, difficulty level 1. Can the World's Largest Popcorn Ball really be 50,000 times bigger than a
normal one? Wow!
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Making a Quilt of Squares
 Annie Fetter

Geometry, difficulty level 4. Figure out the dimensions of some closely packed squares.
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Making Pie Crust
 Annie Fetter

Geometry, difficulty level 3. How would you change the recipe for a 9inch pie crust to make an 11inch pie crust?
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Making the Biggest Triangle
 Lillian Ray

Geometry, difficulty level 2. What length should the third side be to make the triangle with the
maximum possible area?
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The "Middle" Triangle
 Annie Fetter

Geometry, difficulty level 2. In rectangle ABCD, E is the midpoint of BD, F is the midpoint of ED, AD is 10 cm, and AB is 20 cm. What's the area of AEF?
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More Pie, Anyone?
 Annie Fetter

Geometry, difficulty level 2. How big a pie pan would you need if you made twice as much dough as you would need for a 9inch pan?
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Mowing My Lawn
 Annie Fetter

Geometry, difficulty level 3. My lawn is 75 feet by 42 feet, my lawn mower cuts 18 inches at a time. I am mowing in a circular pattern, and have finished part of the lawn. That part took me 8 minutes. How much longer will I have to mow to finish the lawn?
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My Mom Rocks!
 Annie Fetter

Geometry, difficulty level 3. Help my mom decide if she's buried the stones in her new garden deep enough.
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Napoleon's River
 Annie Fetter

Geometry, difficulty level 2. A soldier uses the visor of his cap to measure the distance across the river.
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Nine Congruent Rectangles
 Annie Fetter

Geometry, difficulty level 2. Nine congruent rectangles form a larger rectangle whose area is 720 units^2. What's the perimeter of the large rectangle?
... more>>

Olympic Target Shooting
 Annie Fetter

Geometry, difficulty level 2. Given that a shooter is equally likely to hit any point on a 10inch target, what are the odds that his shot will be in the 6inch circle?
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Overlapping Squares
 Annie Fetter

Geometry, difficulty level 3. Two squares, one 8 cm on a side and the other 10 cm, overlap. A corner of the 10 cm square is anchored at the middle of the 8 cm square and can rotate around that point. What are the maximum and minimum areas they'll overlap?
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Painting a Cube
 Annie Fetter

Geometry, difficulty level 3. A large cube, made up of smaller cubes, has some of its faces painted. When it's taken apart, 24 of the small cubes have no paint on them. How big was the large cube, and how many of its faces were painted?
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Part of a Rectangle
 Annie Fetter

Geometry, difficulty level 3. Given the area of part of a rectangle, find the area of the whole thing.
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Patterning a Shotgun
 Annie Fetter

Geometry, difficulty level 2. Construct a circular target with nine regions of equal area, with one a circle in the middle and spokes going out to the edge. If the whole circle is 6 feet in diameter, what will the diameter of the inner circle be?
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Picking an Area Formula
 Annie Fetter

Geometry, difficulty level 2. Find the area of a polygon on a grid and figure out which of four
formulas, incorporating points on the interior and points on the
boundary, calculates its area correctly.
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Picking Painted Cubes
 Annie Fetter

Geometry, difficulty level 2. A large cube, composed of smaller cubes, is painted on all sides. The edge length is five small cubes. It's then broken apart and the smaller cubes are put in a bag. What are the odds that a chosen cube is painted on three sides? two? one? none?
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A Picturesque Pythgorean Proof
 Annie Fetter

Geometry, difficulty level 3. Write a proof to go with this picture.
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Playing with Blocks
 Annie Fetter

Geometry, difficulty level 4. A large cube is made up of 13 double cubes like the smaller one shown, plus a single cube. What color is the single cube, and where does it have to be?
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Point P Perambulates
 Annie Fetter

Geometry, difficulty level 3. Find the length of the path of a vertex of an equilateral triangle as that triangle rotates around the inside of a square.
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Proofs Without Words I
 Annie Fetter

Geometry, difficulty level 2. Explain how these pictures and algebraic expressions are related. Draw your own picture that illustrates an algebraic concept.
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Proofs Without Words II
 Annie Fetter

Geometry, difficulty level 2. Based on a simple example that illustrates the distributive property, explain what the second picture illustrates.
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The Protean Perimeter
 Annie Fetter

Geometry, difficulty level 3. Given isosceles triangle ABC, with AC=AB=5 inches and angle CAB greater than or equal to 60, what's the perimeter of the triangle if all the edgelengths must be integers?
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The Puzzling Parallelogram
 Annie Fetter

Geometry, difficulty level 3. Draw a parallelogram ABCD with AB=10. Draw EF with E between A and B and F between C and D such that EF divides the area of ABCD in half. If EF=4, what is FD?
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Quad Query
 Annie Fetter

