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

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

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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