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Composite Figures – Area and Volume

You might have noticed various shapes and figures around you. Everything has an area they occupy, from the laptop to your book. There are various shapes whose areas are different from one another. These shapes are known as composite figures. A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc.

To understand the dynamics of composite figures, one needs first to cover the basics of real two-dimensional geometric figures. Because at the end of the day, composite figures are the sum of basic figures.

Let’s understand the basic properties of some of the important two-dimensional figures:

2 D ShapesProperties 
TriangleIt can have no, 2 or 3 equal sidesIt can have no, 2 or 3 equal anglesIt can have up to 2 axes of symmetry
SquareFour equal sidesFour equal angles(90°)Four axes of symmetry
Rectangle2 sets of 2 equal sidesFour equal angles(90°)Two axes of symmetry
CircleConstant diameter and radiusThe total angle of a circle is equal to 360 degreesAlmost infinite axes of symmetry going through the center
TriangleIt can have no, 2 or 3 equal sidesIt can have no, 2 or 3 equal anglesIt can have up to 2 axes of symmetry
pentagon5 sides (can be equal or unequal)5 angles (can be equal or unequal)It can have up to 5 axes of symmetry
hexagon6 sides (can be equal or unequal)6 angles (can be equal or unequal)It can have up to 6 axes of symmetry
Octagon8 sides (can be equal or unequal) 8 angles (can be equal or unequal)It can have up to 8 axes of symmetry
Parallelogram2 sets of 2 equal sides2 sets of 2 equal anglesUsually no axes of symmetry
RhombusAll sides are the same length2 sets of 2 equal angles2 lines of symmetry
TrapeziumAt least 2 parallel sidesCan have pairs of equal anglesIt can have a line of symmetry

Area of composite figures

The area of composite figures is the area that any composite shape covers. A composite shape comprises a few polygons that are joined together to form the desired shape. These shapes or figures can be made up of triangles, squares, and quadrilaterals, among other things. To calculate the area of a composite shape, divide it into basic shapes such as squares, triangles, rectangles, hexagons, etc.

A composite shape is made up of basic shapes that have been combined. It is also referred to as a “composite” or “complex” shape.

Now that we have figured out the basic properties of the important polygons, we can move into understanding the area and perimeter of the polygons:

2d ShapeAreaPerimeter
CircleΠr2 (R is the radius of the circle)2πr
Triangle½ (Base x height)Sum of three sides
SquareSide24(Side)
RectangleLength x Breadth2(Length + Breadth)
Rhombus½ (Product of diagonals)4(Side)
ParallelogramBase x Height2 (Base + Side)

The volume of composite figures

Understanding volume when it comes to finding areas is crucial. You must know the volumes to calculate areas of various sections when limited data is provided. In mathematical terms, volume is the amount of space inside a 3-D object. First, we will understand some basic terms needed to find the volume of composite figures. Then we will get into the formulae.

Solid Figure: Three-dimensional figures of simple geometric shapes are called solid figures. Some examples of solid figures are cubes, prisms, pyramids, and cylinders. The volume of a Solid figure: The volume of a solid figure is the measure of the enclosed space within the figure.

Composite Figure: Figures constructed by connecting different solid figures are composite figures. The volume of a Composite Figure: Sum of the volumes of the given solid figures.

Let us understand the properties of some basic 3D objects like the way we discussed for the 2D objects:

3D ShapesProperties
Sphere (With radius – r)It has no edges or vertices (corners). It has one curved surface. It is perfectly symmetrical. All points on the surface of a sphere are at the same distance (r) from the center.
ConeIt has a flat base. It has one curved side and a one-pointed vertex at the top or bottom known as the apex.
CylinderIt has a flat base and a flat top. The bases are always congruent and parallel. It has one curved side.
CubeIt has six faces in the shape of a square. The sides are of equal lengths.12 diagonals can be drawn on a cube.
PyramidA Pyramid is a polyhedron with a polygon base and an apex with straight lines. Based on their apex alignment with the center of the base, they can be classified into regular and oblique pyramids.
PrismIt has identical ends (polygonal) and flat faces. It has the same cross-section all along its length
CuboidIt has six rectangular faces. All the sides of a cuboid are not equal in length.12 diagonals can be drawn on a cuboid.

How to Find Volume of a Composite Figure?

  • Measure the dimensions of the bottom solid figure and find the volume.
  • Measure the dimensions of the top solid figure and find the volume.
  • Find the volume of the composite figure by adding the volume of the two solid figures.

Some of the important formulae regarding finding the surface areas and volumes of various composite figures:

3D ShapeFormulas
ConeCurved Surface Area = πrl; (where ‘l’ is the slant height and
l = √(h2 + r2))
Total Surface Area = πr(l + r)
Volume = (1/3)πr2h
CylinderTotal Surface Area = 2πr(h+r); (where ‘r’ is the radius and ‘h’ is the height of the cylinder)
Volume = πr2h
SphereDiameter = 2 × r; (where ‘r’ is the radius)
Surface Area = 4πr2
Volume = (4/3)πr3
CubeLateral Surface Area = 4a2; (where ‘a’ is the side length of the cube)
Total Surface Area = 6a2
Volume = a3
CuboidLateral Surface Area = 2h(l + w); (where ‘h’ is the height, ‘l’ is the length and ‘w’ is the width)
Total Surface Area = 2 (lw + wh + lh)
Volume = (l × w × h)
PyramidSurface Area = Base Area + (1/2 × Perimeter × Slant Height)
Volume = [(1/3) × Base Area × Altitude]
PrismSurface Area = [(2 × Base Area) + (Perimeter × Height)]
Volume = (Base Area × Height)

How to find the area of a composite figure?

Finding the areas of composite figures is not much challenging. You have to memorize the formulae by heart and do the calculations. You can score more if you learn the formulae and practice them regularly. The area of the composite figures is the area of one or more simple polygons and circles combined. We can add the areas of all the basic figures together to calculate the area of the composite figures. Find the area of each shape and add them together to find the area of the composite figure. For example, the area of composite shapes is measured in m2, cm2, in2, or ft2.

Example:

  1. Find the area of the composite figure when – 1. There is a rectangle of 4cm and 3cm in height & length, respectively. 2. A triangle with a 5cm base, 10cm height

      A: As we have learned in the past, a composite figure is a summation of basic polygons. So to find the area of composite figures, we have to break the figure into small polygons and then apply the formula as required.

Hence the area of the rectangular- 4 x 3 = 12 square cm

Area of the triangle – ½ x 10 x 5 = 25 square cm

Therefore, the total area of the composite figure = 12 + 25 = 37 square cm 

  1. A composite figure has an area of 100 units square. The shape is composed of a circle and a triangle, and the area of the triangle is 64 units square. What is the area of the circle?

      A: Given the area of the composite figure = 100 units square and the area of the triangle = 64 units square

Using the formula for the area of the composite shape, Area of composite shape = area of triangle + area of the circle.

⇒ 100 = 64 + area of circle

⇒ Area of circle = 100 – 64

⇒ Area of the circle = 36 units square.

Therefore the area of the circle is 36 units square.

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