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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 9-12


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? ... more>>

Approximating Pi - Annie Fetter
Geometry, difficulty level 3. Approximate the value of pi using polygons instead of a circle. ... more>>

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. ... more>>

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? ... more>>

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. ... more>>

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. ... more>>

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. ... more>>

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. ... more>>

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. ... more>>

Clever Cuts - Annie Fetter
Geometry, difficulty level 2. Find three different ways to cut a hexagonal cake into six congruent pieces. ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

Congruent Rectangles Help - Annie Fetter
Geometry, difficulty level 2. Find some possible areas for the large rectangle shown in the given picture. ... more>>

Cutting Corners - Annie Fetter
Geometry, difficulty level 1. Some election signs must have corners cut off them to comply with local election rules. ... more>>

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? ... more>>

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. ... more>>

Equal Area & Congruence - Annie Fetter
Geometry, difficulty level 2. If two triangles are congruent, they have the same area. Is the converse true? ... more>>

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? ... more>>

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? ... more>>

Filling the Hot Tub - Annie Fetter and Steve Risberg
Geometry, difficulty level 3. Given the dimensions of a hot tub and its submerged wood-burning stove, figure out how long it will take to fill the tub using a garden hose. ... more>>

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? ... more>>

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? ... more>>

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. ... more>>

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? ... more>>

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. ... more>>

Fitting In - Annie Fetter
Geometry, difficulty level 3. Figure out which peg takes up a larger percentage of the corresponding hole. ... more>>

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? ... more>>

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? ... more>>

Geoboard Gymnastics - Annie Fetter
Geometry, difficulty level 3. Find squares with different areas on a geoboard. ... more>>

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. ... more>>

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. ... more>>

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? ... more>>

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 semi-circle 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? ... more>>

How Deep Is It? - Annie Fetter
Geometry, difficulty level 2. How deep will the water be when this container is resting its largest face? ... more>>

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. ... more>>

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? ... more>>

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! ... more>>

Making a Quilt of Squares - Annie Fetter
Geometry, difficulty level 4. Figure out the dimensions of some closely packed squares. ... more>>

Making Pie Crust - Annie Fetter
Geometry, difficulty level 3. How would you change the recipe for a 9-inch pie crust to make an 11-inch pie crust? ... more>>

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? ... more>>

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? ... more>>

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 9-inch pan? ... more>>

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? ... more>>

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. ... more>>

Napoleon's River - Annie Fetter
Geometry, difficulty level 2. A soldier uses the visor of his cap to measure the distance across the river. ... more>>

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 10-inch target, what are the odds that his shot will be in the 6-inch circle? ... more>>

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? ... more>>

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? ... more>>

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. ... more>>

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? ... more>>

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. ... more>>

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? ... more>>

A Picturesque Pythgorean Proof - Annie Fetter
Geometry, difficulty level 3. Write a proof to go with this picture. ... more>>

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? ... more>>

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. ... more>>

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. ... more>>

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. ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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 regulation-size NCAA soccer field fit inside an equal-quadrant 400-meter track? ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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. ... more>>

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? ... more>>

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. ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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? ... more>>

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. ... more>>

Tangrams and Perimeter - Annie Fetter
Geometry, difficulty level 2. What are the perimeters of the pieces of a tangram? ... more>>

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. ... more>>

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