Area of Triangles
The area of a triangle is the measure of the space enclosed by the three sides of the triangle. It is calculated using the formula A = 1/2 * base * height, where A represents the area, and the base and height are the respective measurements of the triangle. Understanding the concept of area is fundamental in geometry and practical applications involving shapes.
6 Key excerpts on "Area of Triangles"
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...If the triangle is an isosceles triangle, two lengths will be the same. If it is an equilateral triangle, the perimeter is three times the length of one side because they are all the same. For a right triangle, you may have to calculate the third side by the Pythagorean formula before you can add the three sides. So for any triangle with three sides, a, b, and c, the perimeter (p) is given by The area of a triangle is one-half the base times the height of the triangle. You can choose any of the three sides to be the base, although for an isosceles triangle it is easiest if it is the unequal side, and for a right triangle it should be one of the legs. The height is the perpendicular distance from the base to the opposite angle. This is the tricky part: finding the height of a triangle. It is one of the sides only if the triangle is a right triangle because the two legs are perpendicular. So for any other triangle, remember that it is not one of the other sides. In fact, for an obtuse triangle, the height could be a measurement outside the triangle itself. The formula for the area (A) of a triangle, where b is the base and h is the height to that base, is Where did the come in? For any triangle, we can duplicate it across any side and we will end up with a four-sided figure in which the sides across from each other are equal. We will see shortly that the area of this new four-sided figure (called a parallelogram) is simply its base times its height (or length times width). The two triangles are identical. The area of either triangle is one-half the area of the four-sided figure. So when you figure the area of a triangle, don’t forget the Even though a formula sheet is provided on the GED ® test, the formulas for the perimeter and area of a triangle (and a few others) are not listed there but you are expected to know tem...
- Alfred Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...In geometry, there are other formulas for the Area of Triangles that students may already have learned. For a right triangle, the area is one-half the product of its legs, and for a triangle where the lengths of two sides and the measure of the included angle are given, the area is one-half the product of the lengths of the two given sides and the sine of the included angle (i.e.,). However, how can we find the area of a triangle, if we are only given the lengths of its three sides? The answer is to use Heron’s formula, which students can be motivated to discover by initially giving them the following problems to work out. Present the following three problems at the start of the lesson: Find the area of a triangle whose base length is 8 and height to that base is 6. Find the area of a triangle whose side lengths are 3, 4, and 5. Find the area of a triangle whose sides have lengths 13, 14, and 15. For students to find the area of a triangle with base length 8 and height 6, they can easily use the formula, and find the area to be square units. Next, for students to find the area of a triangle whose sides are lengths 3, 4 and 5, they should recognize that this is a right triangle, with legs 3 and 4. Thus the area is given by square units. Now students are faced with the task of finding the area of a triangle whose sides are 13, 14, and 15. This is not a right triangle, so no side can be thought of as an altitude. There must be another way to find the area. Here the students should recognize a void in their knowledge of finding the area of a triangle. One of the first student reactions is that, since 3-4-5 are sides of a right triangle, what about 13-14-15? Sadly, the triangle is not a right triangle...
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...CHAPTER 11 Areas (1) Learning outcomes: (a) Calculate the areas of triangles, quadrilaterals and circles (b) Identify and use the correct units (c) Solve practical problems involving area calculation 11.11 Introduction Area is defined as the amount of space taken up by a two-dimensional figure. The geometrical properties of triangles, quadrilaterals and circles have been explained in Chapter 10. A summary of the formulae used in calculating the areas and other properties of these geometrical shapes is given in Table 11.1. The units of area used in metric systems are: mm 2, cm 2, m 2 and km 2. Table 11.1 Shape Area and other properties Area = l × b Perimeter = 2 l + 2 b = 2(l + b) Area = l × l = l 2 Perimeter = 4 l Area = l × h Area = π r 2 Circumference = 2π r 11.2 Area of Triangles There are many techniques and formulae that can be used to calculate the Area of Triangles. In this section we consider the triangles with known measurements of the base and the perpendicular height, or where the height can be calculated easily. Example 11.1 Find the area of the triangles shown in Figure 11.1. Figure 11.1 Solution: (a) Base BC = 8 cm We need to calculate height AD, which has not been given. As sides AB and AC are equal, BD must be equal to DC. Therefore, BD = DC = 4 cm. Now we can use Pythagoras’ Theorem to calculate height AD : Therefore (b) 11.3 Area of quadrilaterals A plane figure bounded by four straight lines is called a quadrilateral. The calculation of area of some of the quadrilaterals is explained in this section. Example 11.2 Find the area of the shapes shown in Figure 11.2. Figure 11.2 Solution: (a) Area of a rectangle = length × width Length = 15 cm, and width = 6 cm Area of rectangle ABCD = 15 × 6 = 90 cm 2 (b) In a square, the length is equal to the width...