Solved Name: Per: Date: Area of Regular Polygons (Colorng | Chegg.com - Free Printable

Let’s solve each problem step by step.

We are given the formula:
Area = ½ × a × p
where:
- a = apothem (the perpendicular distance from the center to a side)
- p = perimeter (total length around the polygon)

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Problem 1: Equilateral Triangle

Given:
- Apothem (a) = 5 m
- It’s an equilateral triangle → all sides equal, all angles 60°

But we don’t have the side length or perimeter yet. We need to find the perimeter using the apothem.

In an equilateral triangle, there is a relationship between the apothem and the side length.

Recall: In an equilateral triangle, the apothem is also the height of one of the 3 smaller triangles formed by drawing lines from the center to each vertex.

Actually, let’s think differently.

The apothem in an equilateral triangle relates to the side length s by:

> apothem = s / (2√3)
Wait — that’s not quite right. Let me derive it properly.

Actually, in an equilateral triangle:

Height h = (√3 / 2) × s

And the centroid (which is also the center for regular polygons) divides the height in a 2:1 ratio. The apothem is the shorter part — from center to base — so:

> apothem = h / 3 = [(√3 / 2) × s] / 3 = (√3 / 6) × s

So:

a = (√3 / 6) × s

We’re told a = 5, so:

5 = (√3 / 6) × s
→ s = 5 × 6 / √3 = 30 / √3
Rationalize: (30√3) / 3 = 10√3 ≈ 10 × 1.732 = 17.32 m

Since it’s equilateral, perimeter p = 3 × s = 3 × 17.32 = 51.96 m

Now use area formula:

Area = ½ × a × p = ½ × 5 × 51.96 = 2.5 × 51.96 = 129.9 m²

Round to nearest tenth → 129.9

Wait — let me double-check this because sometimes problems like this expect you to know another way.

Alternative approach: Maybe they expect us to use trigonometry?

In a regular polygon with n sides, area can also be calculated as:

Area = (n × s × a) / 2 → which is same as ½ × a × p since p = n×s.

But we still need s.

Another way: In an equilateral triangle, if apothem is 5, then the radius (distance from center to vertex) is twice the apothem? No — in equilateral triangle, centroid divides median in 2:1, so if apothem is 1 part, then full height is 3 parts → height = 15.

Then, since height h = (√3/2)s → 15 = (√3/2)s → s = 30 / √3 = 10√3 ≈ 17.32 → same as before.

Perimeter = 3 × 17.32 = 51.96

Area = ½ × 5 × 51.96 = 129.9 → correct.

But wait — maybe the diagram implies something else? The dot is at the center, and the 5m is drawn perpendicular to the base — that’s definitely the apothem.

Alternatively, perhaps the problem expects us to assume that the triangle has side lengths such that the apothem is 5, but without more info, we must calculate based on geometry.

I think our calculation is correct.

But let me check online or recall: For equilateral triangle, area in terms of apothem.

There’s a formula: Area = 3√3 × a²

Let me test that.

If a = 5, then Area = 3√3 × 25 = 75√3 ≈ 75 × 1.732 = 129.9 → YES! Same answer.

Perfect. So Area = 3√3 × a² for equilateral triangle.

So 3 × 1.732 × 25 = 129.9 → confirmed.

So Problem 1 answer: 129.9

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Problem 2: Regular Pentagon

Given:
- Apothem (a) = 14
- Side length? Not given directly. But looking at the diagram, it says “14” next to the apothem line, and no other numbers. Wait — actually, re-examining: the label “14” is written near the apothem, and there’s no side length labeled. Hmm.

Wait — look again: In the pentagon, the apothem is labeled as 14, but is there a side length? The image shows “14” along the apothem line, and nothing else. But in the user’s handwritten note above, it says “a=14”, which matches.

But we need the perimeter to use Area = ½ × a × p.

How do we get the perimeter?

For a regular pentagon, if we know the apothem, we can find the side length using trigonometry.

A regular pentagon can be divided into 5 congruent isosceles triangles from the center. Each central angle is 360°/5 = 72°.

The apothem bisects one of those triangles into two right triangles, each with angle 36° at the center (half of 72°).

In that right triangle:
- adjacent side = apothem = 14
- opposite side = half the side length of the pentagon → let’s call it x
- angle at center = 36°

So tan(36°) = opposite / adjacent = x / 14
→ x = 14 × tan(36°)

tan(36°) ≈ 0.7265 (standard value)

So x ≈ 14 × 0.7265 ≈ 10.171

Therefore, full side length s = 2x ≈ 20.342

Perimeter p = 5 × s ≈ 5 × 20.342 = 101.71

Now area = ½ × a × p = ½ × 14 × 101.71 = 7 × 101.71 = 711.97

Round to nearest tenth → 712.0

Wait — let me verify with more precise values.

Use calculator-level precision:

tan(36°) = tan(π/5) ≈ 0.726542528

x = 14 × 0.726542528 ≈ 10.17159539

s = 2x ≈ 20.34319078

p = 5 × s ≈ 101.7159539

Area = 0.5 × 14 × 101.7159539 = 7 × 101.7159539 ≈ 712.0116773

Rounded to nearest tenth: 712.0

Yes.

Alternatively, there’s a formula for area of regular pentagon in terms of apothem:

Area = (5 × a² × tan(36°)) ? Let’s see.

From above: Area = ½ × a × p = ½ × a × (5 × 2 × a × tan(36°)) = 5 × a² × tan(36°)

Wait: p = 5 × s = 5 × 2 × (a × tan(36°)) = 10 a tan(36°)

Then Area = ½ × a × 10 a tan(36°) = 5 a² tan(36°)

So Area = 5 × (14)^2 × tan(36°) = 5 × 196 × 0.726542528 ≈ 980 × 0.726542528 ≈ 712.011677 → same.

So yes, 712.0

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Final Answers:

Problem 1: 129.9
Problem 2: 712.0

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Final Answer:
1. 129.9
2. 712.0