30. Satellites One of the smallest space satellites ever developed has the shape of a pyramid. Each of the four faces of the pyramid is an equilateral triangle with sides about 13 cm long. What is the area of one equilateral triangular face of the satellite? Round your answer to the nearest whole number.
31. Think About a Plan The gazebo in the photo is built in the shape of a regular octagon. Each side is 8 ft long, and the enclosed area is 310 . 4   ft 2 . 310 . 4 , ft squared , .  What is the length of the apothem?
    • How can you draw a diagram to help you solve the problem?
    • How can you use the area of a regular polygon formula?

    

32. A regular hexagon has perimeter 120 m. Find its area.
33. The area of a regular polygon is 36   in . 2 . 36 in , . squared , .  Find the length of a side if the polygon has the given number of sides. Round your answer to the nearest tenth.
    1. 3
    2. 4
    3. 6
    4. Estimation Suppose the polygon is a pentagon. What would you expect the length of a side to be? Explain.
34. A portion of a regular decagon has radii and an apothem drawn. Find the measure of each numbered angle.

    

35. Writing Explain why the radius of a regular polygon is greater than the apothem.
36. Constructions Use a compass to construct a circle.
    1. Construct two perpendicular diameters of the circle.
    2. Construct diameters that bisect each of the four right angles.
    3. Connect the consecutive points where the diameters intersect the circle. What regular polygon have you constructed?
    4. Reasoning How can a circle help you construct a regular hexagon?

Find the area of each regular polygon. Show your answers in simplest radical form and rounded to the nearest tenth.

37. 
38. 
39. 
40. To find the area of an equilateral triangle, you can use the formula A = 1 2 b h eh equals . 1 half , b h  or A = 1 2 a p . eh equals . 1 half , eh p .  A third way to find the area of an equilateral triangle is to use the formula A = 1 4 s 2 3 . eh equals , 1 fourth , s squared , square root of 3 .  Verify the formula A = 1 4 s 2 3 eh equals , 1 fourth , s squared , square root of 3  in two ways as follows:
    1. Find the area of Figure 1 using the formula A = 1 2 b h . eh equals . 1 half , b h .
    2. Find the area of Figure 2 using the formula A = 1 2 a p . eh equals . 1 half , eh p .
    

    Figure 1

    

    Figure 2

41. Proof For Problem 1 on page 629, write a proof that the apothem bisects the vertex angle of an isosceles triangle formed by two radii.

Table of Contents