geometric shapes problem solving

Algebra: Geometry Word Problems

Related Pages Perimeter and Area of Polygons Nets Of 3D Shapes Surface Area Formulas Volume Formulas More Geometry Lessons

In these lessons, we look at geometry word problems, which involves geometric figures and angles described in words. You would need to be familiar with the formulas in geometry .

Making a sketch of the geometric figure is often helpful.

You can see how to solve geometry word problems in the following examples: Problems involving Perimeter Problems involving Area Problems involving Angles

Printable & Online Geometry Worksheets

There is also an example of a geometry word problem that uses similar triangles.

Geometry Word Problems Involving Perimeter

Example 1: A triangle has a perimeter of 50. If 2 of its sides are equal and the third side is 5 more than the equal sides, what is the length of the third side?

Solution: Step 1: Assign variables: Let x = length of the equal side. Sketch the figure.

Step 2: Write out the formula for perimeter of triangle . P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch. 50 = x + x + x + 5

Combine like terms 50 = 3x + 5

Isolate variable x 3x = 50 – 5 3x = 45 x =15

Be careful! The question requires the length of the third side. The length of third side = 15 + 5 =20

Answer: The length of third side is 20

Example 2: Writing an equation and finding the dimensions of a rectangle knowing the perimeter and some information about the about the length and width. The width of a rectangle is 3 feet less than its length. The perimeter of the rectangle is 110 feet. Find its dimensions.

Geometry Word Problems Involving Area

Example 1: A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?

Step 1: Assign variables: Let x = original width of rectangle

Step 2: Write out the formula for area of rectangle. A = lw

Step 3: Plug in the values from the question and from the sketch. 60 = (4x + 4)(x –1)

Use distributive property to remove brackets 60 = 4x 2 – 4x + 4x – 4

Put in Quadratic Form 4x 2 – 4 – 60 = 0 4x 2 – 64 = 0

This quadratic can be rewritten as a difference of two squares (2x) 2 – (8) 2 = 0

Factorize difference of two squares "> (2x) 2 – (8) 2 = 0 (2x – 8)(2x + 8) = 0

Since x is a dimension, it would be positive. So, we take x = 4

The question requires the dimensions of the original rectangle. The width of the original rectangle is 4. The length is 4 times the width = 4 × 4 = 16

Answer: The dimensions of the original rectangle are 4 and 16.

Example 2: This is a geometry word problem that we can solve by writing an equation and factoring. The height of a triangle is 4 inches more than twice the length of the base. The area of the triangle is 35 square inches. Find the height of the triangle.

Geometry Word Problems involving Angles

Example 1: In a quadrilateral two angles are equal. The third angle is equal to the sum of the two equal angles. The fourth angle is 60° less than twice the sum of the other three angles. Find the measures of the angles in the quadrilateral.

Step 1: Assign variables: Let x = size of one of the two equal angles Sketch the figure

Step 2: Write down the sum of angles in quadrilateral . The sum of angles in a quadrilateral is 360°

Step 3: Plug in the values from the question and from the sketch. 360 = x + x + (x + x) + 2(x + x + x + x) – 60

Combine like terms 360 = 4x + 2(4x) – 60 360 = 4x + 8x – 60 360 = 12x – 60

Isolate variable x 12x = 420 x = 35

The question requires the values of all the angles. Substituting x for 35, you will get: 35, 35, 70, 220

Answer: The values of the angles are 35°, 35°, 70° and 220°.

Example 2: The sum of the supplement and the complement of an angle is 130 degrees. Find the measure of the angle.

Geometry Word Problems involving Similar Triangles

Indirect Measurement Using Similar Triangles

This video illustrates how to use the properties of similar triangles to determine the height of a tree.

How to solve problems involving Similar Triangles and Proportions?

Given that triangle ABC is similar to triangle DEF, solve for x and y.

The extendable ramp shown below is used to move crates of fruit to loading docks of different heights. When the horizontal distance AB is 4 feet, the height of the loading dock, BC, is 2 feet. What is the height of the loading dock, DE?

Triangles ABC and A’B’C’ are similar figures. Find the length AB.

How to use similar triangles to solve a geometry word problem?

Examples: Raul is 6 ft tall and he notices that he casts a shadow that’s 5 ft long. He then measures that the shadow cast by his school building is 30 ft long. How tall is the building?

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Basic Geometry Practice Questions with Full Answer Key – Area, Perimeter, Volume & Angles

• Posted by Brian Stocker MA
• Date April 3, 2014
• Comments 17 comments

High School geometry questions similar to what you will find on a standardized test.

Geometry practice test.