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 6 Geometry Topics CHAPTER 6 GEOMETRY TOPICS Plane geometry refers to two-dimensional shapes (that is, shapes that can be drawn on a sheet of paper), such as triangles, parallelograms, trapezoids, and circles. Three-dimensional objects (that is, shapes with depth) are the subjects of solid geometry. TRIANGLES A closed three-sided geometric figure is called a triangle. The points of the intersection of the sides of a triangle are called the vertices of the triangle. A side of a triangle is a line segment whose endpoints are the vertices of two angles of the triangle. The perimeter of a triangle is the sum of the measures of the sides of the triangle. An interior angle of a triangle is an angle formed by two sides and includes the third side within its collection of points. The sum of the measures of the interior angles of a triangle is 180°. A scalene triangle has no equal sides. An isosceles triangle has at least two equal sides. The third side is called the base of the triangle, and the base angles (the angles opposite the equal sides) are equal. An equilateral triangle has all three sides equal.. An equilateral triangle is also equiangular, with each angle equaling 60°. An acute triangle has three acute angles (less than 90°). An obtuse triangle has one obtuse angle (greater than 90°). A right triangle has a right angle. The side opposite the right angle in a right triangle is called the hypotenuse of the right triangle. The other two sides are called the legs (or arms) of the right triangle. By the Pythagorean Theorem, the lengths of the three sides of a right triangle are related by the formula c 2 = a 2 + b 2 where c is the hypotenuse and a and b are the other two sides (the legs). The Pythagorean Theorem is discussed in more detail in the next section. An altitude, or height, of a triangle is a line segment from a vertex of the triangle perpendicular to the opposite side...
eBook - ePub- P.K. Jayasree, K Balan, V Rani(Authors)
- 2021(Publication Date)
- CRC Press(Publisher)
...i.e., it deals with measurement of various parameters of geometric figures. It is all about the method of quantifying. It is done using geometric computations and algebraic equations to deliver data related to depth, width, length, area, or volume of a given entity. However, the measurement results got using mensuration are mere approximations. Hence actual physical measurements are always considered to be accurate. There are two types of geometric shapes: (1) 2D and (2) 3D. 2D regular shapes have a surface area and are categorized as circle, triangle, square, rectangle, parallelogram, rhombus, and trapezium. 3D shapes have surface area as well as volume. They are cube, rectangular prism (cuboid), cylinder, cone, sphere, hemisphere, prism, and pyramid. 3.2.1 Mensuration of Areas 3.2.1.1 Circle For a circle of diameter d as shown in Figure 3.1 having circumference C, Area, A = 1 4 π d 2 (3.1a) = π r 2 (3.1b) = 0.07958 C 2 (3.1c) = 1 4 C × d (3.1d) Circumference, C = π d (3.2a) = 3.5449 area (3.2b) Side of a square with the. same area, A = 0.8862 d (3.3a) = 0.285 C (3.3b) Side of an inscribed square = 0.707 d (3.4a) = 0.225 C (3.4b) Side of an inscribed equilateral triangle = 0.86 d (3.5) Side of a square of equal periphery as a circle = 0.785 d (3.6) 3.2.1.2 Square Area = side 2 = 1.2732 × area of inscribed circle (3.7) Diagonal = √ 2 × side (3.8) Circumference of a circle circumscribing a square = 4.443 × side of square (3.9) Diameter of a...
eBook - ePubDyslexia, Dyscalculia and Mathematics
A practical guide
- Anne Henderson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...The theorem is used to find the length of a side in a right-angled triangle when the lengths of the other two sides are known. The theorem states: In a right-angled triangle the area of the square on the hypotenuse is equal to the sum ofthe areas of the squares on the other two sides. How to help ● Find a picture or drawing depicting Pythagoras and talk about the man. Often students cannot remember the word Pythagoras on its own but they remember the picture and with a little prompting remember his theorem. ● Make a memory card (number 24, see page 144) to show that the longest side, the hypotenuse, is always opposite to the right angle. ● Present the triangle in different ways. Figure 9.21 A right-angled triangle presented in different way to show the hypotenuse A good multi-sensory exercise is to draw a triangle with the two smaller sides with lengths of 3 cm and 4 cm so that the hypotenuse will be 5 cm. Draw squares on the three sides. Cut the 9 cm into 1 cm squares and fit these around the 16 cm square to make a 25 cm square. Figure 9.22 A multi-sensory exercise to explain the Pythagorean theorem To find the length of the hypotenuse of a triangle when the other two sides are known. ● Square the numbers (multiply them by themselves). ● Add the numbers. ● Take the square root to get the answer. Section I: Trigonometry Abbreviations O = opposite side A = adjacent side H = hypotenuse S = sine (angle) C = cosine (angle) T = tangent (angle) Figure 9.23 Identify the sides of the triangle in relation to the angle XZY How to help ● Identify the sides of the triangle, discussing which sides are adjacent and which are opposite. This seemingly simple item can take a great deal of time as many students cannot connect up angles and lines easily. ● Indicate with an arrow the hypotenuse. ● The following mnemonic may help:...
