Video Transcript
Determine, to the nearest tenth, the area of the given figure.
When we’re finding the area of a figure, we’re finding the space that this takes up. In order to answer this question, we’ll need to work out which shapes make up this composite figure. The shape on the right forms half of a circle or a semicircle. The shape on the left is a quadrilateral, and we can say that it’s got a pair of parallel sides. We can be sure that we’ve got a pair of parallel sides as if we consider this dotted line and notice that we have two 90-degree angles, then this means that our dotted line is a transversal between two parallel lines. The name for a quadrilateral with a pair of parallel sides is a trapezoid.
In order to find the area of the whole figure, we need to do the two area calculations. We’ll find the area of the semicircle and the area of the trapezoid and add them together. In order to find the area of a semicircle, we’ll need to recall that the area of a whole circle is found by 𝜋 times the radius squared. As a semicircle is half of a circle, to find the area of a semicircle, we’ll find 𝜋𝑟 squared over two. As we begin to work through this semicircle calculation, we might notice that we have a problem. And that is that we don’t actually know what the radius or even the diameter is. We’ll have to work it out.
In order to find the length or the diameter, we’ll need to look within the trapezoid. Let’s consider this triangle on the left side of our trapezoid. We know that this will be a right triangle, and we also may recall that we can use the Pythagorean theorem in right triangles to help us find an unknown length. We can define the height of this triangle with any letter, but let’s use 𝑥. The length of the base of this triangle can be found by subtracting 39 from 47 centimeters, which is eight centimeters. So now, we have a right triangle, we know two sides, and we want to find the unknown side.
The Pythagorean theorem tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides. Filling in the values into the Pythagorean theorem, the hypotenuse, the longest side, is 20, and the other two sides are eight and 𝑥. So we have 20 squared equals eight squared plus 𝑥 squared. 20 squared is 400, and eight squared is 64. We’ll now rearrange this to find 𝑥 squared. Subtracting 64 from both sides of the equation leaves us with 336 equals 𝑥 squared. In order to find the value of 𝑥, we take the square root of both sides, so we’ll have the square root of 336 is equal to 𝑥.
It’s perfectly fine to keep our answer as the square root of 336. But we can further simplify this square root to give us a neater result. We can split 336 into the factors of 16 and 21, and we have done that because 16 is a square number. Simplifying this then, we’ll have 𝑥 is equal to four root 21 centimeters. Now that we found the height of this trapezoid to be four root 21 centimeters, then we know that the diameter of our semicircle will be of the same length. In order to find the radius for our area calculation, we’ll need to have four root 21. We can simplify the fraction four root 21 over two by dividing through by two, leaving us with a radius that’s two root 21 centimeters.
And now, let’s go back to the area of our semicircle calculation. Filling in the value for the radius, we’ll have 𝜋 times two root 21 all squared over two. Evaluating this gives us 84𝜋 over two, which is 42𝜋 square centimeters. We can keep our answer in this form rather than changing to a decimal as we’ll use it in our next calculation. Now that we found the area of the semicircle, let’s see if we can find the area of the trapezoid. We’ll clear this working on the Pythagorean theorem if you want to pause the video to make any notes.
The formula for the area of a trapezoid is given as a half ℎ times 𝑏 sub one plus 𝑏 sub two, where ℎ is the perpendicular height and 𝑏 sub one and 𝑏 sub two are the lengths of the parallel sides. If we take a look at the diagram, we need to be careful as this length of 20 centimeters is not the perpendicular height. It is instead the value that we worked out of four root 21 centimeters. The lengths of the two parallel sides are 39 and 47 centimeters, so our calculation is a half times four root 21 times 39 plus 47. We can add our values of 39 and 47 to get 86. And as half of 86 is 43, we’ll have 43 times four root 21. 43 multiplied by four gives us 172, so the whole area of the trapezoid is 172 root 21 square centimeters.
Finally, to find the area of the figure, we take the area of our semicircle 42𝜋 and add it to the area of our trapezoid 172 root 21. Using our calculator, we get the answer of 920.149911 and so on square centimeters. And observing that we were asked to give our answer to the nearest tenth, this means that we check our second decimal digit to see if it’s five or more. And as it isn’t, then our answer rounds down to 920.1 square centimeters. And so, we found the area of this figure by finding the area of the semicircle and the area of the trapezoid and adding them together.