Compare Perimeter vs Circumference: Triangle (5-6) vs Circle (Radius 6)

Q: How do I know if the triangle measurement given is radius or side length?

Look at the diagram labels! If the line goes from center to edge of a circle, it's a radius. If it's labeling the side of a triangle, it's a side length. Context matters!

Q: Do I need to find the third side of the triangle?

Yes! The perimeter is the sum of ALL sides. Use the Pythagorean theorem: $ c = \sqrt{a^2 + b^2} = \sqrt{5^2 + 6^2} = \sqrt{61} $

Q: Should I use 3.14 or the π button for circumference?

Use the π button on your calculator for more accuracy! Using 3.14 gives you $ 2 \times 3.14 \times 6 = 37.68 $, while π gives 37.7. Close, but π is more precise.

Q: What if the triangle isn't a right triangle?

Check the diagram for right angle symbols or if two sides are perpendicular. If it's not a right triangle, you'd need all three side lengths given directly to find the perimeter.

Q: How do I compare decimal approximations accurately?

Calculate both values to the same number of decimal places before comparing. Triangle ≈ 18.81, Circle ≈ 37.70. The difference is clear: 37.70 > 18.81!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine which of the following shapes has the larger perimeter

00:04 The perimeter of the triangle equals the sum of its sides

00:11 Side AC equals the sum of its parts (AD+DC)

00:16 Apply the Pythagorean theorem to the triangle ADB

00:23 Substitute in the relevant values and proceed to calculate to determine AD

00:34 This is the length of AD

00:44 Now apply the Pythagorean theorem to the triangle DCB

00:51 Substitute in the relevant values and calculate to determine DC

00:57 This is the length of DC

01:02 Now substitute the side lengths that we found into the equation for calculating perimeter

01:11 This is the perimeter of the triangle

01:17 Apply the formula for calculating the circumference of a circle

01:20 Two x pi multiplied by the radius (R)

01:23 Substitute in our radius value and we should obtain the circumference

01:29 This is the circumference of the circle (larger than the triangle)

01:33 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the triangle and circle below.

Which has the larger perimeter/circumference?

2

Step-by-step solution

To determine which has the larger measurement, the triangle's perimeter or the circle's circumference, we need to compute both values.

Step 1: Calculate the perimeter of the Triangle
We are given two sides of the triangle: 6 and 5. Since it's implied to be a right triangle, we apply the Pythagorean theorem to find the third side, the hypotenuse $c$ :

c = \sqrt{6^2 + 5^2} = \sqrt{36 + 25} = \sqrt{61}

The perimeter $P$ of the triangle is:

P = 6 + 5 + \sqrt{61} \approx 11 + 7.81 = 18.81

Step 2: Calculate the circumference of the Circle
The circumference $C$ of a circle with radius $r$ is given by the formula:

C = 2 \pi r

Assuming the radius of the circle is equivalent to the '6' mentioned for the green line in the SVG:

C = 2 \pi \times 6 \approx 37.7

Step 3: Compare the Triangle's Perimeter and the Circle's Circumference
We compare the values:

Perimeter of the Triangle: $P \approx 18.81$
Circumference of the Circle: $C \approx 37.7$

The circumference of the circle (37.7) is greater than the perimeter of the triangle (18.81).

Therefore, the circle has the larger measurement.

Conclusion: The circle has the larger perimeter.

3

Final Answer

The circle

Key Points to Remember

Essential concepts to master this topic

Formulas: Circumference = $2\pi r$ , Perimeter = sum of all sides
Calculation: Circle: $2\pi(6) \approx 37.7$ , Triangle: $5 + 6 + \sqrt{61} \approx 18.8$
Verification: Compare final values: 37.7 > 18.8, so circle wins ✓

Common Mistakes

Avoid these frequent errors

Confusing radius with diameter in circle calculations
Don't use circumference = $\pi d$ when given radius 6 = diameter 12 formula! This doubles your answer incorrectly. Always use $C = 2\pi r$ when radius is given, or $C = \pi d$ when diameter is given.

Practice Quiz

Test your knowledge with interactive questions

Look at the triangle in the diagram. How long is side AB?

FAQ

Everything you need to know about this question

How do I know if the triangle measurement given is radius or side length?

+

Look at the diagram labels! If the line goes from center to edge of a circle, it's a radius. If it's labeling the side of a triangle, it's a side length. Context matters!

Do I need to find the third side of the triangle?

+

Yes! The perimeter is the sum of ALL sides. Use the Pythagorean theorem: $c = \sqrt{a^2 + b^2} = \sqrt{5^2 + 6^2} = \sqrt{61}$

Should I use 3.14 or the π button for circumference?

+

Use the π button on your calculator for more accuracy! Using 3.14 gives you $2 \times 3.14 \times 6 = 37.68$ , while π gives 37.7. Close, but π is more precise.

What if the triangle isn't a right triangle?

+

Check the diagram for right angle symbols or if two sides are perpendicular. If it's not a right triangle, you'd need all three side lengths given directly to find the perimeter.

How do I compare decimal approximations accurately?

+

Calculate both values to the same number of decimal places before comparing. Triangle ≈ 18.81, Circle ≈ 37.70. The difference is clear: 37.70 > 18.81!

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Compare Perimeter vs Circumference: Triangle (5-6) vs Circle (Radius 6)

Circle Circumference vs Triangle Perimeter

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know if the triangle measurement given is radius or side length?

Do I need to find the third side of the triangle?

Should I use 3.14 or the π button for circumference?

What if the triangle isn't a right triangle?

How do I compare decimal approximations accurately?

🌟 Unlock Your Math Potential

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