Geometry check my answers please :)

Question

Would this be correct??

A pentagon’s radius (measure from center to vertex) measures 4 cm, what is the pentagon’s area?

360 / 5 = 72

sa = 54 

2.5 times tan ( 54 ) 

h = 3.4 

1/2 b ( 7.3 ) = 15

b = 4.1

3.4 times 4.1 / 2 = 6.9

AREA = 6.9 cm^2

Answer 1

Jin, I hope you still need this.

Your reasoning is, unfortunately, incorrect. In your calculations you use tan function. The tan function connects two legs of a right triangle. When we have inscribed regular polygon and we are given the radius, the height of one of the triangles is denoted a and the base is denoted s. These are connected to the radius by sine and cosine functions. We usually use half of the vertex angle, in our case 180^o/5 = 36.

Make a sketch and follow the explanation below.

First step is correct. You found the vertex angle in one of the triangles in which your pentagon can be divided.

The base of your triangle is s, and the height to it is a.

The vertex angle for the isosceles triangle is 72^o. Half of the vertex angle connects half of the base, height of the triangle a and hypotenuse with length R = 4.

The base is

s = 2 R sin(36^o).

The height to it is

a = R cos(36^o).

The area of the pentagon is

A = 5 times 1/2 a s;

A = 5 times 1/2 times 2R sin(36^o) R cos(36^o)

A = 38.04.

This is the well known formula for area of regular polygon given the radius and the number of sides.

The link below will point to improved version of the above, using the double angle formula [2 sin (36^o) cos(36^o) = sin(72⁰) = sin(360^o/5)].

If you have questions, send me a message.

Answer 2

As a check, calculate the area of the circle with radius 4.

The area of an inscribed polygon can be shown to equal n/2 r² sin(2 π/n). (You have n triangles, each with altitude r cos π/n and base 2 r sin π/n, so the area of each triangle is 1/2 bh = 1/2 r² cos π/n * 2 sin π/n = r² cos π/n sin π/n = 1/2 r² sin 2π/n. Thus, the area of the polygon is n/2 r² sin 2π/n. To see that the formula makes sense, in the limit as n goes to infinity, sin x goes to x, so you have n/2 r² * 2 π/n = πr²

Since we have a pentagon inscribed in a circle with radius 4, the area is 5/2 * 4² sin(2 π/5) = 38.0422606518061

If you want the exact value, you can also calculate

As sin(72°) = sqrt(10+2*sqrt(5))/4, the area is 5/2 * 4² * sqrt(10+2*sqrt(5))/4 = 10 sqrt(10+2*sqrt(5))