# Math Help - Area of shapes?

1. ## Area of shapes?

These problems are actually in my Calculus book, we're starting to learn area/volume tomorrow, and I got these problems assigned to me, yet there's not explanation in the book..

1. a square with sides of length x
2. a square with diagonals of length x
3. a semicircle of radius x
4. a semicircle of diameter x
5. an equilateral triangle with sides of length x
6. an isosceles right triangle with legs of length x
7. an isosceles right triangle with hypotenuse x
8. an isosceles triangle with two sides of length 2x and one side of length x
9. a triangle with sides 3x, 4x, and 5x
10. a regular hexagon with sides of length x

The answers are:
1. x^2
2. x^2/2
3. (pi*x^2)/2
4. (pi*x^2)/8
5. ((√3)/4)*x^2
6. x^2/2
7. x^2/4
8. ((√15)/4)*x^2
9. 6x^2
10. (3((√3)/2))*x^2

Will someone explain how to do these? I only under stand 1/3/4.. the basic ones. thanks

Am I going to have to find formulas for every one of em? I'm pretty unsure about the triangle-shaped ones.

2. Originally Posted by releki
These problems are actually in my Calculus book, we're starting to learn area/volume tomorrow, and I got these problems assigned to me, yet there's not explanation in the book..

1. a square with sides of length x
2. a square with diagonals of length x
3. a semicircle of radius x
4. a semicircle of diameter x
5. an equilateral triangle with sides of length x
6. an isosceles right triangle with legs of length x
7. an isosceles right triangle with hypotenuse x
8. an isosceles triangle with two sides of length 2x and one side of length x
9. a triangle with sides 3x, 4x, and 5x
10. a regular hexagon with sides of length x

The answers are:
1. x^2
2. x^2/2
3. (pi*x^2)/2
4. (pi*x^2)/8
5. ((√3)/4)*x^2
6. x^2/2
7. x^2/4
8. ((√15)/4)*x^2
9. 6x^2
10. (3((√3)/2))*x^2

Will someone explain how to do these? I only under stand 1/3/4.. the basic ones. thanks

Am I going to have to find formulas for every one of em? I'm pretty unsure about the triangle-shaped ones.
You're expected to know the properties of these shapes, as well as the formulas for them. I will walk you through an example so hopefully you can finish the rest yourself.

2) a square with diagonals of length x

The diagonal cuts the square into two congruent right triangles.
The diagonal forms one side of the triangle. The other two sides are also the sides of the square, so they are equal.
The angle between the two equal sides is the same angle as an edge of a square, so it's 90 degrees.
Let a = length of a side of square. By pythagoras, $a^2 + a^2 = 2a^2 = x^2 \Rightarrow a^2 = \frac{x^2}{2}$
Now area of square is $a x a = a^2 = \frac{x^2}{2}$

Now try some for yourself. Here are some hints:

5) Equliateral triangle= 3 sides equal and all interior angles are 60 degrees. An easy way is to use the formula $A = \frac{1}{2} ab \sin C$. But you can also split the triangle into two isoceles triangles are use pythagoras as in question 2).

6,7,8) I dealt with the method to solve these in part in my explantaion in 2).

9) the ratio 3:4:5 is a pythagorean triple (i.e. they form the sides of a right angle triangle)

10) Any polygon (including hexagons) can be cut into triangular pieces.