Find the area between the circumcircle and incircle of an equilateral triangle with area of 43

We are tasked with finding the area between the circumcircle and the incircle of an equilateral triangle, given that the area of the triangle is \( 43 \).

### Step 1: Find the side length of the triangle
The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
We are given that the area is 43, so:
\[
43 = \frac{\sqrt{3}}{4} s^2
\]
Multiplying both sides by 4:
\[
172 = \sqrt{3} s^2
\]
Now, divide by \( \sqrt{3} \):
\[
s^2 = \frac{172}{\sqrt{3}} = \frac{172 \sqrt{3}}{3}
\]
Taking the square root of both sides:
\[
s = \sqrt{\frac{172 \sqrt{3}}{3}} \approx 9.98
\]
So the side length \( s \) is approximately \( 9.98 \).

### Step 2: Find the radii of the circumcircle and incircle
The radius \( R \) of the circumcircle of an equilateral triangle is related to the side length \( s \) by the formula:
\[
R = \frac{s}{\sqrt{3}}
\]
Substituting \( s \approx 9.98 \):
\[
R = \frac{9.98}{\sqrt{3}} \approx 5.76
\]

The radius \( r \) of the incircle is given by:
\[
r = \frac{s}{2\sqrt{3}}
\]
Substituting \( s \approx 9.98 \):
\[
r = \frac{9.98}{2\sqrt{3}} \approx 2.88
\]

### Step 3: Find the areas of the circumcircle and incircle
The area of the circumcircle is:
\[
\text{Area of circumcircle} = \pi R^2 = \pi (5.76)^2 \approx 104.24
\]

The area of the incircle is:
\[
\text{Area of incircle} = \pi r^2 = \pi (2.88)^2 \approx 26.07
\]

### Step 4: Find the area between the circumcircle and incircle
The area between the circumcircle and the incircle is the difference between their areas:
\[
\text{Area between} = 104.24 - 26.07 \approx 78.17
\]

Thus, the area between the circumcircle and the incircle is approximately \( \boxed{78.17} \).

Would you like further details or have any questions?

---

Here are 5 related questions to explore:
1. How does the area between the circumcircle and incircle change if the side length of the triangle is doubled?
2. What is the ratio of the circumcircle's radius to the incircle's radius for any equilateral triangle?
3. Can you derive the area formulas for the circumcircle and incircle in terms of the triangle's area instead of side length?
4. How would the result change if the triangle were not equilateral but isosceles?
5. What would the area be if the area of the triangle were instead \(100 \, \text{units}^2\)?

**Tip**: Always check if simplifying complex roots in geometry problems can lead to quicker results!

Find the area between the circumcircle and incircle of an equilateral triangle with area of 43.

This solution explains how to find the area between the circumcircle and incircle of an equilateral triangle with a given area of 43. Learn step-by-step geometry concepts, including calculating the side length, circumradius, inradius, and the areas of the circles. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation