# Geometry word problems

A variety of geometry word problems along with step by step solutions will help you practice lots of skills in geometry.

Word problem #1:

The measure of one supplementary angle is twice the measure of the second. What is the measure of each angle?

Let x be the measure of the first angle. Then, the second angle is 2x

Since the angle are supplementary, they add up to 180°

x + 2x = 180°

3x = 180°

Since 3 × 60 = 180, x = 60

The measure of the first angle is 60°

The measure of the second is 2x = 2 × 60 = 120°

Word problem #2:

Without plotting the points, say if the points (2, 4), (2, 0), and (2, -6) are colinear

If the x-coordinate or the y-coordinate is the same for all points, then the points are colinear

After a close inspection, we see that the x-coordinate is the same for all points. Therefore, the points are colinear

Word problem #3:

The perimeter of a square is 8 cm. What is the area?

If the perimeter is 8 cm, then the length of one side is 2 cm since 2 cm + 2 cm + 2 cm + 2 cm = 8 cm

Area = 2 cm × 2 cm = 4 cm2

Word problem #4:

A right triangle has acute angles whose measures are in the ratio 1:3

Find the measure of these acute angles.

Thing to know: The sum of the angles in a triangle is equal to 180°

Meaning of the ratio 1:3

This means that the second acute angle is 3 times bigger than the first acute angle

Let x be the first acute angle, then the second acute angle will be 3x

x + 3x + 90° = 180°

4x + 90° = 180°

4x + 90° - 90° = 180° - 90°

4x = 90°

Since 4 × 22.5 = 90°, x = 22.5°

The second angle is 3x = 3 × 22.5 = 67.5

The measure of the two acute angles are 22.5 and 67.5

A tricky geometry word problem

Word problem #5:

The midpoint of a segment is (3, 6). If one endpoint is (4, 7), what is the other endpoint?

Suppose x1 is the missing x-coordinate of the other endpoint.

To get the x-coordinate of the midpoint, you will need to do the math below:

x1 + 4 / 2
= 3

x1 = 2 since 2 + 4 = 6 and 6 divided by 2 = 3

Suppose y1 is the missing y-coordinate of the other endpoint.

To get the y-coordinate of the midpoint, you will need to do the math below:

y1 + 7 / 2
= 6

y1 = 5 since 5 + 7 = 12 and 12 divided by 2 = 6

The other endpoint is (2, 5)

Word problem #6:

The sum of the measures of the angles of an n-gon is 2340°. How many sides does this n-gon have?

To solve this problem, you need to know the following formula:

Sum of the angles in an n-gon = (n - 2)× 180°

n is the number of sides. So just plug in the numbers and solve

2340° = (n - 2)× 180°

2340° = 180°n - 360°

2340° + 360° = 180°n - 360° + 360°

2700° = 180°n

Divide both sides 180°

(2700° ÷ 180°) = (180° ÷ 180°)n

15 = n

The n-gon has 15 sides

Word problem #7:

If two lines are perpendicular, what is the slope of the first line if the second line has a slope of 5

When two lines are perpendicular, the following equation is true

Let m1 × m2 = -1

m1 is the slope of the first line and m2 is the slope of the second line

Thus, m1 × 5 = -1

Divide both sides of this equation by 5

(m1 × 5 ÷ 5) = (-1 ÷ 5)

m1 =
-1 / 5
= - 0.20

An interesting geometry word problem

Word problem #8:

The diameter of a penny is 0.750 inch and the diameter of a quarter is 0.955 inch.

You put the penny on top and exactly in the middle of the quarter. Since the coin is smaller, it will not cover completely the quarter.

What is the area of the portion that is not covered? Will the area change if the coin is not centered?

We can use A = πr2 since the coin is shaped like a circle

Let B stand for the area of the portion not covered

B = area of quarter - area of penny

r = 0.375 inch for the penny and r = 0.4775 for the quarter

B = 3.14 × 0.4775 × 0.4775 - 3.14 × 0.375 × 0.375

B = 0.715 - 0.441

B = 0.274 inch2

As long as the coin remains inside the quarter, the area should stay the same

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