10.6.3 Tangram Areas Again

Activity Symbol Activity: Tangram Areas Again

The figure below shows a 4 cm square tangram puzzle.

Note that the image on your monitor may not be drawn to scale.

By using the formulas for the area of a triangle and for the area of a parallelogram, calculate the areas of the following shapes.

Check that your answers agree with those you obtained by counting the squares.

(a) cap delta times cap a times cap f times cap d with AD as the base

Solution Symbol

Answer

(a) cap a times cap d equals four cm. The perpendicular height from F onto cap a times cap d equals two cm. The area of a triangle = one divided by two postfix multiplication base multiplication height

So the area of cap delta equation left hand side cap a times cap f times cap d equals right hand side one divided by two multiplication four cm multiplication two cm equals four cm super two.

How many other ways could you calculate this area? Think about what information you have about the whole space.

(b) cap delta times cap d times cap e times cap h with DH as the base

Solution Symbol

Answer

(b) cap d times cap h equals two cm. The perpendicular height from E onto cap d times cap h equals one cm. So, the area of cap delta equation left hand side cap d times cap e times cap h equals right hand side one divided by two multiplication two cm prefix multiplication of one cm equals one cm super two.

(c) Parallelogram GBJI with BJ as the base

Solution Symbol

Answer

(c) cap b times cap j equals two cm. The perpendicular height from G onto cap b times cap j equals one cm.

the area of a parallelogram equals base multiplication height

That means the area of cap g times cap b times cap j times cap i equals two cm prefix multiplication of one cm equals two cm super two.

(d) cap delta times cap j times cap h times cap c

Solution Symbol

Answer

(d) The base cap h times cap c equals two cm and the corresponding height cap j times cap c equals two cm. So, the area of cap delta equation left hand side cap j times cap h times cap c equals right hand side one divided by two multiplication two cm prefix multiplication of two cm equals two cm super two.

Many area problems can be calculated by using combinations of squares, rectangles, and triangles. However, you often need to find circular areas, too. The formula for the area of a circle is:

area of a circle equation left hand side equals right hand side pi multiplication open radius close squared

We will look at more on this on the next page.

Now that you know how to calculate the areas of basic shapes, you can calculate more complicated areas by breaking each shape into basic shapes and adding the individual areas.

Even the surface areas of some containers that have curved surfaces like cylinders can be broken down into circles and rectangles. [ How would you find the surface area of a box or a cylinder? For the cylinder, think about what shape you would get if you unrolled the tube. ]

10.6.2 Formulas for Areas

10.6.4 Calculator Exploration: Areas of Circles