1. What is measurement of the indicated angle assuming the figure is a square?

a. 45 o b. 90 o c. 60 o d. 30 o

2. What is the sum of all the angles in the rectangle above?

a. 180 o b. 360 o c. 90 o d. 120 o

3. What is the measurement of the indicated angle?

a. 45 o b. 90 o c. 60 o d. 50 o

4. If the line m is parallel to the side AB of ? ABC, what is angle a ?

a. 130 o b. 25 o c. 65 o d. 50 o

5. What is perimeter of the above shape?

a. 12 cm b. 16 cm c. 6 cm d. 20 cm

6. What is (area of large circle) – (area of small circle) in the figure above?

a. 8 п cm 2 b. 10 п cm 2 c. 12 п cm 2 d. 16 п cm 2

7. What is perimeter of ? ABC in the above shape?

8. What is the volume of the figure above?

a. 125 cm 3 b. 875 cm 3 c. 1000 cm 3 d. 500 cm 3

9. What is the volume of the above solid made by a hollow cylinder with half in size of the larger cylinder?

a. 1440 п in 3 b. 1260 п in 3 c. 1040 п in 3 d. 960 п in 3

1. B The diagonals of a square intersect perpendicularly with each other so each angle measures 90 o x =90 o

2. B a+b+c+d = ? The sum of angles around a point is 360 o    a+b+c+d = 360 o

Video Solution

3. C The sum of angles around a point is 360 o d+300 = 360 o d = 60 o

4. D Two parallel lines( m & side AB) intersected by side AC a= 50 o (interior angles)

5. B The square with 2 cm side common to the rectangle apart from 4cm side Perimeter = 2+2+2+4+2+4 = 16 cm

6. C In the figure, we are given a large circle and a small circle inside it; with the diameter equal to the radius of the large one. The diameter of the small circle is 4 cm. This means that its radius is 2 cm. Since the diameter of the small circle is the radius of the large circle, the radius of the large circle is 4 cm. The area of a circle is calculated by: пr 2 where r is the radius.  Area of the small circle: п(2) 2 = 4п Area of the large circle: п(4) 2 2 = 16п

The difference area is found by: Area of the large circle – Area of the small circle = 16п – 4п = 12п

7. D Perimeter of a shape with two squares and triangle ABC. Perimeter = 8.5+8.5+6+6 Perimeter = 29 cm.

8. C Large cube is made up of 8 smaller cubes of 5 cm sides. Volume = Volume of small cube x 8 Volume = (5 x 5 x 5) x 8, 125 x 8 Volume = 1000cm 3

9. B Volume= Volume of large cylinder – Volume of small cylinder (Volume of cylinder = area of  base x height) We know the small cylinder is 1/2 the size of the large cylinder So, Volume of the small cylinder = Large Cylinder (п 12 2 x 10) – Small Cylinder(п 6 2 x 5) = 1440п – 180п Volume of small cylinder = 1260п in 3

See also our tutorial on Complex Shapes

Most Popular Geometry Questions

Common geometry questions on on standardized tests :

• Solve for the missing angle or side
• Finding the area or perimeter of different shapes (e.g. triangles, rectangles, circles)
• Problems using the Pythagorean Theorem
• Calculate properties of geometric shapes such as angles, right angles or parallel sides
• Calculating volume or surface area of complex shapes for example spheres, cylinders or cones
• Solve geometric transformations such as rotation, translation or reflections

Most Common Geometry Mistakes on a Test

• Not clearly labeling or identifying the given and unknown information in a problem
• Not understanding the properties and definitions of basic geometric figures (e.g. line, angle, triangle, etc.)
• Incorrectly using basic formulas (e.g. area of a triangle, Pythagorean theorem)
• Incorrectly interpreting geometric diagrams
• Not understanding the relationship between parallel lines and transversals
• Not understanding the relationship between angles and their degree measures
• Not understanding the relationship between perimeter and area

Tag: Basic Math , Geometry , Practice Questions

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17 Comments

answer to question five should be 18, you forgot to include the 2cm outer side of the rectangle

Divide the figure into a square and a rectangle – perimeter of the square – 3 X 2 = 6 note 3 sides are included in the perimeter. Upper rectangle 2 X 4 = 8 + 2 = 10 total perimeter = 10 + 6 = 16

Try again – the question asks for the PERIMITER – 3 sides of the square – 2 + 2 + 2 = 6; 2 sides of the rectangle – 4 + 4 = 8; 1 side of the rectangle (top) = 2 6 + 8 + 2 = 16

The line between the square and the rectangle isn’t counted for perimeter, perimeter is the amount of units around the OUTSIDE. Since the square and rectangle are connected, try thinking of them as one big rectangle and forget about the line in between.

Correct – the bottom square has 2 cm each side on 3 sides (2 + 2 + 2 = 6). The top rectangle has 2, 4 cm sides (4 + 4 = 8) and 1, 2 cm side. So the total is 6 + 8 + 2 = 16.

Please in no 6 ,42 -22 is 20,so how comes about 16 and how did u get 122 in no 10

The answers are correct – I have updated and expanded the solutions – also the п symbol wasn’t displaying properly – thanks!

102°-2x +78°=180°

Love this. Super helpful!

In #9, since it is a cylinder; shouldn’t you use Pi in calculating the area of the base??

hi – yes the answer does use pi – the web rendering is a little funny tho – looks like an ‘n’ but is is pi – e. g. Volume= (п 122x 10) – (п 62x 5), 1440? – 180п

in question #9 what is the formula of a cylinder if the diameter is given and not the radius?

I understand how you got the answer in question 5, however the way it is displayed is unclear. It is not clear that the number 4 is attached to just the rectangle and not to the line as a whole. This means there are two interpretations of the figure.

1) 4*2+2*2 = 12 2) 2(4+2) + 2*2 = 16

Thanks! this was super helpful!

Can you explain the question, the question isn’t asked properly. They are asking the volume made by the small cylinder, but the answer is solved for the remaining volume removing the small cylinder volume.

Not quite – the small cylinder is 1/2 the size of the larger cylinder, so Large Cylinder – Small Cylinder = Size of Small Cylinder. I have updated the answer explanation.

Tahnk you for all the wonderful explanations and answers

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Grade 5 geometry worksheets.

These geometry worksheets review 2-D geometry  including classifying angles, triangles and quadrilaterals, calculating perimeters and areas and working with circles. 3D geometry is introduced with rectangular prisms.  Reading and plotting points on a coordinate grid is also covered.

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