Probability can also relate to the areas of geometric shapes. The following are some examples of probability problems that involve areas of geometric shapes.
^{2}

Area of non-shaded circle = 3.142 × 7 2 = 153.99 cm^{2}

Area of shaded region = 615.83 – 153.99 = 461.84 cm^{2} = 462 cm^{2} (rounded to whole number)

** Probability of hitting the shaded region: **

=

b) in the green sector.

c) in any sector**except** the green sector.

**Probability that the point lies on red sector** =

b) Area of green sector = × area of circle

**Probability that the point lies on green sector** =

c) in any sector**except** the green sector.

**Probability that the point does not lie in the green sector** =

**Example:**

An arrow is shot at random onto the rectangle*PQRS*. Calculate the probability that the arrow strikes:

a) triangle*AQB*.

b) a shaded region.

c) either triangle*BRC* or the unshaded region.

* Solution: *

a) Let*PQ* = 2*x* and *QR* = 2*y*. Then, *AQ* = *x* and *QB *= *y*.

Area of rectangle*PQRS* = 2*x* × 2*y* = 4*xy*

Area*AQB* = *xy *

**Probability of striking triangle ***AQB* = *xy *÷ 4*xy* =

b) All the shaded triangles are equal.

Total area of shaded regions = 4 ×*xy* = 2*xy*

**Probability of striking a shaded region** = 2*xy* ÷ 4*xy* =

c) Area of unshaded region = 4*xy* – 2*xy* = 2*xy*

Probability of striking unshaded region = 2*xy* ÷ 4*xy* =

Area of triangle*BRC* = *xy *

Probability of striking triangle*BRC*=

**Probability of striking triangle ***BRC* or unshaded region =

Geometric Probability with Area

Example 1: A circle with radius 2 lies within a square with side length 6. A dart lands randomly inside the square. What is the probability that the dart lands inside the circle? Give the exact probability and the probability as a percent rounded to the nearest tenths.

Example 2: A point is chosen at random on this figure. What is the probability that the point is in the yellow region?

Example 3: A square is inscribed inside a circle. What is the probability that a point chosen at random inside the circle will be inside the square?

Example 4: A circle is inscribed in an equilateral triangle. What is the probability that a point chosen at random in the triangle will be inside the circle?

Problem 1: Find the probability that a point chosen at random inside the circle will be inside the shaded region.

Problem 2: Find the probability that a point chosen at random inside the square will be inside the shaded region.
Area Probability Problem 1

Find the probability that a point randomly selected from a figure would land in the shaded area.

Area Probability Problem 2
Area Probability Problem 3

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

Related Topics: More examples of Probability

* Example: *

A dart is thrown at random onto a board that has the shape of a circle as shown below. Calculate the probability that the dart will hit the shaded region. (Use π = 3.142)

* Solution: *

Area of non-shaded circle = 3.142 × 7 2 = 153.99 cm

Area of shaded region = 615.83 – 153.99 = 461.84 cm

=

* Example: *

The figure shows a circle divided into sectors of different colors.

If a point is selected at random in the circle, calculate the probability that it lies:

a) in the red sectorb) in the green sector.

c) in any sector

* Solution:*

b) Area of green sector = × area of circle

c) in any sector

In the figure below, *PQRS* is a rectangle, and *A*, *B*, *C*, *D* are the midpoints of the respective sides as shown.

An arrow is shot at random onto the rectangle

a) triangle

b) a shaded region.

c) either triangle

a) Let

Area of rectangle

Area

b) All the shaded triangles are equal.

Total area of shaded regions = 4 ×

c) Area of unshaded region = 4

Probability of striking unshaded region = 2

Area of triangle

Probability of striking triangle

Example 1: A circle with radius 2 lies within a square with side length 6. A dart lands randomly inside the square. What is the probability that the dart lands inside the circle? Give the exact probability and the probability as a percent rounded to the nearest tenths.

Example 2: A point is chosen at random on this figure. What is the probability that the point is in the yellow region?

Example 3: A square is inscribed inside a circle. What is the probability that a point chosen at random inside the circle will be inside the square?

Example 4: A circle is inscribed in an equilateral triangle. What is the probability that a point chosen at random in the triangle will be inside the circle?

Problem 1: Find the probability that a point chosen at random inside the circle will be inside the shaded region.

Problem 2: Find the probability that a point chosen at random inside the square will be inside the shaded region.

Find the probability that a point randomly selected from a figure would land in the shaded area.

Area Probability Problem 2

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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