Angles in Regular Polygons Worksheet | Fun and Engaging Geometry ... - 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°

In an equilateral triangle, the apothem relates to the side length.

Recall: In an equilateral triangle with side length s, the height h is:
h = (√3 / 2) × s

The apothem is 1/3 of the height (because centroid divides median in 2:1 ratio, and apothem goes to midpoint of side — so it’s the shorter part).

Wait — actually, let me double-check that.

Actually, in any regular polygon, the apothem is the radius of the inscribed circle. For an equilateral triangle:

The centroid (which is also the incenter) divides the height into two parts:
- From vertex to centroid: 2/3 of height
- From centroid to base (apothem): 1/3 of height

So if apothem = 5 m, then total height h = 3 × 5 = 15 m

Now, since h = (√3 / 2) × s
→ 15 = (√3 / 2) × s
→ s = 15 × 2 / √3 = 30 / √3
Rationalize: (30√3)/3 = 10√3 ≈ 10 × 1.732 = 17.32 m

Perimeter p = 3 × s = 3 × 10√3 = 30√3 ≈ 51.96 m

Now area = ½ × a × p = ½ × 5 × 30√3 = (5 × 30√3)/2 = (150√3)/2 = 75√3

Calculate numerically:
√3 ≈ 1.732
75 × 1.732 = let's compute:

75 × 1.7 = 127.5
75 × 0.032 = 2.4
Total ≈ 129.9

But wait — let me check this another way because maybe I made a mistake.

Alternative approach for equilateral triangle area using apothem:

There’s a direct formula? Or we can use trigonometry.

In a regular polygon, area = (1/2) × n × s × a, where n = number of sides, s = side length, a = apothem.

But we don’t have s yet.

From geometry: In an equilateral triangle, the apothem a = s × √3 / 6

Why? Because height h = s√3 / 2, and apothem = h/3 = (s√3 / 2)/3 = s√3 / 6

So if a = 5 = s√3 / 6
→ s = 5 × 6 / √3 = 30 / √3 = 10√3 (same as before)

Then perimeter p = 3s = 30√3

Area = ½ × a × p = ½ × 5 × 30√3 = 75√3 ≈ 75 × 1.7320508 ≈ ?

Compute 75 × 1.732:

70 × 1.732 = 121.24
5 × 1.732 = 8.66
Total = 129.9

So approximately 129.9 m²

But let’s verify with standard area formula for equilateral triangle:

Area = (√3 / 4) × s²
s = 10√3 → s² = 100 × 3 = 300
Area = (√3 / 4) × 300 = 75√3 → same! So correct.

So Problem 1 answer: 129.9 m² (rounded to nearest tenth)

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Problem 2: Regular Pentagon

Given:
- Apothem (a) = 11 ft
- Side length (s) = 14 ft (written as “a=14” but looking at diagram, it’s labeled on a side, and apothem is dotted line with right angle — so likely “a=14” means side length, not apothem. Wait — confusion!)

Look again at image description:

Problem 2 has a pentagon. There’s a dotted line from center to side, labeled “11ft” with right angle → that’s the apothem.

And next to one side, it says “a=14”. But in the formula written above, “a” is used for apothem. However, in the diagram, they probably mean side length = 14 ft.

Because if apothem were 14, why label 11? The 11 is clearly the apothem (dotted line with right angle mark).

So interpretation:

- Apothem (let’s call it ‘ap’) = 11 ft
- Side length (s) = 14 ft
- Number of sides (n) = 5 (pentagon)

Perimeter p = n × s = 5 × 14 = 70 ft

Area = ½ × apothem × perimeter = ½ × 11 × 70

Compute:
½ × 70 = 35
35 × 11 = 385

So area = 385.0 ft²

No rounding needed — it’s exact.

But let’s confirm: Is “a=14” really side length?

In the student’s note above, they wrote “a = ½ a p” — which suggests they’re using ‘a’ for apothem in the formula. But in problem 2, they wrote “a=14” next to a side. That must be a labeling conflict.

Looking at context: In problem 1, apothem was given as 5m inside the triangle. In problem 2, the 11ft is drawn as apothem (with right angle), and “a=14” is written beside a side — so very likely, “a=14” here means side length = 14 ft, even though they used ‘a’ earlier for apothem. Probably just inconsistent notation in the worksheet.

To avoid error: If we mistakenly took apothem=14 and side=11, that wouldn’t make sense because the diagram shows 11 as apothem.

Moreover, in regular pentagon, if apothem is 11, side should be calculable, but here they give both — so we trust the diagram: apothem = 11 ft, side = 14 ft.

Thus, perimeter = 5 × 14 = 70 ft
Area = 0.5 × 11 × 70 = 385 ft²

Final Answer for Problem 2: 385.0 ft²

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Double-check calculations:

Problem 1:
Apothem = 5 m
For equilateral triangle:
apothem = s√3 / 6 → s = 6×5 / √3 = 30/√3 = 10√3 ≈ 17.3205 m
Perimeter = 3 × 17.3205 ≈ 51.9615 m
Area = 0.5 × 5 × 51.9615 ≈ 2.5 × 51.9615 ≈ 129.90375 → rounds to 129.9 m²

Problem 2:
Apothem = 11 ft, side = 14 ft, n=5
Perimeter = 70 ft
Area = 0.5 × 11 × 70 = 385 → 385.0 ft²

All good.

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Final Answer:
1. 129.9
2. 385.0