Geometry, difficulty level 2. Given quadrilateral ABCD, with E the midpoint of AB and F the midpoint of AD. The area of FAEC is 13. What's the area of ABCD?
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Regional Ratios
 Annie Fetter

Geometry, difficulty level 2. A regular hexagon and an equilateral triangle have the same perimeter. What's the ratio of their areas?
... more>>

Replacing the Football Field
 Annie Fetter

Geometry, difficulty level 2. Will a regulationsize NCAA soccer field fit inside an equalquadrant 400meter track?
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Returning to Napoleon's River
 Annie Fetter

Geometry, difficulty level 3. A soldier stands on the bank of a river and uses his cap to measure its width. How far off will he be if he drops his head even one degree?
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The Return of the Parallelogram I
 Annie Fetter

Geometry, difficulty level 3. Given parallelogram ABCD, with AB=10, and E on AB and F on CD. If EF is 4, what is the area of ABCD?
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The Return of the Parallelogram II
 Annie Fetter

Geometry, difficulty level 3. Draw a parallelogram ABCD with AB=10. Draw EF with E between A and B and F between C and D such that EF divides the area of ABCD in half. If EF=4, what is the area of the parallelogram?
... more>>

Running a Mile on a Metric Track
 Annie Fetter

Geometry, difficulty level 4. How far out from the rail on a 400 meter track would you have to run to cover 440 yards?
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Save the Lake!
 Annie & Tom Fetter

Geometry, difficulty level 2. How far does the water level drop if you take 900,000 gallons of water out of a lake with a surface area of 334 acres?
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Scintillating Similarity
 Annie Fetter

Geometry, difficulty level 2. Given a triangle with sides of 3, 4, and 6. What is the perimeter of the smallest triangle that is similar to the first one and has one side with length 12?
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Seven Congruent Rectangles
 Annie Fetter

Geometry, difficulty level 2. Seven congruent rectangles are arranged to form a larger rectangle. If the area of the large rectangle is 336 units^2, what's the perimeter of the large rectangle?
... more>>

Shotgun Targets
 Annie Fetter

Geometry, difficulty level 2. Figure out how large to make the inner circle of this shotgun target.
... more>>

Slicing a Cube
 Annie Fetter

Geometry, difficulty level 3. Given a cube with a surface area of 54 cm^2, find the perimeter of the square cross section and that of the largest rectangular cross section of that cube.
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Snow Removal
 Annie Fetter

Geometry, difficulty level 4. How many times would the snow in the Math Forum parking lot fill my office? How many pounds does all that snow weigh?
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So, What Do You Know About Tangents?
 Annie Fetter

Geometry, difficulty level 3. AB and CB are tangents to a circle, with A and C on the circle and B their intersection. D is a point on the minor arc AC, and a tangent is drawn at D, which intersects the AB at E and CB at F. What is the perimeter of EFB?
... more>>

The Speed Skating Oval
 Annie Fetter

Geometry, difficulty level 2. Find the radii of the turns of a 400 meter speed skating oval.
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Splitting a Parallelogram
 Annie Fetter

Geometry, difficulty level 3. Draw parallelogram ABCD with AB = 10. Draw EF with E between A and B and F between C and D such that EF divides the area of ABCD in half. If EF = 4, what is FD?
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Splitting up a Triangle
 Annie Fetter

Geometry, difficulty level 4. Take any triangle ABC. Construct D and E as the midpoints of BC and
AB, respectively. Now construct DF and EF, where F is any point on AC.
How are the areas of the triangles related to the area of the
quadrilateral?
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Square Segments
 Annie Fetter

Geometry, difficulty level 2. A square has been split into four rectangles, one of whose areas is 24
cm^2. The area of the square is no more than 125 cm^2. What are all of
the possible combinations of lengths for the four segments if the
lengths of all of the segments are integers?
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Squares on Squares
 Annie Fetter

Geometry, difficulty level 3. Let P be the center of a 16" x 16" square as well as a vertex of
another 16" x 16" square that pivots freely around P. What is the
largest possible value for the area where these two squares overlap?
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St. George's Banner
 Annie Fetter

Geometry, difficulty level 2. Find out how wide the cross on this flag needs to be in order for the red and white regions to have the same area. If the flag is x feet by y feet, how wide is the red stripe?
... more>>

The Sun vs. the Earth
 Annie Fetter

Geometry, difficulty level 1. Would one million Earths really fit inside the sun?
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Tangrams
 Abram Falk

Geometry, difficulty level 3. Explain which triangles in a tangram are congruent to each other and what the area is of each piece.
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Tangrams and Perimeter
 Annie Fetter

Geometry, difficulty level 2. What are the perimeters of the pieces of a tangram?
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Tangrams, Congruent Triangles, and Area
 Annie Fetter

Geometry, difficulty level 3. Tell me which pieces of a tangram are congruent, and tell me the area of each of the pieces.
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